Engineering analysis

First Edition, 2009

ISBN 978 93 80168 25 8

© All rights reserved.

Published by: Global Media 1819, Bhagirath Palace, Chandni Chowk, Delhi-110 006 Email: [email protected]

Table of Contents 1. Linear Algebra 2. Matrix Calculus 3. Wavelet Analysis 4. Stochastic Processes 5. Optimization 6. Integral Transforms 7. Mathematical Tables 8. Statistical Tables

Vectors and Scalars A scalar is a single number value, such as 3, 5, or 10. A vector is an ordered set of scalars. A vector is typically described as a matrix with a row or column size of 1. A vector with a column size of 1 is a row vector, and a vector with a row size of 1 is a column vector. [Column Vector]

[Row Vector]

A "common vector" is another name for a column vector, and this book will simply use the word "vector" to refer to a common vector.

Vector Spaces A vector space is a set of vectors and two operations (addition and multiplication, typically) that follow a number of specific rules. We will typically denote vector spaces with a capital-italic letter: V, for instance. A space V is a vector space if all the following requirements are met. We will be using x and y as being arbitrary vectors in V. We will also use c and d as arbitrary scalar values. There are 10 requirements in all: Given: 1. There is an operation called "Addition" (signified with a "+" sign) between two vectors, x + y, such that if both the operands are in V, then the result is also in V. 2. The addition operation is commutative for all elements in V. 3. The addition operation is associative for all elements in V. 4. There is a neutral element, φ, in V, such that x + φ = x. This is also called a one element. 5. For every x in V, then there is a negative element -x in V. 6. 7. c(x + y) = cx + cy 8. (c + d)x = cx + dx 9. c(dx) = cdx

10. 1 × x = x Some of these rules may seem obvious, but that's only because they have been generally accepted, and have been taught to people since they were children.

Vector Basics Scalar Product A scalar product is a special type of operation that acts on two vectors, and returns a scalar result. Scalar products are denoted as an ordered pair between angle-brackets: <x,y>. A scalar product between vectors must satisify the following four rules: 1. 2. 3. 4.

, only if x = 0

If an operation satisifes all these requirements, then it is a scalar product.

Examples One of the most common scalar products is the dot product, that is discussed commonly in Linear Algebra

Norm The norm is an important scalar quantity that indicates the magnitude of the vector. Norms of a vector are typically denoted as . To be a norm, an operation must satisfy the following four conditions: 1. 2. 3. 4.

only if x = 0.

A vector is called normal if it's norm is 1. A normal vector is sometimes also referred to as a unit vector. Both notations will be used in this book. To make a vector normal, but keep it pointing in the same direction, we can divide the vector by it's norm:

Examples One of the most common norms is the cartesian norm, that is defined as the square-root of the sum of the squares:

Unit Vector A vector is said to be a unit vector if the norm of that vector is 1.

Orthogonality Two vectors x and y are said to be orthogonal if the scalar product of the two is equal to zero:

Two vectors are said to be orthonormal if their scalar product is zero, and both vectors are unit vectors.

Cauchy-Schwartz Inequality The cauchy-schwartz inequality is an important result, and relates the norm of a vector to the scalar product:

Metric (Distance) The distance between two vectors in the vector space V, called the metric of the two vectors, is denoted by d(x, y). A metric operation must satisfy the following four conditions: 1. 2. d(x,y) = 0 only if x = y 3. d(x,y) = d(y,x) 4.

Examples A common form of metric is the distance between points a and b in the cartesian plane:

Linear Independence and Basis Linear Independance A set of vectors are said to be linearly dependant on one another if any vector v from the set can be constructed from a linear combination of the other vectors in the set. Given the following linear equation:

The set of vectors V is linearly independant only if all the a coefficients are zero. If we combine the v vectors together into a single row vector:

And we combine all the a coefficients into a single column vector:

We have the following linear equation:

We can show that this equation can only be satisifed for invertable:

, the matrix

must be

Remember that for the matrix to be invertable, the determinate must be non-zero.

Non-Square Matrix V If the matrix is not square, then the determinate can not be taken, and therefore the matrix is not invertable. To solve this problem, we can premultiply by the transpose matrix:

And then the square matrix

must be invertable:

Rank The rank of a matrix is the largest number of linearly independant rows or columns in the matrix. To determine the Rank, typically the matrix is reduced to row-echelon form. From the reduced form, the number of non-zero rows, or the number of non-zero colums (whichever is smaller) is the rank of the matrix. If we multiply two matrices A and B, and the result is C: AB = C Then the rank of C is the minimum value between the ranks A and B:

Span A Span of a set of vectors V is the set of all vectors that can be created by a linear combination of the vectors.

Basis A basis is a set of linearly-independant vectors that span the entire vector space.

Basis Expansion If we have a vector , and V has basis vectors write y in terms of a linear combination of the basis vectors:

, by definition, we can

or

If is invertable, the answer is apparent, but if the following technique:

is not invertable, then we can perform

And we call the quantity

the left-pseudoinverse of

.

Change of Basis Frequently, it is useful to change the basis vectors to a different set of vectors that span the set, but have different properties. If we have a space V, with basis vectors and a vector in V called x, we can use the new basis vectors to represent x:

or,

If V is invertable, then the solution to this problem is simple.

Grahm-Schmidt Orthogonalization If we have a set of basis vectors that are not orthogonal, we can use a process known as orthogonalization to produce a new set of basis vectors for the same space that are orthogonal: Given: Find the new basis Such that We can define the vectors as follows: 1. w1 = v1 2. Notice that the vectors produced by this technique are orthogonal to each other, but they are not necessarily orthonormal. To make the w vectors orthonormal, you must divide each one by it's norm:

Reciprocal Basis A Reciprocal basis is a special type of basis that is related to the original basis. The reciprocal basis can be defined as:

Linear Transformations A linear transformation is a matrix M that operates on a vector in space V, and results in a vector in a different space W. We can define a transformation as such:

In the above equation, we say that V is the domain space of the transformation, and W is the range space of the transformation. Also, we can use a "function notation" for the transformation, and write it as: M(x) = Mx = y Where x is a vector in V, and y is a vector in W. To be a linear transformation, the principle of superposition must hold for the transformation: M(av1 + bv2) = aM(v1) + bM(v2) Where a and b are arbitary scalars.

Null Space The Nullspace of an equation is the set of all vectors x for which the following relationship holds: Mx = 0 Where M is a linear transformation matrix. Depending on the size and rank of M, there may be zero or more vectors in the nullspace. Here are a few rules to remember: 1. If the matrix M is invertable, then there is no nullspace. 2. The number of vectors in the nullspace (N) is the difference between the rank(R) of the matrix and the number of columns(C) of the matrix: N=R−C

If the matrix is in row-eschelon form, the number of vectors in the nullspace is given by the number of rows without a leading 1 on the diagonal. For every column where there is not a leading one on the diagonal, the nullspace vectors can be obtained by placing a negative one in the leading position for that column vector. We denote the nullspace of a matrix A as:

Linear Equations If we have a set of linear equations in terms of variables x, scalar coefficients a, and a scalar result b, we can write the system in matrix notation as such: Ax = b Where x is a m × 1 vector, b is an n × 1 vector, and A is an n × m matrix. Therefore, this is a system of n equations with m unknown variables. There are 3 possibilities: 1. If Rank(A) is not equal to Rank([A b]), there is no solution 2. If Rank(A) = Rank([A b]) = n, there is exactly one solution 3. If Rank(A) = Rank([A b]) < n, there are infinately many solutions.

Complete Solution The complete solution of a linear equation is given by the sum of the homogeneous solution, and the particular solution. The homogeneous solution is the nullspace of the transformation, and the particular solution is the values for x that satisfy the equation: A(x) = b A(xh + xp) = b Where xh is the homogeneous solution, and is the nullspace of A that satisfies the equation A(xh) = 0 xp is the particular solution that satisfies the equation A(xp) = b

Minimum Norm Solution If Rank(A) = Rank([A b]) < n, then there are infinately many solutions to the linear equation. In this situation, the solution called the minimum norm solution must be found. This solution represents the "best" solution to the problem. To find the minimum norm solution, we must minimize the norm of x subject to the constraint of:

Ax − b = 0 There are a number of methods to minimize a value according to a given constraint, and we will talk about them later.

Least-Squares Curve Fit If Rank(A) doesnt equal Rank([A b]), then the linear equation has no solution. However, we can find the solution which is the closest. This "best fit" solution is known as the Least-Squares curve fit. We define an error quantity E, such that:

Our job then is to find the minimum value for the norm of E:

We do this by differentiating with respect to x, and setting the result to zero:

Solving, we get our result: x = (ATA) − 1ATb

Minimization Khun-Tucker Theorem The Khun-Tucker Theorem is a method for minimizing a function f(x) under the constraint g(x). We can define the theorem as follows:

Where Λ is the lagrangian vector, and < , > denotes the scalar product operation. We will discuss scalar products more later. If we differentiate this equation with respect to x first, and then with respect to Λ, we get the following two equations:

We have the final result: x = AT[AAT] − 1b

Projection The projection of a vector onto the vector space is the minimum distance between v and the space W. In other words, we need to minimize the distance between vector v, and an arbitrary vector :

[Projection onto space W]

For every vector there exists a vector called the projection of v onto W such that = 0, where p is an arbitrary element of W.

Orthogonal Complement

Distance between v and W The distance between v and an arbitrary

and the space W is given as the minimum distance between :

Intersections Given two vector spaces V and W, what is the overlapping area between the two? We define an arbitrary vector z that is a component of both V, and W:

Where N is the nullspace.

Linear Spaces Linear Spaces are like Vector Spaces, but are more general. We will define Linear Spaces, and then use that definition later to define Function Spaces. If we have a space X, elements in that space f and g, and scalars a and b, the following rules must hold for X to be a linear space: 1. 2. f + g = g + f 3. There is a null element φ such that φ + f = f. 4. 5. f + (-f) = φ

Matrices Derivatives Consider the following set of linear equations: a = bx1 + cx2 d = ex1 + fx2 We can define the matrix A to represent the coefficients, the vector B as the results, and the vector x as the variables:

And rewriting the equation in terms of the matrices, we get: B = Ax Now, let's say we want the derivative of this equation with respect to the vector x:

We know that the first term is constant, so the derivative of the left-hand side of the equation is zero. Analyzing the right side shows us:

Pseudo-Inverses There are special matrices known as pseudo-inverses, that satisfies some of the properties of an inverse, but not others. To recap, If we have two square matrices A and B, that are both n × n, then if the following equation is true, we say that A is the inverse of B, and B is the inverse of A: AB = BA = I

Right Pseudo-Inverse Consider the following matrix: R = AT[AAT] − 1 We call this matrix R the right pseudo-inverse of A, because: AR = I but

We will denote the right pseudo-inverse of A as

Left Pseudo-Inverse Consider the following matrix: L = [ATA] − 1AT We call L the left pseudo-inverse of A because LA = I but

We will denote the left pseudo-inverse of A as

Matrix Forms Matrices that follow certain predefined formats are useful in a number of computations. We will discuss some of the common matrix formats here. Later chapters will show how these formats are used in calculations and analysis.

Diagonal Matrix A diagonal matrix is a matrix such that:

In otherwords, all the elements off the main diagonal are zero, and the diagonal elements may be (but don't need to be) non-zero.

Companion Form Matrix If we have the following characteristic polynomial for a matrix:

We can create a companion form matrix in one of two ways:

Or, we can also write it as:

Jordan Canonical Form To discuss the Jordan canonical form, we first need to introduce the idea of the Jordan Block:

Jordan Blocks A jordan block is a square matrix such that all the diagonal elements are equal, and all the super-diagonal elements (the elements directly above the diagonal elements) are all 1. To illustrate this, here is an example of an n-dimensional jordan block:

Canonical Form

A square matrix is in Jordan Canonical form, if it is a diagonal matrix, or if it has one of the following two block-diagonal forms:

Or:

where the D element is a diagonal block matrix, and the J blocks are in Jordan block form.

Quadratic Forms If we have an n × 1 vector x, and an n × n symmetric matrix M, we can write: xTMx = a Where a is a scalar value. Equations of this form are called quadratic forms.

Matrix Definiteness Based on the quadratic forms of a matrix, we can create a certain number of categories for special types of matrices: 1. if xTMx > 0 for all x, then the matrix is positive definate. 2. if for all x, then the matrix is positive semi-definate. 3. if xTMx < 0 for all x, then the matrix is negative definate. 4. if

for all x, then the matrix is negative semi-definate.

These classifications are used commonly in control engineering.

Eigenvalues and Eigenvectors The Eigen Problem This page is going to talk about the concept of Eigenvectors and Eigenvalues, which are important tools in linear algebra, and which play an important role in State-Space control systems. The "Eigen Problem" stated simply, is that given a square matrix A which is n × n, there exists a set of n scalar values λ and n corresponding non-trivial vectors v such that: Av = λv We call λ the eigenvalues of A, and we call v the corresponding eigenvectors of A. We can rearrange this equation as: (A − λI)v = 0 For this equation to be satisfied so that v is non-trivial, the matrix (A - λI) must be singular. That is: | A − λI | = 0

Characteristic Equation The characteristic equation of a square matrix A is given by: [Characteristic Equation]

| A − λI | = 0 Where I is the identity matrix, and λ is the set of eigenvalues of matrix A. From this equation we can solve for the eigenvalues of A, and then using the equations discussed above, we can calculate the corresponding eigenvectors. In general, we can expand the characteristic equation as: [Characteristic Polynomial]

This equation satisfies the following properties: 1. | A | = ( − 1)nc0 2. A is nonsingular if c0 is non-zero.

Example: 2 × 2 Matrix Let's say that X is a square matrix of order 2, as such:

Then we can use this value in our characteristic equation:

(a − λ)(d − λ) − (b)(c) = 0 The roots to the above equation (the values for λ that satisifies the equality) are the eigenvalues of X.

Eigenvalues The solutions, λ, of the characteristic equation for matrix X are known as the eigenvalues of the matrix X. Eigenvalues satisfy the following properties: 1. If λ is an eigenvalue of A, λn is an eigenvalue of An. 2. If λ is a complex eigenvalue of A, then λ* (the complex conjugate) is also an eigenvalue of A. 3. If any of the eigenvalues of A are zero, then A is singular. If A is non-singular, all the eigenvalues of A are nonzero.

Eigenvectors The characteristic equation can be rewritten as such: Xv = λv Where X is the matrix under consideration, and λ are the eigenvalues for matrix X. For every unique eigenvalue, there is a solution vector v to the above equation, known as an eigenvector. The above equation can also be rewritten as: | X − λI | v = 0 Where the resulting values of v for each eigenvalue λ is an eigenvector of X. There is a unique eigenvector for each unique eigenvalue of X. From this equation, we can see that the eigenvectors of A form the nullspace:

And therefore, we can find the eigenvectors through row-reduction of that matrix. Eigenvectors satisfy the following properties: 1. If v is a complex eigenvector of A, then v* (the complex conjugate) is also an eigenvector of A. 2. Distinct eigenvectors of A are linearly independant. 3. If A is n × n, and if there are n distinct eigenvectors, then the eigenvectors of A form a complete basis set for

Generalized Eigenvectors Let's say that matrix A has the following characteristic polynomial:

Where d1, d2, ... , ds are known as the algebraic multiplicity of the eigenvalue λi. Also note that d1 + d2 + ... + ds = n, and s < n. In other words, the eigenvalues of A are repeated. Therefore, this matrix doesnt have n distinct eigenvectors. However, we can create vectors known as generalized eigenvectors to make up the missing eigenvectors by satisfying the following equations: (A − λI)kvk = 0 (A − λI)k − 1vk = 0

Right and Left Eigenvectors The equation for determining eigenvectors is: (A − λI)v = 0 And because the eigenvector v is on the right, these are more appropriately called "right eigenvectors". However, if we rewrite the equation as follows: u(A − λI) = 0 The vectors u are called the "left eigenvectors" of matrix A.

Diagonalization Similarity Matrices A and B are said to be similar to one another if there exists an invertable matrix T such that: T − 1AT = B If there exists such a matrix T, the matrices are similar. Similar matrices have the same eigenvalues. If A has eigenvectors v1, v2 ..., then B has eigenvectors u given by: ui = Tvi

Matrix Diagonalization Some matricies are similar to diagonal matrices using a transition matrix, T. We will say that matrix A is diagonalizable if the following equation can be satisfied: T − 1AT = D Where D is a diagonal matrix. An n × n square matrix is diagonalizable if and only if it has n linearly independant eigenvectors.

Transition Matrix If an n × n square matrix has n distinct eigenvalues λ, and therefore n distinct eigenvectors v, we can create a transition matrix T as: T = [v1v2...vn] And transforming matrix X gives us:

Therefore, if the matrix has n distinct eigenvalues, the matrix is diagonalizable, and the diagonal entries of the diagonal matrix are the corresponding eigenvalues of the matrix.

Complex Eigenvalues

Consider the situation where a matrix A has 1 or more complex conjugate eigenvalue pairs. The eigenvectors of A will also be complex. The resulting diagonal matrix D will have the complex eigenvalues as the diagonal entries. In engineering situations, it is often not a good idea to deal with complex matrices, so other matrix transformations can be used to create matrices that are "nearly diagonal".

Generalized Eigenvectors If the matrix A does not have a complete set of eigenvectors, that is, that they have d eigenvectors and n - d generalized eigenvectors, then the matrix A is not diagonalizable. However, the next best thing is acheived, and matrix A can be transformed into a Jordan Cannonical Matrix. Each set of generalized eigenvectors that are formed from a single eigenvector basis will create a jordan block. All the distinct eigenvectors that do not spawn any generalized eigenvectors will form a diagonal block in the Jordan matrix.

Spectral Decomposition If λi are are the n distinct eigenvalues of matrix A, and vi are the corresponding n distinct eigenvectors, and if wi are the n distinct left-eigenvectors, then the matrix A can be represented as a sum:

this is known as the spectral decomposition of A.

Error Estimation Consider a scenario where the matrix representation of a system A differs from the actual implementation of the system by a factor of ∆A. In other words, our system uses the matrix: A + ∆A From the study of Control Systems, we know that the values of the eigenvectors can affect the stability of the system. For that reason, we would like to know how a small error in A will affect the eigenvalues. First off, we assume that ∆A is a small shift. The definition of "small" in this sense is arbitrary, and will remained undefined. Keep in mind that the techniques discussed here are more accurate the smaller ∆A is. If ∆A is the error in the matrix A, then ∆λ is the error in the eigenvalues and ∆v is the error in the eigenvectors. The characteristic equation becomes:

(A + ∆A)(v + ∆v) = (λ + ∆λ)(v + ∆v) We have an equation now with two unknowns: ∆λ and ∆v. In other words, we dont know how a small change in A will affect the eigenvalues and eigenvectors. If we multiply out both sides, we get: Av + ∆Av + v∆A + ∆v∆A = λv + ∆λv + v∆λ + ∆λ∆v This situation seems hopeless, until we pre-multiply both sides by the corresponding lefteigenvalue w: wTAv + wT∆Av + wTv∆A + wT∆v∆A = wTλv + wT∆λv + wTv∆λ + wT∆λ∆v Terms where two ∆ errors (which are known to be small, by definition) are multipled together, we can say are negligible, and set them to zero. Also, we know from our righteigenvalue equation that: wTA = λwT Another fact is that the right-eigenvalues and left eigenvalues are orthogonal to each other, so the following result holds: wTv = 0 Substituting these results, where necessary, into our long equation above, we get the following simplification: wT∆Av = ∆λwT∆v And solving for the change in the eigenvalue gives us:

This approximate result is only good for small values of ∆A, and the result is less precise as the error increases.

Matrix Functions If we have functions, and we use a matrix as the input to those functions, the output values are not always intuitive. For instance, if we have a function f(x), and as the input argument we use matrix A, the output matrix is not necessarily the function f applied to the individual elements of A.

Diagonal Matrix In the special case of diagonal matrices, the result of f(A) is the function applied to each element of the diagonal matrix:

Then the function f(A) is given by:

Jordan Cannonical Form Matrices in Jordan Cannonical form also have an easy way to compute the functions of the matrix. However, this method is not nearly as easy as the diagonal matrices described above. If we have a matrix in Jordan Block form, A, the function f(A) is given by:

The matrix indices have been removed, because in Jordan block form, all the diagonal elements must be equal. If the matrix is in Jordan Block form, the value of the function is given as the function applied to the individual diagonal blocks.

Cayley Hamilton Theorem If the characteristic equation of matrix A is given by:

Then the Cayley-Hamilton theorem states that the matrix A itself is also a valid solution to that equation:

Another theorem worth mentioning here (and by "worth mentioning", we really mean "fundamental for some later topics") is stated as: If λ are the eigenvalues of matrix A, and if there is a function f that is defined as a linear combination of powers of λ:

If this function has a radius of convergence S, and if all the eigenvectors of A have magnitudes less then S, then the matrix A itself is also a solution to that function:

Matrix Exponential If we have a matrix A, we can raise that matrix to a power of e as follows: eA It is important to note that this is not necessarily (not usually) equal to each individual element of A being raised to a power of e. Using taylor-series expansion of exponentials, we can show that:

. In other words, the matrix exponential can be reducted to a sum of powers of the matrix. This follows from both the taylor series expansion of the exponential function, and the cayley-hamilton theorem discussed previously. However, this infinite sum is expensive to compute, and because the sequence is infinite, there is no good cut-off point where we can stop computing terms and call the answer a "good approximation". To alleviate this point, we can turn to the Cayley-Hamilton Theorem. Solving the Theorem for An, we get:

Multiplying both sides of the equation by A, we get:

We can substitute the first equation into the second equation, and the result will be An+1 in terms of the first n - 1 powers of A. In fact, we can repeat that process so that Am, for any arbitrary high power of m can be expressed as a linear combination of the first n - 1 powers of A. Applying this result to our exponential problem:

Where we can solve for the α terms, and have a finite polynomial that expresses the exponential.

Inverse The inverse of a matrix exponential is given by: (eA) − 1 = e − A

Derivative The derivative of a matrix exponential is:

Notice that the exponential matrix is commutative with the matrix A. This is not the case with other functions, necessarily.

Sum of Matrices If we have a sum of matrices in the exponent, we cannot separate them:

Differential Equations If we have a first-degree differential equation of the following form: x'(t) = Ax(t) + f(x) With initial conditions x(t0) = c Then the solution to that equation is given in terms of the matrix exponential:

This equation shows up frequently in control engineering.

Laplace Transform As a matter of some interest, we will show the Laplace Transform of a matrix exponential function:

We will not use this result any further in this book, although other books on engineering might make use of it.

Lyapunov Equation [Lyapunov's Equation]

AM + MB = C Where A, B and C are constant square matrices, and M is the solution that we are trying to find. If A, B, and C are of the same order, and if A and B have no eigenvalues in common, then the solution can be given in terms of matrix exponentials:

Function Spaces A function space is a linear space where all the elements of the space are functions. A function space that has a norm operation is known as a normed function space. The spaces we consider will all be normed.

Continuity f(x) is continuous at x0 if, for every ε > 0 there exists a δ(ε) > 0 such that |f(x) - f(x0)| < &epsilon when |x - x0| < δ(ε).

Common Function Spaces Here is a listing of some common function spaces. This is not an exhaustive list.

C Space The C function space is the set of all functions that are continuous. The metric for C space is defined as:

Consider the metric of sin(x) and cos(x):

Cp Space The Cp is the set of all continuous functions for which the first p derivatives are also continuous. If the function is called "infinitely continuous. The set is the set of all such functions. Some examples of functions that are infinitely continuous are exponentials, sinusoids, and polynomials.

L Space The L space is the set of all functions that are finitely integrable over a given interval [a, b].

f(x) is in L(a, b) if:

L p Space The Lp space is the set of all functions that are finitely integrable over a given interval [a, b] when raised to the power p:

Most importantly for engineering is the L2 space, or the set of functions that are "square integrable".

L2 Space The L2 space is very important to engineers, because functions in this space do not need to be continuous. Many discontinuous engineering functions, such as the delta (impulse) function, the unit step function, and other discontinuous finctions are part of this space.

L2 Functions A large number of functions qualify as L2 functions, including uncommon, discontinuous, piece-wise, and other functions. A function which, over a finite range, has a finite number of discontinuties is an L2 function. For example, a unit step and an impulse function are both L2 functions. Also, other functions useful in signal analysis, such as square waves, triangle waves, wavelets, and other functions are L2 functions. In practice, most physical systems have a finite amount of noise associated with them. Noisy signals and random signals, if finite, are also L2 functions: this makes analysis of those functions using the techniques listed below easy.

Null Function The null functions of L2 are the set of all functions φ in L2 that satisfy the equation:

for all a and b.

Norm The L2 norm is defined as follows: [L2 Norm]

If the norm of the function is 1, the function is normal. We can show that the derivative of the norm squared is:

Scalar Product The scalar product in L2 space is defined as follows: [L2 Scalar Product]

If the scalar product of two functions is zero, the functions are orthogonal. We can show that given coefficient matrices A and B, and variable x, the derivative of the scalar product can be given as:

We can recognize this as the product rule of differentiation. Generalizing, we can say that:

We can also say that the derivative of a matrix A times a vector x is:

Metric The metric of two functions (we will not call it the "distance" here, because that word has no meaning in a function space) will be denoted with ρ(x,y). We can define the metric of an L2 function as follows: [L2 Metric]

Cauchy-Schwartz Inequality The Cauchy-Schwartz Inequality still holds for L2 functions, and is restated here:

Linear Independance A set of functions in L2 are linearly independant if:

If and only if all the a coefficients are 0.

Grahm-Schmidt Orthogonalization The Grahm-Schmidt technique that we discussed earlier still works with functions, and we can use it to form a set of linearly independant, orthogonal functions in L2. For a set of functions φ, we can make a set of orthogonal functions ψ that space the same space but are orthogonal to one another:

[Grahm-Schmidt Orthogonalization]

ψ1 = φ1

Basis

The L2 is an infinite-basis set, which means that any basis for the L2 set will require an infinite number of basis functions. To prove that an infinite set of orthogonal functions is a basis for the L2 space, we need to show that the null function is the only function in L2 that is orthogonal to all the basis functions. If the null function is the only function that satisfies this relationship, then the set is a basis set for L2. By definition, we can express any function in L2 as a linear sum of the basis elements. If we have basis elements φ, we can define any other function ψ as a linear sum:

We will explore this important result in the section on Fourier Series.

Banach and Hilbert Spaces There are some special spaces known as Banach spaces, and Hilbert spaces.

Convergent Functions Let's define the piece-wise function φ(x) as:

We can see that as we set , this function becomes the unit step function. We can say that as n approaches infinity, that this function converges to the unit step function. Notice that this function only converges in the L2 space, because the unit step function does not exist in the C space (it is not continuous).

Convergence We can say that a function φ converges to a function φ* if:

We can call this sequences, and all such sequences that converge to a given function as n approaches infinity a cauchy sequence.

Complete Function Spaces

A function space is called complete if all sequences in that space converge to another function in that space.

Banach Space A Banach Space is a complete normed function space.

Hilbert Space A Hilbert Space is a Banach Space with respect to a norm induced by the scalar product. That is, if there is a scalar product in the space X, then we can say the norm is induced by the scalar product if we can write:

That is, that the norm can be written as a function of the scalar product. In the L2 space, we can define the norm as:

If the scalar product space is a Banach Space, if the norm space is also a Banach space. In a Hilbert Space, the Parallelogram rule holds for all members f and g in the function space:

The L2 space is a Hilbert Space. The C space, however, is not.

Fourier Series The L2 space is an infinite function space, and therefore a linear combination of any infinite set of orthogonal functions can be used to represent any single member of the L2 space. The decomposition of an L2 function in terms of an infinite basis set is a technique known as the Fourier Decomposition of the function, and produces a result called the Fourier Series.

Fourier Basis Let's consider a set of L2 functions, φ as follows: φ = 1,sin(nπx),cos(nπx),n = 1,2,...

We can prove that over a range [a, b] = [0, 2\pi], all of these functions are orthogonal:

And both the sinusoidal functions are orthogonal with the function φ(x) = 1. Because this serves as an infinite orthogonal set in L2, this is also a valid basis set in that space. Therefore, we can decompose any function in L2 as the following sum: [Classical Fourier Series]

However, the difficulty occurs when we need to calculate the a and b coefficients. We will show the method to do this below:

a0: The Constant Term Calculation of a0 is the easiest, and therefore we will show how to calculate it first. We first create an error function, E, that is equal to the squared norm of the difference between the function f(x) and the infinite sum above:

For ease, we will write all the basis functions as the set φ, described above:

Combining the last two functions together, and writing the norm as an integral, we can say:

We attempt to minimize this error function with respect to the constant term. To do this, we differentiate both sides with respect to a0, and set the result to zero:

The &phi0 term comes out of the sum because of the chain rule: it is the only term in the entire sum dependant on a0. We can separate out the integral above as follows:

All the other terms drop out of the infinite sum because they are all orthogonal to φ0. Again, we can rewrite the above equation in terms of the scalar product:

And solving for a0, we get our final result:

Sin Coefficients Using the above method, we can solve for the an coefficients of the sin terms:

Cos Coefficients Also using the above method, we can solve for the bn terms of the cos term.

Arbitrary Basis Expansion The classical Fourier series uses the following basis: φ(x) = 1,sin(nπx),cos(nπx),n = 1,2,... However, we can generalize this concept to extend to any orthogonal basis set from the L2 space.

We can say that if we have our orthogonal basis set that is composed of an infinite set of arbitrary, orthogonal L2 functions:

We can define any L2 function f(x) in terms of this basis set: [Generalized Fourier Series]

Using the method from the previous chapter, we can solve for the coefficients as follows: [Generalized Fourier Coefficient]

Bessel Equation and Parseval Theorem Bessel's equation relates the original function to the fourier coefficients an: [Bessel's Equation]

If the basis set is infinitely orthogonal, and if an infinite sum of the basis functions perfectly reproduces the function f(x), then the above equation will be an equality, known as Parseval's Theorem: [Parseval's Theorem]

Engineers may recognize this as a relationship between the energy of the signal, as represented in the time and frequency domains. However, parseval's rule applies not only to the classical Fourier series coefficients, but also to the generalized series coefficients as well.

Multi-Dimensional Fourier Series The concept of the fourier series can be expanded to include 2-dimensional and ndimensional function decomposition as well. Let's say that we have a function in terms of independant variables x and y. We can decompose that function as a double-summation as follows:

Where φij is a 2-dimensional set of orthogonal basis functions. We can define the coefficients as:

This same concept can be expanded to include series with n-dimensions.

Wavelets Wavelets are orthogonal basis functions that only exist for certain windows in time. This is in contrast to sinusoidal waves, which exist for all times t. A wavelet, because it is dependant on time, can be used as a basis function. A wavelet basis set gives rise to wavelet decomposition, which is a 2-variable decomposition of a 1-variable function. Wavelet analysis allows us to decompose a function in terms of time and frequency, while fourier decomposition only allows us to decompose a function in terms of frequency.

Mother Wavelet If we have a basic wavelet function ψ(t), we can write a 2-dimensional function known as the mother wavelet function as such: ψjk = 2j / 2ψ(2jt − k)

Wavelet Series If we have our mother wavelet function, we can write out a fourier-style series as a double-sum of all the wavelets:

Scaling Function Sometimes, we can add in an additional function, known as a scaling function:

The idea is that the scaling function is larger then the wavelet functions, and occupies more time. In this case, the scaling function will show long-term changes in the signal, and the wavelet functions will show short-term changes in the signal.

Random Variables A random variable is a variable that takes a random value at any particular point t in time. The properties of the random variable are known as the distribution of the random variable. We will denote random variables by the abbreviation "r.v.", or simply "rv". This is a common convention used in the literature concerning this subject.

Probability Function The probability function, P[], will denote the probability of a particular occurance happening. Here are some examples: • • •

P[X < x], the probability that the random variable X has a value less then some variable x. P[X = x], the probability that the random variable X has a value equal to some variable x. P[X < x,Y > y], the probability that the random variable X has a value equal to x, and the random variable Y has a value equal to y.

Example: Fair Coin Consider the example that a fair coin is flipped. We will define X to be the random variable, and we will define "head" to be 1, and "tail" to be 0. What is the probability that the coin is a head? P[X = 1] = 50%

Example: Fair Dice Consider now a fair 6-sided dice. X is the r.v., and the numerical value on the face of the die is the value that X can take. What is the probability that when the dice is rolled, the value is less then 4? P[X < 4] = 50% What is the probability that the value will be even? P[X is even] = 50%

Notation We will typically write random variables as upper-case letters, such as Z, X, Y, etc. Lower-case letters will be used to denote variables that are related with the random variables. For instance, we will use "x" as a variable that is related to "X", the random variable.

When we are using random variables in conjunction with matrices, we will use the following conventions: 1. Random variables, and random vectors or matrices will be denoted with letters from the end of the alphabet, such as W, X, Y, and Z. Also, Θ and Ω will be used as a random variables, especially when we talk about random frequencies. 2. A random matrix or vector, will be denoted with a capital letter. The entries in that random vector or matrix will be denoted with capital letters and subscripts. These matrices will also use letters from the end of the alphabet, or the greek letters Θ and Ω. 3. A regular coefficient vector or matrix that is not random will use a capital matrix from the beginning of the alphbet, such as A, B, C, or D. 4. Special vectors or matrices that are derived from random variables, such as correlation matrices, or covariance matrices, will use capital letters from the middle of the alphabet, such as K, M, N, P, or Q. Any other variables or notations will be explained in the context of the page where it appears.

Conditional Probability A conditional probability is the probability measure of one event happening given that another event already has happened. For instance, what are the odds that your computer system will suddenly break while you are reading this page? P[computer breaks] = small The odds that your computer will suddenly stop working are very small. However, what are the odds that your computer will break given that it just got struck by lightning? P[computer breaks | struck by lightning] = large The vertical bar separates the things that haven't happened yet (the a priori probabilities, on the left) from the things that have already happened and might affect our outcome (the a posteriori probabilities, on the right). As another example, what are the odds that a dice rolled will be a 2, assuming that we know the number is less then 4? P[X = 2 | X < 4] = 33.33% If X is less then 4, we know it can only be one of the values 1, 2, or 3. Or another example, what if a person asks you "I'm thinking of a number between 1 and 10", what are your odds of guessing the right number? P[X = x | 0 < X < 10] = 10% Where x is the correct number that you are trying to guess.

Probability Functions Probability Density Function The probability density function, or pdf of a random variable is the function defined by: fX(x) = P[X = x] Remember here that X is the random variable, and x is a related variable (but is not random). The subscript X on fX denotes that this is the pdf for the X variable. pdf's follow a few simple rules: 1. The pdf is always non-negative. 2. The area under the pdf curve is 1.

Cumulative Distribution Function The cumulative distribution function, (CDF), is also known as the Probability Distribution Function, (PDF). to reduce confusion with the pdf of a random variable, we will use the acronym CDF to denote this function. The CDF of a random variable is the function defined by:

The CDF and the pdf of a random variable are related:

The CDF is the function corresponding to the probability that a given value x is less then the value of the random variable X. The CDF is a non-decreasing function, and is always non-negative.

Example: X between two bounds

To determine whether our random variable X lies between two bounds, [a, b], we can take the CDF functions:

Distributions There are a number of common distributions that are used in conjunction with random variables.

Uniform Distribution The uniform distribution is one of the easiest distributions to analyze. Also, uniform distributions of random numbers are easy to generate on computers, so they are typically used in computer software.

Gaussian Distribution The gaussian distribution, or the "normal distribution" is one of the most common random distributions. A gaussian random variable is typically called a "normal" random variable.

Where µ is the mean of the function, and σ2 is the variance of the function. we will discuss both these terms later.

Expectation and Entropy Expectation The expectation operator of a random variable is defined as:

This operator is very useful, and we can use it to derive the moments of the random variable.

Moments A moment is a value that contains some information about the random variable. The nmoment of a random variable is defined as:

Mean The mean value, or the "average value" of a random variable is defined as the first moment of the random variable:

We will use the greek letter µ to denote the mean of a random variable.

Central Moments A central moment is similar to a moment, but it is also dependant on the mean of the random variable:

The first central moment is always zero.

Variance The variance of a random variable is defined as the second central moment: E[(x − µX)2] = σ2 The square-root of the variance, σ, is known as the standard-deviation of the random variable

Mean and Variance The mean and variance of a random variable can be related by: σ2 = µ2 + E[x2] This is an important function, and we will use it later.

Entropy The entropy of a random variable X is defined as:

SISO Transformations Let's say that we have a random variable X that is the input into a given system. The system output, Y is then also a random variable that is related to the input X by the response of the system. In other words, we can say that: Y = g(X) Where g is the mathematical relationship between the system input and the system output. To discover information about Y, we can use the information we know about the r.v. X, and the relationship g:

Where xi are the roots of g.

MISO Transformations Consider now a system with two inputs, both of which are random (or pseudorandom, in the case of non-deterministic data). For instance, let's consider a system with the following inputs and outputs: • •

X: non-deterministic data input Y: disruptive noise

•

Z: System output

Our system satisfies the following mathematical relationship: Z = g(X,Y) Where g is the mathematical relationship between the system input, the disruptive noise, and the system output. By knowing information about the distributions of X and Y, we can determine the distribution of Z.

Correlation Independance Two random variables are called independant if changes in one do not affect, and are not affected by, changes in the other.

Correlation Two random variables are said to have correlation if they take the same values, or similar values, at the same point in time. Independance implies that two random variables will be uncorrelated, but two random variables being uncorrelated does not imply that they are independant.

Random Vectors Many of the concepts that we have learned so far have been dealing with random variables. However, these concepts can all be translated to deal with vectors of random numbers. A random vector X contains N elements, Xi, each of which is a distinct random variable. The individual elements in a random vector may or may not be correlated or dependent on one another.

Expectation The expectation of a random vector is a vector of the expectation values of each element of the vector. For instance:

Using this definition, the mean vector of random vector X, denoted µX is the vector composed of the means of all the individual elements of X:

Correlation Matrix The correlation matrix of a random vector X is defined as: RX = E[XXT] Where each element of the correlation matrix corresponds to the correlation between the row element of X, and the column element of XT. The correlation matrix is a realsymmetric matrix. If the off-diagonal elements of the correlation matrix are all zero, the random vector is said to be uncorrelated. If the R matrix is an identity matrix, the random vector is said to be "white". For instance, "white noise" is uncorrelated, and each element of the vector has an equal correlation value.

Matrix Diagonalization As discussed earlier, we can diagonalize a matrix by constructing the V matrix from the eigenvectors of that matrix. If X is our non-diagonal matrix, we can create a diagonal matrix D by: D = V − 1XV If the X matrix is real symmetric (as is always the case with the correlation matrix), we can simplify this to be: D = VTXV

Whitening A matrix can be whitened by constructing a matrix W that contains the inverse squareroots of the eigenvalues of X on the diagonal:

Using this W matrix, we can convert X into the identity matrix: I = WTVTXVW

Simultaneous Diagonalization If we have two matrices, X and Y, we can construct a matrix A that will satisfy the following relationships: ATXA = I ATYA = D Where I is an identity matrix, and D is a diagonal matrix. This process is known as simultaneous diagonalization. If we have the V and W matrices described above such that I = WTVTXVW, We can then construct the B matrix by applying this same transformation to the Y matrix: WTVTYVW = B We can combine the eigenvalues of B into a transformation matrix Z such that: ZTBZ = D We can then define our A matrix as: A = VWZ AT = ZTWTVT This A matrix will satisfy the simultaneous diagonalization proceedure, outlined above.

Covariance Matrix The Covariance Matrix of two random vectors, X and Y, is defined as:

QX = E[(X − µX)(Y − µY)T] = E[(Y − µY)(X − µX)T] Where each element of the covariance matrix expresses the variance relationship between the row element of X, and the column element of Y. The covariance matrix is real symmetric. We can relate the correlation matrix and the covariance matrix through the following formula:

Cumulative Distribution Function An N-vector X has a cumulative distribution function Fx of N variables that is defined as:

Probability Density Function The probability density function of a random vector can be defined in terms of the Nth partial derivative of the cumulative distribution function:

If we know the density function, we can find the mean of the ith element of X using N-1 integrations:

Optimization Optimization is an important concept in engineering. Finding any solution to a problem is not nearly as good as finding the one "optimal solution" to the problem. Optimization problems are typically reformatted so they become minimization problems, which are well-studied problems in the field of mathematics. Typically, when optimizing a system, the costs and benefits of that system are arranged into a cost function. It is the engineers job then to minimize this cost function (and thereby minimize the cost of the system). It is worth noting at this point that the word "cost" can have multiple meanings, depending on the particular problem. For instance, cost can refer to the actual monetary cost of a system (number of computer units to host a website, amount of cable needed to connect Philadelphia and New York), the delay of the system (loading time for a website, transmission delay for a communication network), the reliability of the system (number of dropped calls in a cellphone network, average lifetime of a car transmission), or any other types of factors that reduce the effectiveness and efficiency of the system. Because optimization typically becomes a mathematical minimization problem, we are going to discuss minimization here.

Minimization Minimization is the act of finding the numerically lowest point in a given function, or in a particular range of a given function. Students of mathematics and calculus may remember using the derivative of a function to find the maxima and minima of a function. If we have a function f(x), we can find the maxima, minima, or saddle-points (points where the function has zero slope, but is not a maxima or minima) by solving for x in the following equation:

In other words, we are looking for the roots of the derivative of the function f. Once we have the roots of the function (if any), we can test those points to see if they are relatively high (maxima), or relatively low (minima). Some other words to remember are: Global Minima: A global minima of a function is the lowest value of that function anywhere. Local Minima: A local minima of a function is the lowest value of that function within a given range A < x < B. If the function derivative has no roots in that range, then the minima occurs at either A, or B. We will discuss some other techniques for finding minima below.

Unconstrained Minimization Unconstrained Minimization refers to the minimization of the given function without having to worry about any other rules or caveats. Constrained Minimization, on the other hand, refers to minimization problems where there are other factors or constraints that must be satisfied. Besides the method above (where we take the derivative of the function and set that equal to zero), there are several numerical methods that we can use to find the minima of a function. These methods are useful when using computational tools such as Matlab.

Hessian Matrix The function has a local minima at a point x if the Hessian matrix H(x) is positive definite:

Where x is a vector of all the independant variables of the function. If x is a scalar variable, the hessian matrix reduces to the second derivative of the function f.

Newton-Raphson Method The Newton-Raphson Method of computing the minima of a function, f uses an iterative computation. We can define the scheme:

Where

As we repeat the above equation, plugging in consecutive values for n, our solution will converge on the true solution. However, this process will take infinitely many iterations to converge, so oftentimes an approximation of the true solution will suffice.

Steepest Descent Method The Newton-Raphson method can be tricky because it relies on the second derivative of the function f, and this can oftentimes be difficult (if not impossible) to accurately calculate. The Steepest Descent Method, however, does not require the second derivative, but it does require the selection of an appropriate scalar quantity ε, which cannot be chosen arbitrarily (but which can also not be calculated using a set formula). The Steepest Descent method is defined by the following iterative computation:

Where epsilon needs to be sufficiently small. If epsilon is too large, the iteration may diverge. If this happens, a new epsilon value needs to be chosen, and the process needs to be repeated.

Constrained Minimization Constrained Minimization' is the process of finding the minimum value of a function under a certain number of additional rules or constraints. For instance, we could say "Find the minium value of f(x), but g(x) must equal 10". These kinds of problems are difficult, but fortunately we can utilize the Khun-Tucker theorem, and also the Karush=Khun-Tucker theorem to solve for them. There are two different types of constraints: equality constraints and inequality constraints. We will consider them individually, and then we will consider them together.

Equality Constraints The Khun-Tucker Theorem is a method for minimizing a function f(x) under the equality constraint g(x). We can define the theorem as follows: If we have a function f, and an equality constraint g in the following form: g(x) = 0, Then we can convert this problem into an unconstrained minimization problem by constructing the Lagrangian function of f and g:

Where Λ is the lagrangian vector, and < , > denotes the scalar product operation of the Rn vector space (where n is the number of equality constraints). Λ is the Lagrangian Multipler vector, with one entry in Λ for each equality constraint on the equation. We will discuss scalar products more later. If we differentiate this equation with respect to x,

we can find the minimum of this whole function L(x), and that will be the minimum of our function f.

This is a set of n equations with 2n unknown variables (Λ and x vectors). We can create additional equations by differentiating with respect to each element of Λ and x.

Inequality Constraints Similar to the method above, let's say that we have a cost function f, and an inequality constraint in the following form:

Then we can take the Lagrangian of this again:

But we now must also use the following two equations in determining our solution:

These last two equations can be interpreted in the following way: if g(x) < 0, then Λ = 0 if , then Using these two additional equations, we can solve for our minimization answer in a similar manner as above.

Equality and Inequality Constraints If we have a set of inequality and equality constraints: g(x) = 0 We can combine them into a single Lagrangian with two additional conditions:

The last two conditions can be interpreted in the same manner as above to find the solution.

Infinite Dimension Minimization The above methods work well if the variables involved in the analysis are finitedimensional vectors, especially those in the RN space. However, what if we are trying to minimize something that is more complex then a vector, such as a function? If we consider the L2 space, we have an infinite-dimensional space where the members of that space are all functions. We will define the term functional as follows: Functional A functional is a function that takes one or more functions as arguments, and which returns a scalar value. Let's say that we have a function x of time t. We can define the functional f as: f(x(t)) With that function, we can associate a cost function J:

Where we are explicitly taking account of t in the definition of f. To minimize this function, like all minimization problems, we need to take the derivative of the function, and set the derivative to zero. However, we are not able to take a standard derivative of J with respect to x, because x is a function that varies with time. However, we can define a new type of derivative, the Gateaux Derivative that can handle this special case.

Gateaux Derivative We can define the Gateaux Derivative in terms of the following limit:

Which is similar to the classical definition of the derivative, except with the inclusion of the term ε. In english, above we took the derivative of F with respect to x, in the direction of h. h is an arbitrary function of time, in the same space as x (here we are talking about the L2 space). We can use the Gateaux derivative to find the minimization of our function above.

Euler-Lagrange Equation The Euler-Lagrange Equation uses the Gateaux derivative, discussed above, to find the minimization of the following types of function:

We want to find the solutions to this problem: δJ(x) = 0 And the solution is:

The partial derivatives can be done in an ordinary way ignoring the fact that x is a function of t. Solutions to this equation are either the maxima or minima of the cost function J.

Example: Shortest Distance We've heard colloquially that the shortest distance between two points is a straight line. We can use the Euler-Lagrange equation to prove this rule. If we have two points in R2 space, a, and b, we would like to find the minimum function that joins these two points. We can define the differential ds as the differential along the function that joins points a and b:

Our function that we are trying to minimize then is defined as:

or:

We can take the Gateaux derivative of the function J and set it equal to zero to find the minimum function between these two points.

Laplace Transform Table Time Domain

Laplace Domain

δ(t)

1

δ(t − a)

e − as

u(t)

u(t − a)

tu(t)

tnu(t)

eatu(t)

tneatu(t)

cos(ωt)u(t)

sin(ωt)u(t)

cosh(ωt)u(t)

sinh(ωt)u(t)

eatcos(ωt)u(t)

eatsin(ωt)u(t)

Laplace Transform Table 2

ID

1

Function

Time domain Failed to parse Laplace domain (Can't write to or Region of Failed to parse (Can't write to or create maths convergence create maths output directory): output directory): for causal X(s) = \mathcal{L}\left\{ x(t) x(t) = systems \right\} \mathcal{L}^{-1} \left\{ X(s) \right\}

ideal delay

Failed to parse (Can't write to or Failed to parse (Can't write to or create maths create maths output directory): output directory): e^{-\tau s} \ \delta(t-\tau) \

unit impulse

Failed to parse (Can't Failed to parse write to or (Can't write to or Failed to parse (Can't write to or create maths create maths create maths output directory): output output directory): 1\ directory): \delta(t) \ \mathrm{all} \ s \,

2

delayed nth power with frequency shift

Failed to parse Failed to (Can't write to or parse (Can't create maths Failed to parse (Can't write to or write to or output directory): create maths output directory): create maths \frac{(t\frac{e^{-\tau output \tau)^n}{n!} e^{s}}{(s+\alpha)^{n+1}} directory): s \alpha (t-\tau)} > 0 \, \cdot u(t-\tau)

2a

nth power

1a

Failed to parse Failed to Failed to parse (Can't write to or (Can't write to or parse (Can't create maths output directory): { create maths write to or

output directory): { t^n \over n! } \cdot u(t)

1 \over s^{n+1} }

create maths output directory): s > 0 \,

qth power

Failed to parse Failed to (Can't write to or parse (Can't create maths Failed to parse (Can't write to or write to or output directory): { create maths output directory): { create maths t^q \over 1 \over s^{q+1} } output \Gamma(q+1) } directory): s \cdot u(t) > 0 \,

unit step

Failed to Failed to parse parse (Can't (Can't write to or Failed to parse (Can't write to or write to or create maths create maths output directory): { create maths output directory): 1 \over s } output u(t) \ directory): s > 0 \,

delayed unit step

Failed to Failed to parse parse (Can't (Can't write to or Failed to parse (Can't write to or write to or create maths create maths output directory): { create maths output directory): e^{-\tau s} \over s } output directory): s u(t-\tau) \ > 0 \,

2c

ramp

Failed to Failed to parse parse (Can't (Can't write to or Failed to parse (Can't write to or write to or create maths create maths output directory): create maths output directory): t \frac{1}{s^2} output \cdot u(t)\ directory): s > 0 \,

2d

nth power with frequency

Failed to parse (Can't write to or Failed to Failed to parse (Can't write to or create maths output directory): parse (Can't \frac{1}{(s+\alpha)^{n+1}} create maths write to or

2a.1

2a.2

2b

shift

output directory): \frac{t^{n}}{n!}e^{\alpha t} \cdot u(t)

create maths output directory): s > - \alpha \,

exponential decay

Failed to Failed to parse parse (Can't (Can't write to or Failed to parse (Can't write to or write to or create maths create maths output directory): { create maths output directory): 1 \over s+\alpha } output e^{-\alpha t} \cdot directory): s u(t) \ > - \alpha \

exponential approach

Failed to Failed to parse parse (Can't (Can't write to or Failed to parse (Can't write to or write to or create maths create maths output directory): create maths output directory): ( \frac{\alpha}{s(s+\alpha)} output 1-e^{-\alpha t}) directory): s \cdot u(t) \ > 0\

sine

Failed to Failed to parse parse (Can't (Can't write to or Failed to parse (Can't write to or write to or create maths create maths output directory): { create maths output directory): \omega \over s^2 + \omega^2 } output \sin(\omega t) \cdot directory): s u(t) \ >0\

5

cosine

Failed to Failed to parse parse (Can't (Can't write to or Failed to parse (Can't write to or write to or create maths create maths output directory): { create maths output directory): s \over s^2 + \omega^2 } output \cos(\omega t) \cdot directory): s u(t) \ >0\

6

hyperbolic sine

Failed to parse (Can't write to or Failed to Failed to parse create maths output directory): { parse (Can't (Can't write to or \alpha \over s^2 - \alpha^2 } create maths write to or

2d.1

3

4

output directory): \sinh(\alpha t) \cdot u(t) \

7

hyperbolic cosine

create maths output directory): s > | \alpha | \

Failed to Failed to parse parse (Can't (Can't write to or Failed to parse (Can't write to or write to or create maths create maths output directory): { create maths output directory): s \over s^2 - \alpha^2 } output \cosh(\alpha t) \cdot directory): s u(t) \ > | \alpha | \

8

Failed to parse Failed to (Can't write to or parse (Can't Failed to parse (Can't write to or Exponentiallycreate maths write to or create maths output directory): { decaying output directory): create maths \omega \over (s+\alpha )^2 + sine wave e^{-\alpha t} output \omega^2 } \sin(\omega t) \cdot directory): s u(t) \ > -\alpha \

9

Failed to parse Failed to (Can't write to or parse (Can't Failed to parse (Can't write to or Exponentiallycreate maths write to or create maths output directory): { decaying output directory): create maths s+\alpha \over (s+\alpha )^2 + cosine wave e^{-\alpha t} output \omega^2 } \cos(\omega t) \cdot directory): s u(t) \ > -\alpha \

10

nth root

11

natural logarithm

Failed to parse (Can't write to or Failed to parse (Can't write to or create maths create maths output directory): output directory): s^{-(n+1)/n} \cdot \sqrt[n]{t} \cdot \Gamma\left(1+\frac{1}{n}\right) u(t)

Failed to parse (Can't write to or create maths output directory): s > 0 \,

Failed to parse Failed to parse (Can't write to or Failed to (Can't write to or create maths output directory): - parse (Can't create maths { t_0 \over s} \ [ \ \ln(t_0 write to or

output directory): \ln \left ( { t \over t_0 } \right ) \cdot u(t)

12

Bessel function of the first kind, of order n

s)+\gamma \ ]

create maths output directory): s > 0 \,

Failed to parse (Can't write to or create maths output Failed to parse Failed to parse (Can't write to or directory): s (Can't write to or create maths output directory): > 0 \, create maths \frac{ \omega^n output directory): \left(s+\sqrt{s^2+ Failed to J_n( \omega t) \cdot \omega^2}\right)^{parse u(t) n}}{\sqrt{s^2 + \omega^2}}

(Can't write to or create maths output directory): (n > -1) \,

13

Modified Bessel function of the first kind, of order n

Failed to Failed to parse Failed to parse (Can't write to or parse (Can't (Can't write to or create maths output directory): write to or create maths \frac{ \omega^n create maths output directory): \left(s+\sqrt{s^2output I_n(\omega t) \cdot \omega^2}\right)^{directory): s u(t) n}}{\sqrt{s^2-\omega^2}} > | \omega | \,

14

Failed to parse Bessel (Can't write to or function create maths of the second output directory): kind, Y_0(\alpha t) \cdot of order 0 u(t)

15

Modified Failed to parse Bessel (Can't write to or function create maths of the second output directory): kind, K_0(\alpha t) \cdot of order 0 u(t)

16 Error function

Failed to parse Failed to parse (Can't write to or Failed to (Can't write to or create maths output directory): parse (Can't create maths {e^{s^2/4} \operatorname{erfc} write to or

output directory): \mathrm{erf}(t) \cdot u(t)

\left(s/2\right) \over s}

create maths output directory): s > 0 \,

Explanatory notes: •

Failed to parse (Can't write to or create maths output directory): u(t) \,

represents the Heaviside step function. •

Failed to parse (Can't write to or create maths output directory): \delta(t) \,

represents the Dirac delta function. •

Failed to parse (Can't write to or create maths output directory): \Gamma (z) \,

represents the Gamma function. •

Failed to parse (Can't write to or create maths output directory): \gamma \,

•

Failed to parse (Can't write to or create maths output directory): t \,

, a real number, typically represents time, although it can represent any independent dimension. •

Failed to parse (Can't write to or create maths output directory): s \,

is the complex angular frequency. •

Failed to parse (Can't write to or create maths output directory): \alpha \,

, Failed to parse (Can't write to or create maths output directory): \beta \, , Failed to parse (Can't write to or create maths output directory): \tau \, , and Failed to parse (Can't write to or create maths output directory): \omega \, are real numbers.

•

Failed to parse (Can't write to or create maths output directory): n \,

is an integer.

is the Euler-Mascheroni constant. •

A causal system is a system where the impulse response h(t) is zero for all time t prior to t = 0. In general, the ROC for causal systems is not the same as the ROC for anticausal systems. See also causality.

Laplace Transform Properties Property

Linearity

Differentiation

Frequency Division Frequency Integration Time Integration Scaling Initial value theorem Final value theorem Frequency Shifts

Time Shifts Convolution Theorem Where:

Definition

s = σ + jω

Fourier Transform Table Time Domain

1

Fourier Domain

2πδ(ω)

− 0.5 + u(t)

δ(t)

1

δ(t − c)

e − jωc

u(t)

e − btu(t)

cosω0t

π[δ(ω + ω0) + δ(ω − ω0)]

cos(ω0t + θ)

π[e − jθδ(ω + ω0) + ejθδ(ω − ω0)]

sinω0t

jπ[δ(ω + ω0) − δ(ω − ω0)]

sin(ω0t + θ)

jπ[e − jθδ(ω + ω0) − ejθδ(ω − ω0)]

2πpτ(ω)

Note: sinc(x) = sin(x) / x ; pτ(t) is the rectangular pulse function of width τ

Fourier Transform Table 2

Signal

Fourier transform unitary, angular frequency

Fourier transform unitary, ordinary frequency

Remarks

10

The rectangular pulse and the normalized sinc function

11

Dual of rule 10. The rectangular function is an idealized low-pass filter, and the sinc function is the noncausal impulse response of such a filter.

12

tri is the triangular function

13

Dual of rule 12.

14

Shows that the Gaussian function exp( − αt2) is its own Fourier transform. For this to be integrable we must have Re(α) > 0.

common in optics

a>0

the transform is the function itself

J0(t) is the Bessel function of first kind of order 0, rect is the rectangular function it's the generalizatio n of the previous transform; Tn (t) is the Chebyshev polynomial of the first kind.

Un (t) is the Chebyshev polynomial of the second kind

Fourier Transform Properties

Signal

Fourier transform unitary, angular frequency

Fourier transform unitary, ordinary frequency

Remarks

1

Linearity

2

Shift in time domain

3

Shift in frequency domain, dual of 2

If 4

is large, then is concentrated around 0 and spreads

out and flattens

5

Duality property of the Fourier transform. Results from swapping "dummy" variables of and .

6

Generalized derivative property of the Fourier transform

7

This is the dual to 6

8

denotes the convolution of and — this rule is the convolution theorem

9

This is the dual of 8

DTFT Transform Table Time domain

Frequency domain

Remarks

integer k

real number a

real number a

real number a

integer M

real number a

real number W

real numbers W, a

real numbers W, a

real numbers A, B complex C

DTFT Transform Properties

Property

Time domain

Frequency domain

Remarks

Linearity

Shift in time

integer k

Shift in frequency

real number a

Time reversal Time conjugation Time reversal & conjugation

Derivative in frequency

Integral in frequency

Convolve in time

Multiply in time

Correlation

Where: • • •

is the convolution between two signals is the complex conjugate of the function x[n] represents the correlation between x[n] and y[n].

DFT Transform Table Time-Domain x[n]

Frequency Domain X[k]

Notes

DFT Definition

Shift theorem

Real DFT

Z Transform Table Signal, x[n]

1

2

3

4

5

Z-transform, X(z)

ROC

6

7

8

9

10

11

Z Transform Properties Time domain

Z-domain

ROC

Notation

ROC:

Linearity

At least the intersection of ROC1 and ROC2

Time shifting

ROC, except and

if if

Scaling in the zdomain

Time reversal

Conjugat ion

ROC

Real part

ROC

Imaginar y part

ROC

Differenti ation

ROC

Convolut ion

At least the intersection of ROC1 and ROC2

Correlati on

At least the intersection of ROC of X1(z) and X2(z − 1)

At least

Multiplic ation

Parseval' s relation

•

Inital value theorem , If

•

causal

Final value theorem , Only if poles of circle

Hilbert Transform Table Signal

Sinc function

Hilbert transform

are inside unit

Rectangular function

δ(t) Dirac delta function

Properties of Integrals Property

Integral

Homogeniety

Associativity

Integration by Parts

:

Table of Integrals This is a small summary of the identities found Here.

Integral

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

Properties of Derivatives Properties of Derivation

Product Rule

Quotient rule

Functional Power Rule

Chain Rule

Logarithm Rule

Table of Derivatives Table of Derivatives

where both xc and cxc-1 are defined.

x>0

c > 0

c > 0,

Trigonometric Identities sin2 + cos2 = 1

1 + tan2 = sec2

sin( − θ) = − sinθ

cos( − θ) = sinθ

sin2θ = 2sinθcosθ

cos2θ = cos2 − sin2 = 2cos2θ − 1 = 1 − 2sin2θ

1 + cot2 = csc2

ejθ = cosθ + jsinθ

tan( − θ) = cotθ

Normal Distribution The normal distibution is an extremely important family of continuous probability distributions. It has applications in every engineering discipline. Each member of the family may be defined by two parameters, location and scale: the mean ("average", μ) and variance (standard deviation squared, σ2) respectively. The probability density function, or pdf, of the distribution is given by:

The cumulative distibution function, or cdf, of the normal distribution is:

These functions are often inpractical to evaluate quickly, and therefore tables of values are used to allow fast look-up of required data. The family or normal distibutions is infinite in size, but all can be "normalised" to the case with mean of 0 and SD of 1: Given a normal distibution distribution, Z, is:

, the standardised normal

Due to this relationship, all tables refer to the standardised distibution, Z.

Probability Content from –∞ to Z (Z≤0) Table of Probability Content between –∞ and z in the Standardised Normal Distribution Z~N(0,1) for z≤0

z

0.0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.0 0.50000 0.49601 0.49202 0.48803 0.48405 0.48006 0.47608 0.47210 0.46812 0.46414 0.46017 0.45620 0.45224 0.44828 0.44433 0.44038 0.43644 0.43251 0.42858 0.42465 0.1 0.42074 0.41683 0.41294 0.40905 0.40517 0.40129 0.39743 0.39358 0.38974 0.38591 0.2 0.38209 0.37828 0.37448 0.37070 0.36693 0.36317 0.35942 0.35569 0.35197 0.34827 0.3 0.34458 0.34090 0.33724 0.33360 0.32997 0.32636 0.32276 0.31918 0.31561 0.31207 0.4 0.30854 0.30503 0.30153 0.29806 0.29460 0.29116 0.28774 0.28434 0.28096 0.27760 0.5 0.27425 0.27093 0.26763 0.26435 0.26109 0.25785 0.25463 0.25143 0.24825 0.24510 0.6 0.24196 0.23885 0.23576 0.23270 0.22965 0.22663 0.22363 0.22065 0.21770 0.21476 0.7 0.21186 0.20897 0.20611 0.20327 0.20045 0.19766 0.19489 0.19215 0.18943 0.18673 0.8 0.18406 0.18141 0.17879 0.17619 0.17361 0.17106 0.16853 0.16602 0.16354 0.16109 0.9 0.15866 0.15625 0.15386 0.15151 0.14917 0.14686 0.14457 0.14231 0.14007 0.13786 1.0

0.13567 0.13350 0.13136 0.12924 0.12714 0.12507 0.12302 0.12100 0.11900 0.11702 1.1 0.11507 0.11314 0.11123 0.10935 0.10749 0.10565 0.10383 0.10204 0.10027 0.09853 1.2 0.09680 0.09510 0.09342 0.09176 0.09012 0.08851 0.08691 0.08534 0.08379 0.08226 1.3 0.08076 0.07927 0.07780 0.07636 0.07493 0.07353 0.07215 0.07078 0.06944 0.06811 1.4 0.06681 0.06552 0.06426 0.06301 0.06178 0.06057 0.05938 0.05821 0.05705 0.05592 1.5 0.05480 0.05370 0.05262 0.05155 0.05050 0.04947 0.04846 0.04746 0.04648 0.04551 1.6 0.04457 0.04363 0.04272 0.04182 0.04093 0.04006 0.03920 0.03836 0.03754 0.03673 1.7 0.03593 0.03515 0.03438 0.03362 0.03288 0.03216 0.03144 0.03074 0.03005 0.02938 1.8 0.02872 0.02807 0.02743 0.02680 0.02619 0.02559 0.02500 0.02442 0.02385 0.02330 1.9 0.02275 0.02222 0.02169 0.02118 0.02068 0.02018 0.01970 0.01923 0.01876 0.01831 2.0 0.01786 0.01743 0.01700 0.01659 0.01618 0.01578 0.01539 0.01500 0.01463 0.01426 2.1

0.01390 0.01355 0.01321 0.01287 0.01255 0.01222 0.01191 0.01160 0.01130 0.01101 2.2 0.01072 0.01044 0.01017 0.00990 0.00964 0.00939 0.00914 0.00889 0.00866 0.00842 2.3 0.00820 0.00798 0.00776 0.00755 0.00734 0.00714 0.00695 0.00676 0.00657 0.00639 2.4 0.00621 0.00604 0.00587 0.00570 0.00554 0.00539 0.00523 0.00508 0.00494 0.00480 2.5 0.00466 0.00453 0.00440 0.00427 0.00415 0.00402 0.00391 0.00379 0.00368 0.00357 2.6 0.00347 0.00336 0.00326 0.00317 0.00307 0.00298 0.00289 0.00280 0.00272 0.00264 2.7 0.00256 0.00248 0.00240 0.00233 0.00226 0.00219 0.00212 0.00205 0.00199 0.00193 2.8 0.00187 0.00181 0.00175 0.00169 0.00164 0.00159 0.00154 0.00149 0.00144 0.00139 2.9 0.00135 0.00131 0.00126 0.00122 0.00118 0.00114 0.00111 0.00107 0.00104 0.00100 3.0 0.00097 0.00094 0.00090 0.00087 0.00084 0.00082 0.00079 0.00076 0.00074 0.00071 3.1 0.00069 0.00066 0.00064 0.00062 0.00060 0.00058 0.00056 0.00054 0.00052 0.00050 3.2

0.00048 0.00047 0.00045 0.00043 0.00042 0.00040 0.00039 0.00038 0.00036 0.00035 3.3 0.00034 0.00032 0.00031 0.00030 0.00029 0.00028 0.00027 0.00026 0.00025 0.00024 3.4 0.00023 0.00022 0.00022 0.00021 0.00020 0.00019 0.00019 0.00018 0.00017 0.00017 3.5 0.00016 0.00015 0.00015 0.00014 0.00014 0.00013 0.00013 0.00012 0.00012 0.00011 3.6 0.00011 0.00010 0.00010 0.00010 0.00009 0.00009 0.00008 0.00008 0.00008 0.00008 3.7 0.00007 0.00007 0.00007 0.00006 0.00006 0.00006 0.00006 0.00005 0.00005 0.00005 3.8 0.00005 0.00005 0.00004 0.00004 0.00004 0.00004 0.00004 0.00004 0.00003 0.00003 3.9 0.00003 0.00003 0.00003 0.00003 0.00003 0.00003 0.00002 0.00002 0.00002 0.00002 4.0

Probability Content from –∞ to Z (Z≥0) Table of Probability Content between –∞ and z in the Standardised Normal Distribution Z~N(0,1) for z≥0

Z

0.0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 0.50000 0.50399 0.50798 0.51197 0.51595 0.51994 0.52392 0.52790 0.53188 0.53586 0.1 0.53983 0.54380 0.54776 0.55172 0.55567 0.55962 0.56356 0.56749 0.57142 0.57535 0.2 0.57926 0.58317 0.58706 0.59095 0.59483 0.59871 0.60257 0.60642 0.61026 0.61409 0.3 0.61791 0.62172 0.62552 0.62930 0.63307 0.63683 0.64058 0.64431 0.64803 0.65173 0.4 0.65542 0.65910 0.66276 0.66640 0.67003 0.67364 0.67724 0.68082 0.68439 0.68793 0.5 0.69146 0.69497 0.69847 0.70194 0.70540 0.70884 0.71226 0.71566 0.71904 0.72240 0.6 0.72575 0.72907 0.73237 0.73565 0.73891 0.74215 0.74537 0.74857 0.75175 0.75490 0.7 0.75804 0.76115 0.76424 0.76730 0.77035 0.77337 0.77637 0.77935 0.78230 0.78524 0.8 0.78814 0.79103 0.79389 0.79673 0.79955 0.80234 0.80511 0.80785 0.81057 0.81327 0.9 0.81594 0.81859 0.82121 0.82381 0.82639 0.82894 0.83147 0.83398 0.83646 0.83891 1.0 0.84134 0.84375 0.84614 0.84849 0.85083 0.85314 0.85543 0.85769 0.85993 0.86214 1.1 0.86433 0.86650 0.86864 0.87076 0.87286 0.87493 0.87698 0.87900 0.88100 0.88298 1.2 0.88493 0.88686 0.88877 0.89065 0.89251 0.89435 0.89617 0.89796 0.89973 0.90147 1.3 0.90320 0.90490 0.90658 0.90824 0.90988 0.91149 0.91309 0.91466 0.91621 0.91774

1.4 0.91924 0.92073 0.92220 0.92364 0.92507 0.92647 0.92785 0.92922 0.93056 0.93189 1.5 0.93319 0.93448 0.93574 0.93699 0.93822 0.93943 0.94062 0.94179 0.94295 0.94408 1.6 0.94520 0.94630 0.94738 0.94845 0.94950 0.95053 0.95154 0.95254 0.95352 0.95449 1.7 0.95543 0.95637 0.95728 0.95818 0.95907 0.95994 0.96080 0.96164 0.96246 0.96327 1.8 0.96407 0.96485 0.96562 0.96638 0.96712 0.96784 0.96856 0.96926 0.96995 0.97062 1.9 0.97128 0.97193 0.97257 0.97320 0.97381 0.97441 0.97500 0.97558 0.97615 0.97670 2.0 0.97725 0.97778 0.97831 0.97882 0.97932 0.97982 0.98030 0.98077 0.98124 0.98169 2.1 0.98214 0.98257 0.98300 0.98341 0.98382 0.98422 0.98461 0.98500 0.98537 0.98574 2.2 0.98610 0.98645 0.98679 0.98713 0.98745 0.98778 0.98809 0.98840 0.98870 0.98899 2.3 0.98928 0.98956 0.98983 0.99010 0.99036 0.99061 0.99086 0.99111 0.99134 0.99158 2.4 0.99180 0.99202 0.99224 0.99245 0.99266 0.99286 0.99305 0.99324 0.99343 0.99361 2.5 0.99379 0.99396 0.99413 0.99430 0.99446 0.99461 0.99477 0.99492 0.99506 0.99520 2.6 0.99534 0.99547 0.99560 0.99573 0.99585 0.99598 0.99609 0.99621 0.99632 0.99643 2.7 0.99653 0.99664 0.99674 0.99683 0.99693 0.99702 0.99711 0.99720 0.99728 0.99736 2.8 0.99744 0.99752 0.99760 0.99767 0.99774 0.99781 0.99788 0.99795 0.99801 0.99807

2.9 0.99813 0.99819 0.99825 0.99831 0.99836 0.99841 0.99846 0.99851 0.99856 0.99861 3.0 0.99865 0.99869 0.99874 0.99878 0.99882 0.99886 0.99889 0.99893 0.99896 0.99900 3.1 0.99903 0.99906 0.99910 0.99913 0.99916 0.99918 0.99921 0.99924 0.99926 0.99929 3.2 0.99931 0.99934 0.99936 0.99938 0.99940 0.99942 0.99944 0.99946 0.99948 0.99950 3.3 0.99952 0.99953 0.99955 0.99957 0.99958 0.99960 0.99961 0.99962 0.99964 0.99965 3.4 0.99966 0.99968 0.99969 0.99970 0.99971 0.99972 0.99973 0.99974 0.99975 0.99976 3.5 0.99977 0.99978 0.99978 0.99979 0.99980 0.99981 0.99981 0.99982 0.99983 0.99983 3.6 0.99984 0.99985 0.99985 0.99986 0.99986 0.99987 0.99987 0.99988 0.99988 0.99989 3.7 0.99989 0.99990 0.99990 0.99990 0.99991 0.99991 0.99992 0.99992 0.99992 0.99992 3.8 0.99993 0.99993 0.99993 0.99994 0.99994 0.99994 0.99994 0.99995 0.99995 0.99995 3.9 0.99995 0.99995 0.99996 0.99996 0.99996 0.99996 0.99996 0.99996 0.99997 0.99997 4.0 0.99997 0.99997 0.99997 0.99997 0.99997 0.99997 0.99998 0.99998 0.99998 0.99998

Far-Right Tail Probability Content

Table of Probability Content between z and +∞ in the Standardised Normal Distribution Z~N(0,1) for z>2

Z

P(Z>z)

z

P(Z>z)

z

P(Z>z)

z

P(Z>z)

2.0 0.02275

3.0 0.001350

4.0 0.00003167

5.0 2.867 E-7

2.1 0.01786

3.1 0.0009676

4.1 0.00002066

5.5 1.899 E-8

2.2 0.01390

3.2 0.0006871

4.2 0.00001335

6.0 9.866 E-10

2.3 0.01072

3.3 0.0004834

4.3 0.00000854

6.5 4.016 E-11

2.4 0.00820

3.4 0.0003369

4.4 0.000005413 7.0 1.280 E-12

2.5 0.00621

3.5 0.0002326

4.5 0.000003398 7.5 3.191 E-14

2.6 0.004661 3.6 0.0001591

4.6 0.000002112 8.0 6.221 E-16

2.7 0.003467 3.7 0.0001078

4.7 0.000001300 8.5 9.480 E-18

2.8 0.002555 3.8 0.00007235 4.8 7.933 E-7

9.0 1.129 E-19

2.9 0.001866 3.9 0.00004810 4.9 4.792 E-7

9.5 1.049 E-21

Student's T-Distribution

Table of Critical Values, tα,ν, in a Student T-Distribution with ν degrees of freedom and a confidence limit p where α=1–p.

Confidence Limits (top) and α (bottom) for a One-Tailed Test.

ν

60%

75%

80%

85%

90%

95%

97.5%

98%

99%

99.5%

99.75%

99.9%

99.95%

0.4

0.25

0.2

0.15

0.1

0.05

0.025

0.02

0.01

0.005

0.0025

0.001

0.0005

1

0.32492 1.00000 1.37638 1.96261 3.07768 6.31375 12.70620 15.89454 31.82052 63.65674 127.32134 318.30884 636.61925

2

0.28868 0.81650 1.06066 1.38621 1.88562 2.91999

4.30265

4.84873

6.96456

9.92484

14.08905

22.32712

31.59905

3

0.27667 0.76489 0.97847 1.24978 1.63774 2.35336

3.18245

3.48191

4.54070

5.84091

7.45332

10.21453

12.92398

4

0.27072 0.74070 0.94096 1.18957 1.53321 2.13185

2.77645

2.99853

3.74695

4.60409

5.59757

7.17318

8.61030

5

0.26718 0.72669 0.91954 1.15577 1.47588 2.01505

2.57058

2.75651

3.36493

4.03214

4.77334

5.89343

6.86883

6

0.26483 0.71756 0.90570 1.13416 1.43976 1.94318

2.44691

2.61224

3.14267

3.70743

4.31683

5.20763

5.95882

7

0.26317 0.71114 0.89603 1.11916 1.41492 1.89458

2.36462

2.51675

2.99795

3.49948

4.02934

4.78529

5.40788

8

0.26192 0.70639 0.88889 1.10815 1.39682 1.85955

2.30600

2.44898

2.89646

3.35539

3.83252

4.50079

5.04131

9

0.26096 0.70272 0.88340 1.09972 1.38303 1.83311

2.26216

2.39844

2.82144

3.24984

3.68966

4.29681

4.78091

10

0.26018 0.69981 0.87906 1.09306 1.37218 1.81246

2.22814

2.35931

2.76377

3.16927

3.58141

4.14370

4.58689

11

0.25956 0.69745 0.87553 1.08767 1.36343 1.79588

2.20099

2.32814

2.71808

3.10581

3.49661

4.02470

4.43698

12

0.25903 0.69548 0.87261 1.08321 1.35622 1.78229

2.17881

2.30272

2.68100

3.05454

3.42844

3.92963

4.31779

13

0.25859 0.69383 0.87015 1.07947 1.35017 1.77093

2.16037

2.28160

2.65031

3.01228

3.37247

3.85198

4.22083

14

0.25821 0.69242 0.86805 1.07628 1.34503 1.76131

2.14479

2.26378

2.62449

2.97684

3.32570

3.78739

4.14045

15

0.25789 0.69120 0.86624 1.07353 1.34061 1.75305

2.13145

2.24854

2.60248

2.94671

3.28604

3.73283

4.07277

16

0.25760 0.69013 0.86467 1.07114 1.33676 1.74588

2.11991

2.23536

2.58349

2.92078

3.25199

3.68615

4.01500

17

0.25735 0.68920 0.86328 1.06903 1.33338 1.73961

2.10982

2.22385

2.56693

2.89823

3.22245

3.64577

3.96513

18

0.25712 0.68836 0.86205 1.06717 1.33039 1.73406

2.10092

2.21370

2.55238

2.87844

3.19657

3.61048

3.92165

19

0.25692 0.68762 0.86095 1.06551 1.32773 1.72913

2.09302

2.20470

2.53948

2.86093

3.17372

3.57940

3.88341

20

0.25674 0.68695 0.85996 1.06402 1.32534 1.72472

2.08596

2.19666

2.52798

2.84534

3.15340

3.55181

3.84952

21

0.25658 0.68635 0.85907 1.06267 1.32319 1.72074

2.07961

2.18943

2.51765

2.83136

3.13521

3.52715

3.81928

22

0.25643 0.68581 0.85827 1.06145 1.32124 1.71714

2.07387

2.18289

2.50832

2.81876

3.11882

3.50499

3.79213

23

0.25630 0.68531 0.85753 1.06034 1.31946 1.71387

2.06866

2.17696

2.49987

2.80734

3.10400

3.48496

3.76763

24

0.25617 0.68485 0.85686 1.05932 1.31784 1.71088

2.06390

2.17154

2.49216

2.79694

3.09051

3.46678

3.74540

25

0.25606 0.68443 0.85624 1.05838 1.31635 1.70814

2.05954

2.16659

2.48511

2.78744

3.07820

3.45019

3.72514

26

0.25595 0.68404 0.85567 1.05752 1.31497 1.70562

2.05553

2.16203

2.47863

2.77871

3.06691

3.43500

3.70661

27

0.25586 0.68368 0.85514 1.05673 1.31370 1.70329

2.05183

2.15782

2.47266

2.77068

3.05652

3.42103

3.68959

28

0.25577 0.68335 0.85465 1.05599 1.31253 1.70113

2.04841

2.15393

2.46714

2.76326

3.04693

3.40816

3.67391

29

0.25568 0.68304 0.85419 1.05530 1.31143 1.69913

2.04523

2.15033

2.46202

2.75639

3.03805

3.39624

3.65941

30

0.25561 0.68276 0.85377 1.05466 1.31042 1.69726

2.04227

2.14697

2.45726

2.75000

3.02980

3.38518

3.64596

40

0.25504 0.68067 0.85070 1.05005 1.30308 1.68385

2.02108

2.12291

2.42326

2.70446

2.97117

3.30688

3.55097

50

0.25470 0.67943 0.84887 1.04729 1.29871 1.67591

2.00856

2.10872

2.40327

2.67779

2.93696

3.26141

3.49601

60

0.25447 0.67860 0.84765 1.04547 1.29582 1.67065

2.00030

2.09936

2.39012

2.66028

2.91455

3.23171

3.46020

70

0.25431 0.67801 0.84679 1.04417 1.29376 1.66691

1.99444

2.09273

2.38081

2.64790

2.89873

3.21079

3.43501

80

0.25419 0.67757 0.84614 1.04320 1.29222 1.66412

1.99006

2.08778

2.37387

2.63869

2.88697

3.19526

3.41634

90

0.25410 0.67723 0.84563 1.04244 1.29103 1.66196

1.98667

2.08394

2.36850

2.63157

2.87788

3.18327

3.40194

100 0.25402 0.67695 0.84523 1.04184 1.29007 1.66023

1.98397

2.08088

2.36422

2.62589

2.87065

3.17374

3.39049

500 0.25348 0.67498 0.84234 1.03751 1.28325 1.64791

1.96472

2.05912

2.33383

2.58570

2.81955

3.10661

3.31009

1000 0.25341 0.67474 0.84198 1.03697 1.28240 1.64638

1.96234

2.05643

2.33008

2.58075

2.81328

3.09840

3.30028

0.25335 0.67449 0.84162 1.03643 1.28155 1.64485

1.95996

2.05375

2.32635

2.57583

2.80703

3.09023

3.29053

∞

Explanatory Notes

• •

For a Two-Tailed Test, use the α here that corresponds to half the two-tailed α. o For example if a two-tailed confidence limit of 90% is desired (α=0.1), use a one-tailed α from this table of 0.05 In the limit ν=∞, this distribution is equivalent to a normal distribution X~N(0,1)

Chi-Squared Distibution

Table of values of χ2 in a Chi-Squared Distribution with k degrees of freedom such that p is the area between χ2 and +∞

Probability Content, p, between χ2 and +∞ k 0.995

0.99

0.975

1

3.927e5

1.570e4

2

0.0100

0.0201

0.0506

3

0.0717

0.115

4

0.207

5

0.95

0.75

0.5

0.0157

0.102

0.455

1.323

2.706

3.841

5.024

6.635

7.879

9.550

10.828

0.103

0.211

0.575

1.386

2.773

4.605

5.991

7.378

9.210

10.597

12.429

13.816

0.216

0.352

0.584

1.213

2.366

4.108

6.251

7.815

9.348

11.345

12.838

14.796

16.266

0.297

0.484

0.711

1.064

1.923

3.357

5.385

7.779

9.488

11.143

13.277

14.860

16.924

18.467

0.412

0.554

0.831

1.145

1.610

2.675

4.351

6.626

9.236

11.070

12.833

15.086

16.750

18.907

20.515

6

0.676

0.872

1.237

1.635

2.204

3.455

5.348

7.841

10.645

12.592

14.449

16.812

18.548

20.791

22.458

7

0.989

1.239

1.690

2.167

2.833

4.255

6.346

9.037

12.017

14.067

16.013

18.475

20.278

22.601

24.322

8

1.344

1.646

2.180

2.733

3.490

5.071

7.344

10.219

13.362

15.507

17.535

20.090

21.955

24.352

26.124

9

1.735

2.088

2.700

3.325

4.168

5.899

8.343

11.389

14.684

16.919

19.023

21.666

23.589

26.056

27.877

10

2.156

2.558

3.247

3.940

4.865

6.737

9.342

12.549

15.987

18.307

20.483

23.209

25.188

27.722

29.588

11

2.603

3.053

3.816

4.575

5.578

7.584

10.341

13.701

17.275

19.675

21.920

24.725

26.757

29.354

31.264

12

3.074

3.571

4.404

5.226

6.304

8.438

11.340

14.845

18.549

21.026

23.337

26.217

28.300

30.957

32.909

13

3.565

4.107

5.009

5.892

7.042

9.299

12.340

15.984

19.812

22.362

24.736

27.688

29.819

32.535

34.528

9.820e0.00393 4

0.9

0.25

0.1

0.05

0.025

0.01

0.005

0.002

0.001

14

4.075

4.660

5.629

6.571

7.790

10.165

13.339

17.117

21.064

23.685

26.119

29.141

31.319

34.091

36.123

15

4.601

5.229

6.262

7.261

8.547

11.037

14.339

18.245

22.307

24.996

27.488

30.578

32.801

35.628

37.697

16

5.142

5.812

6.908

7.962

9.312

11.912

15.338

19.369

23.542

26.296

28.845

32.000

34.267

37.146

39.252

17

5.697

6.408

7.564

8.672

10.085

12.792

16.338

20.489

24.769

27.587

30.191

33.409

35.718

38.648

40.790

18

6.265

7.015

8.231

9.390

10.865

13.675

17.338

21.605

25.989

28.869

31.526

34.805

37.156

40.136

42.312

19

6.844

7.633

8.907

10.117

11.651

14.562

18.338

22.718

27.204

30.144

32.852

36.191

38.582

41.610

43.820

20

7.434

8.260

9.591

10.851

12.443

15.452

19.337

23.828

28.412

31.410

34.170

37.566

39.997

43.072

45.315

21

8.034

8.897

10.283

11.591

13.240

16.344

20.337

24.935

29.615

32.671

35.479

38.932

41.401

44.522

46.797

22

8.643

9.542

10.982

12.338

14.041

17.240

21.337

26.039

30.813

33.924

36.781

40.289

42.796

45.962

48.268

23

9.260

10.196

11.689

13.091

14.848

18.137

22.337

27.141

32.007

35.172

38.076

41.638

44.181

47.391

49.728

24

9.886

10.856

12.401

13.848

15.659

19.037

23.337

28.241

33.196

36.415

39.364

42.980

45.559

48.812

51.179

25

10.520

11.524

13.120

14.611

16.473

19.939

24.337

29.339

34.382

37.652

40.646

44.314

46.928

50.223

52.620

26

11.160

12.198

13.844

15.379

17.292

20.843

25.336

30.435

35.563

38.885

41.923

45.642

48.290

51.627

54.052

27

11.808

12.879

14.573

16.151

18.114

21.749

26.336

31.528

36.741

40.113

43.195

46.963

49.645

53.023

55.476

28

12.461

13.565

15.308

16.928

18.939

22.657

27.336

32.620

37.916

41.337

44.461

48.278

50.993

54.411

56.892

29

13.121

14.256

16.047

17.708

19.768

23.567

28.336

33.711

39.087

42.557

45.722

49.588

52.336

55.792

58.301

30

13.787

14.953

16.791

18.493

20.599

24.478

29.336

34.800

40.256

43.773

46.979

50.892

53.672

57.167

59.703

31

14.458

15.655

17.539

19.281

21.434

25.390

30.336

35.887

41.422

44.985

48.232

52.191

55.003

58.536

61.098

32

15.134

16.362

18.291

20.072

22.271

26.304

31.336

36.973

42.585

46.194

49.480

53.486

56.328

59.899

62.487

33

15.815

17.074

19.047

20.867

23.110

27.219

32.336

38.058

43.745

47.400

50.725

54.776

57.648

61.256

63.870

34

16.501

17.789

19.806

21.664

23.952

28.136

33.336

39.141

44.903

48.602

51.966

56.061

58.964

62.608

65.247

35

17.192

18.509

20.569

22.465

24.797

29.054

34.336

40.223

46.059

49.802

53.203

57.342

60.275

63.955

66.619

36

17.887

19.233

21.336

23.269

25.643

29.973

35.336

41.304

47.212

50.998

54.437

58.619

61.581

65.296

67.985

37

18.586

19.960

22.106

24.075

26.492

30.893

36.336

42.383

48.363

52.192

55.668

59.893

62.883

66.633

69.346

38

19.289

20.691

22.878

24.884

27.343

31.815

37.335

43.462

49.513

53.384

56.896

61.162

64.181

67.966

70.703

39

19.996

21.426

23.654

25.695

28.196

32.737

38.335

44.539

50.660

54.572

58.120

62.428

65.476

69.294

72.055

40

20.707

22.164

24.433

26.509

29.051

33.660

39.335

45.616

51.805

55.758

59.342

63.691

66.766

70.618

73.402

41

21.421

22.906

25.215

27.326

29.907

34.585

40.335

46.692

52.949

56.942

60.561

64.950

68.053

71.938

74.745

42

22.138

23.650

25.999

28.144

30.765

35.510

41.335

47.766

54.090

58.124

61.777

66.206

69.336

73.254

76.084

43

22.859

24.398

26.785

28.965

31.625

36.436

42.335

48.840

55.230

59.304

62.990

67.459

70.616

74.566

77.419

44

23.584

25.148

27.575

29.787

32.487

37.363

43.335

49.913

56.369

60.481

64.201

68.710

71.893

75.874

78.750

45

24.311

25.901

28.366

30.612

33.350

38.291

44.335

50.985

57.505

61.656

65.410

69.957

73.166

77.179

80.077

46

25.041

26.657

29.160

31.439

34.215

39.220

45.335

52.056

58.641

62.830

66.617

71.201

74.437

78.481

81.400

47

25.775

27.416

29.956

32.268

35.081

40.149

46.335

53.127

59.774

64.001

67.821

72.443

75.704

79.780

82.720

48

26.511

28.177

30.755

33.098

35.949

41.079

47.335

54.196

60.907

65.171

69.023

73.683

76.969

81.075

84.037

49

27.249

28.941

31.555

33.930

36.818

42.010

48.335

55.265

62.038

66.339

70.222

74.919

78.231

82.367

85.351

50

27.991

29.707

32.357

34.764

37.689

42.942

49.335

56.334

63.167

67.505

71.420

76.154

79.490

83.657

86.661

51

28.735

30.475

33.162

35.600

38.560

43.874

50.335

57.401

64.295

68.669

72.616

77.386

80.747

84.943

87.968

52

29.481

31.246

33.968

36.437

39.433

44.808

51.335

58.468

65.422

69.832

73.810

78.616

82.001

86.227

89.272

53

30.230

32.018

34.776

37.276

40.308

45.741

52.335

59.534

66.548

70.993

75.002

79.843

83.253

87.507

90.573

54

30.981

32.793

35.586

38.116

41.183

46.676

53.335

60.600

67.673

72.153

76.192

81.069

84.502

88.786

91.872

55

31.735

33.570

36.398

38.958

42.060

47.610

54.335

61.665

68.796

73.311

77.380

82.292

85.749

90.061

93.168

56

32.490

34.350

37.212

39.801

42.937

48.546

55.335

62.729

69.919

74.468

78.567

83.513

86.994

91.335

94.461

57

33.248

35.131

38.027

40.646

43.816

49.482

56.335

63.793

71.040

75.624

79.752

84.733

88.236

92.605

95.751

58

34.008

35.913

38.844

41.492

44.696

50.419

57.335

64.857

72.160

76.778

80.936

85.950

89.477

93.874

97.039

59

34.770

36.698

39.662

42.339

45.577

51.356

58.335

65.919

73.279

77.931

82.117

87.166

90.715

95.140

98.324

60

35.534

37.485

40.482

43.188

46.459

52.294

59.335

66.981

74.397

79.082

83.298

88.379

91.952

96.404

99.607

61

36.301

38.273

41.303

44.038

47.342

53.232

60.335

68.043

75.514

80.232

84.476

89.591

93.186

97.665

100.888

62

37.068

39.063

42.126

44.889

48.226

54.171

61.335

69.104

76.630

81.381

85.654

90.802

94.419

98.925

102.166

63

37.838

39.855

42.950

45.741

49.111

55.110

62.335

70.165

77.745

82.529

86.830

92.010

95.649

100.182

103.442

64

38.610

40.649

43.776

46.595

49.996

56.050

63.335

71.225

78.860

83.675

88.004

93.217

96.878

101.437

104.716

65

39.383

41.444

44.603

47.450

50.883

56.990

64.335

72.285

79.973

84.821

89.177

94.422

98.105

102.691

105.988

66

40.158

42.240

45.431

48.305

51.770

57.931

65.335

73.344

81.085

85.965

90.349

95.626

99.330

103.942

107.258

67

40.935

43.038

46.261

49.162

52.659

58.872

66.335

74.403

82.197

87.108

91.519

96.828

100.554

105.192

108.526

68

41.713

43.838

47.092

50.020

53.548

59.814

67.335

75.461

83.308

88.250

92.689

98.028

101.776

106.440

109.791

69

42.494

44.639

47.924

50.879

54.438

60.756

68.334

76.519

84.418

89.391

93.856

99.228

102.996

107.685

111.055

70

43.275

45.442

48.758

51.739

55.329

61.698

69.334

77.577

85.527

90.531

95.023

100.425

104.215

108.929

112.317

71

44.058

46.246

49.592

52.600

56.221

62.641

70.334

78.634

86.635

91.670

96.189

101.621

105.432

110.172

113.577

72

44.843

47.051

50.428

53.462

57.113

63.585

71.334

79.690

87.743

92.808

97.353

102.816

106.648

111.412

114.835

73

45.629

47.858

51.265

54.325

58.006

64.528

72.334

80.747

88.850

93.945

98.516

104.010

107.862

112.651

116.092

74

46.417

48.666

52.103

55.189

58.900

65.472

73.334

81.803

89.956

95.081

99.678

105.202

109.074

113.889

117.346

75

47.206

49.475

52.942

56.054

59.795

66.417

74.334

82.858

91.061

96.217

100.839

106.393

110.286

115.125

118.599

76

47.997

50.286

53.782

56.920

60.690

67.362

75.334

83.913

92.166

97.351

101.999

107.583

111.495

116.359

119.850

77

48.788

51.097

54.623

57.786

61.586

68.307

76.334

84.968

93.270

98.484

103.158

108.771

112.704

117.591

121.100

78

49.582

51.910

55.466

58.654

62.483

69.252

77.334

86.022

94.374

99.617

104.316

109.958

113.911

118.823

122.348

79

50.376

52.725

56.309

59.522

63.380

70.198

78.334

87.077

95.476

100.749

105.473

111.144

115.117

120.052

123.594

80

51.172

53.540

57.153

60.391

64.278

71.145

79.334

88.130

96.578

101.879

106.629

112.329

116.321

121.280

124.839

81

51.969

54.357

57.998

61.261

65.176

72.091

80.334

89.184

97.680

103.010

107.783

113.512

117.524

122.507

126.083

82

52.767

55.174

58.845

62.132

66.076

73.038

81.334

90.237

98.780

104.139

108.937

114.695

118.726

123.733

127.324

83

53.567

55.993

59.692

63.004

66.976

73.985

82.334

91.289

99.880

105.267

110.090

115.876

119.927

124.957

128.565

84

54.368

56.813

60.540

63.876

67.876

74.933

83.334

92.342

100.980

106.395

111.242

117.057

121.126

126.179

129.804

85

55.170

57.634

61.389

64.749

68.777

75.881

84.334

93.394

102.079

107.522

112.393

118.236

122.325

127.401

131.041

86

55.973

58.456

62.239

65.623

69.679

76.829

85.334

94.446

103.177

108.648

113.544

119.414

123.522

128.621

132.277

87

56.777

59.279

63.089

66.498

70.581

77.777

86.334

95.497

104.275

109.773

114.693

120.591

124.718

129.840

133.512

88

57.582

60.103

63.941

67.373

71.484

78.726

87.334

96.548

105.372

110.898

115.841

121.767

125.913

131.057

134.745

89

58.389

60.928

64.793

68.249

72.387

79.675

88.334

97.599

106.469

112.022

116.989

122.942

127.106

132.273

135.978

90

59.196

61.754

65.647

69.126

73.291

80.625

89.334

98.650

107.565

113.145

118.136

124.116

128.299

133.489

137.208

91

60.005

62.581

66.501

70.003

74.196

81.574

90.334

99.700

108.661

114.268

119.282

125.289

129.491

134.702

138.438

92

60.815

63.409

67.356

70.882

75.100

82.524

91.334

100.750

109.756

115.390

120.427

126.462

130.681

135.915

139.666

93

61.625

64.238

68.211

71.760

76.006

83.474

92.334

101.800

110.850

116.511

121.571

127.633

131.871

137.127

140.893

94

62.437

65.068

69.068

72.640

76.912

84.425

93.334

102.850

111.944

117.632

122.715

128.803

133.059

138.337

142.119

95

63.250

65.898

69.925

73.520

77.818

85.376

94.334

103.899

113.038

118.752

123.858

129.973

134.247

139.546

143.344

96

64.063

66.730

70.783

74.401

78.725

86.327

95.334

104.948

114.131

119.871

125.000

131.141

135.433

140.755

144.567

97

64.878

67.562

71.642

75.282

79.633

87.278

96.334

105.997

115.223

120.990

126.141

132.309

136.619

141.962

145.789

98

65.694

68.396

72.501

76.164

80.541

88.229

97.334

107.045

116.315

122.108

127.282

133.476

137.803

143.168

147.010

99

66.510

69.230

73.361

77.046

81.449

89.181

98.334

108.093

117.407

123.225

128.422

134.642

138.987

144.373

148.230

100

67.328

70.065

74.222

77.929

82.358

90.133

99.334

109.141

118.498

124.342

129.561

135.807

140.169

145.577

149.449

101

68.146

70.901

75.083

78.813

83.267

91.085 100.334

110.189

119.589

125.458

130.700

136.971

141.351

146.780

150.667

102

68.965

71.737

75.946

79.697

84.177

92.038 101.334

111.236

120.679

126.574

131.838

138.134

142.532

147.982

151.884

103

69.785

72.575

76.809

80.582

85.088

92.991 102.334

112.284

121.769

127.689

132.975

139.297

143.712

149.183

153.099

104

70.606

73.413

77.672

81.468

85.998

93.944 103.334

113.331

122.858

128.804

134.111

140.459

144.891

150.383

154.314

105

71.428

74.252

78.536

82.354

86.909

94.897 104.334

114.378

123.947

129.918

135.247

141.620

146.070

151.582

155.528

106

72.251

75.092

79.401

83.240

87.821

95.850 105.334

115.424

125.035

131.031

136.382

142.780

147.247

152.780

156.740

107

73.075

75.932

80.267

84.127

88.733

96.804 106.334

116.471

126.123

132.144

137.517

143.940

148.424

153.977

157.952

108

73.899

76.774

81.133

85.015

89.645

97.758 107.334

117.517

127.211

133.257

138.651

145.099

149.599

155.173

159.162

109

74.724

77.616

82.000

85.903

90.558

98.712 108.334

118.563

128.298

134.369

139.784

146.257

150.774

156.369

160.372

110

75.550

78.458

82.867

86.792

91.471

99.666 109.334

119.608

129.385

135.480

140.917

147.414

151.948

157.563

161.581

111

76.377

79.302

83.735

87.681

92.385 100.620 110.334

120.654

130.472

136.591

142.049

148.571

153.122

158.757

162.788

112

77.204

80.146

84.604

88.570

93.299 101.575 111.334

121.699

131.558

137.701

143.180

149.727

154.294

159.950

163.995

113

78.033

80.991

85.473

89.461

94.213 102.530 112.334

122.744

132.643

138.811

144.311

150.882

155.466

161.141

165.201

114

78.862

81.836

86.342

90.351

95.128 103.485 113.334

123.789

133.729

139.921

145.441

152.037

156.637

162.332

166.406

115

79.692

82.682

87.213

91.242

96.043 104.440 114.334

124.834

134.813

141.030

146.571

153.191

157.808

163.523

167.610

116

80.522

83.529

88.084

92.134

96.958 105.396 115.334

125.878

135.898

142.138

147.700

154.344

158.977

164.712

168.813

117

81.353

84.377

88.955

93.026

97.874 106.352 116.334

126.923

136.982

143.246

148.829

155.496

160.146

165.900

170.016

118

82.185

85.225

89.827

93.918

98.790 107.307 117.334

127.967

138.066

144.354

149.957

156.648

161.314

167.088

171.217

119

83.018

86.074

90.700

94.811

99.707 108.263 118.334

129.011

139.149

145.461

151.084

157.800

162.481

168.275

172.418

120

83.852

86.923

91.573

95.705 100.624 109.220 119.334

130.055

140.233

146.567

152.211

158.950

163.648

169.461

173.617

121

84.686

87.773

92.446

96.598 101.541 110.176 120.334

131.098

141.315

147.674

153.338

160.100

164.814

170.647

174.816

122

85.520

88.624

93.320

97.493 102.458 111.133 121.334

132.142

142.398

148.779

154.464

161.250

165.980

171.831

176.014

123

86.356

89.475

94.195

98.387 103.376 112.089 122.334

133.185

143.480

149.885

155.589

162.398

167.144

173.015

177.212

124

87.192

90.327

95.070

99.283 104.295 113.046 123.334

134.228

144.562

150.989

156.714

163.546

168.308

174.198

178.408

125

88.029

91.180

95.946 100.178 105.213 114.004 124.334

135.271

145.643

152.094

157.839

164.694

169.471

175.380

179.604

126

88.866

92.033

96.822 101.074 106.132 114.961 125.334

136.313

146.724

153.198

158.962

165.841

170.634

176.562

180.799

127

89.704

92.887

97.698 101.971 107.051 115.918 126.334

137.356

147.805

154.302

160.086

166.987

171.796

177.743

181.993

128

90.543

93.741

98.576 102.867 107.971 116.876 127.334

138.398

148.885

155.405

161.209

168.133

172.957

178.923

183.186

129

91.382

94.596

99.453 103.765 108.891 117.834 128.334

139.440

149.965

156.508

162.331

169.278

174.118

180.103

184.379

130

92.222

95.451 100.331 104.662 109.811 118.792 129.334

140.482

151.045

157.610

163.453

170.423

175.278

181.282

185.571

131

93.063

96.307 101.210 105.560 110.732 119.750 130.334

141.524

152.125

158.712

164.575

171.567

176.438

182.460

186.762

132

93.904

97.163 102.089 106.459 111.652 120.708 131.334

142.566

153.204

159.814

165.696

172.711

177.597

183.637

187.953

133

94.746

98.020 102.968 107.357 112.573 121.667 132.334

143.608

154.283

160.915

166.816

173.854

178.755

184.814

189.142

134

95.588

98.878 103.848 108.257 113.495 122.625 133.334

144.649

155.361

162.016

167.936

174.996

179.913

185.990

190.331

135

96.431

99.736 104.729 109.156 114.417 123.584 134.334

145.690

156.440

163.116

169.056

176.138

181.070

187.165

191.520

136

97.275 100.595 105.609 110.056 115.338 124.543 135.334

146.731

157.518

164.216

170.175

177.280

182.226

188.340

192.707

137

98.119 101.454 106.491 110.956 116.261 125.502 136.334

147.772

158.595

165.316

171.294

178.421

183.382

189.514

193.894

138

98.964 102.314 107.372 111.857 117.183 126.461 137.334

148.813

159.673

166.415

172.412

179.561

184.538

190.688

195.080

139

99.809 103.174 108.254 112.758 118.106 127.421 138.334

149.854

160.750

167.514

173.530

180.701

185.693

191.861

196.266

140

100.655 104.034 109.137 113.659 119.029 128.380 139.334

150.894

161.827

168.613

174.648

181.840

186.847

193.033

197.451

141

101.501 104.896 110.020 114.561 119.953 129.340 140.334

151.934

162.904

169.711

175.765

182.979

188.001

194.205

198.635

142

102.348 105.757 110.903 115.463 120.876 130.299 141.334

152.975

163.980

170.809

176.882

184.118

189.154

195.376

199.819

143

103.196 106.619 111.787 116.366 121.800 131.259 142.334

154.015

165.056

171.907

177.998

185.256

190.306

196.546

201.002

144

104.044 107.482 112.671 117.268 122.724 132.219 143.334

155.055

166.132

173.004

179.114

186.393

191.458

197.716

202.184

145

104.892 108.345 113.556 118.171 123.649 133.180 144.334

156.094

167.207

174.101

180.229

187.530

192.610

198.885

203.366

146

105.741 109.209 114.441 119.075 124.574 134.140 145.334

157.134

168.283

175.198

181.344

188.666

193.761

200.054

204.547

147

106.591 110.073 115.326 119.979 125.499 135.101 146.334

158.174

169.358

176.294

182.459

189.802

194.912

201.222

205.727

148

107.441 110.937 116.212 120.883 126.424 136.061 147.334

159.213

170.432

177.390

183.573

190.938

196.062

202.390

206.907

149

108.291 111.802 117.098 121.787 127.349 137.022 148.334

160.252

171.507

178.485

184.687

192.073

197.211

203.557

208.086

150

109.142 112.668 117.985 122.692 128.275 137.983 149.334

161.291

172.581

179.581

185.800

193.208

198.360

204.723

209.265

151

109.994 113.533 118.871 123.597 129.201 138.944 150.334

162.330

173.655

180.676

186.914

194.342

199.509

205.889

210.443

152

110.846 114.400 119.759 124.502 130.127 139.905 151.334

163.369

174.729

181.770

188.026

195.476

200.657

207.054

211.620

153

111.698 115.266 120.646 125.408 131.054 140.866 152.334

164.408

175.803

182.865

189.139

196.609

201.804

208.219

212.797

154

112.551 116.134 121.534 126.314 131.980 141.828 153.334

165.446

176.876

183.959

190.251

197.742

202.951

209.383

213.973

155

113.405 117.001 122.423 127.220 132.907 142.789 154.334

166.485

177.949

185.052

191.362

198.874

204.098

210.547

215.149

156

114.259 117.869 123.312 128.127 133.835 143.751 155.334

167.523

179.022

186.146

192.474

200.006

205.244

211.710

216.324

157

115.113 118.738 124.201 129.034 134.762 144.713 156.334

168.561

180.094

187.239

193.584

201.138

206.390

212.873

217.499

158

115.968 119.607 125.090 129.941 135.690 145.675 157.334

169.599

181.167

188.332

194.695

202.269

207.535

214.035

218.673

159

116.823 120.476 125.980 130.848 136.618 146.637 158.334

170.637

182.239

189.424

195.805

203.400

208.680

215.197

219.846

160

117.679 121.346 126.870 131.756 137.546 147.599 159.334

171.675

183.311

190.516

196.915

204.530

209.824

216.358

221.019

161

118.536 122.216 127.761 132.664 138.474 148.561 160.334

172.713

184.382

191.608

198.025

205.660

210.968

217.518

222.191

162

119.392 123.086 128.651 133.572 139.403 149.523 161.334

173.751

185.454

192.700

199.134

206.790

212.111

218.678

223.363

163

120.249 123.957 129.543 134.481 140.331 150.486 162.334

174.788

186.525

193.791

200.243

207.919

213.254

219.838

224.535

164

121.107 124.828 130.434 135.390 141.260 151.449 163.334

175.825

187.596

194.883

201.351

209.047

214.396

220.997

225.705

165

121.965 125.700 131.326 136.299 142.190 152.411 164.334

176.863

188.667

195.973

202.459

210.176

215.539

222.156

226.876

166

122.823 126.572 132.218 137.209 143.119 153.374 165.334

177.900

189.737

197.064

203.567

211.304

216.680

223.314

228.045

167

123.682 127.445 133.111 138.118 144.049 154.337 166.334

178.937

190.808

198.154

204.675

212.431

217.821

224.472

229.215

168

124.541 128.318 134.003 139.028 144.979 155.300 167.334

179.974

191.878

199.244

205.782

213.558

218.962

225.629

230.383

169

125.401 129.191 134.897 139.939 145.909 156.263 168.334

181.011

192.948

200.334

206.889

214.685

220.102

226.786

231.552

170

126.261 130.064 135.790 140.849 146.839 157.227 169.334

182.047

194.017

201.423

207.995

215.812

221.242

227.942

232.719

171

127.122 130.938 136.684 141.760 147.769 158.190 170.334

183.084

195.087

202.513

209.102

216.938

222.382

229.098

233.887

172

127.983 131.813 137.578 142.671 148.700 159.154 171.334

184.120

196.156

203.602

210.208

218.063

223.521

230.253

235.053

173

128.844 132.687 138.472 143.582 149.631 160.117 172.334

185.157

197.225

204.690

211.313

219.189

224.660

231.408

236.220

174

129.706 133.563 139.367 144.494 150.562 161.081 173.334

186.193

198.294

205.779

212.419

220.314

225.798

232.563

237.385

175

130.568 134.438 140.262 145.406 151.493 162.045 174.334

187.229

199.363

206.867

213.524

221.438

226.936

233.717

238.551

176

131.430 135.314 141.157 146.318 152.425 163.009 175.334

188.265

200.432

207.955

214.628

222.563

228.074

234.870

239.716

177

132.293 136.190 142.053 147.230 153.356 163.973 176.334

189.301

201.500

209.042

215.733

223.687

229.211

236.023

240.880

178

133.157 137.066 142.949 148.143 154.288 164.937 177.334

190.337

202.568

210.130

216.837

224.810

230.347

237.176

242.044

179

134.020 137.943 143.845 149.056 155.220 165.901 178.334

191.373

203.636

211.217

217.941

225.933

231.484

238.328

243.207

180

134.884 138.820 144.741 149.969 156.153 166.865 179.334

192.409

204.704

212.304

219.044

227.056

232.620

239.480

244.370

181

135.749 139.698 145.638 150.882 157.085 167.830 180.334

193.444

205.771

213.391

220.148

228.179

233.755

240.632

245.533

182

136.614 140.576 146.535 151.796 158.018 168.794 181.334

194.480

206.839

214.477

221.251

229.301

234.891

241.783

246.695

183

137.479 141.454 147.432 152.709 158.951 169.759 182.334

195.515

207.906

215.563

222.353

230.423

236.026

242.933

247.857

184

138.344 142.332 148.330 153.623 159.883 170.724 183.334

196.550

208.973

216.649

223.456

231.544

237.160

244.084

249.018

185

139.210 143.211 149.228 154.538 160.817 171.688 184.334

197.586

210.040

217.735

224.558

232.665

238.294

245.234

250.179

186

140.077 144.090 150.126 155.452 161.750 172.653 185.334

198.621

211.106

218.820

225.660

233.786

239.428

246.383

251.339

187

140.943 144.970 151.024 156.367 162.684 173.618 186.334

199.656

212.173

219.906

226.761

234.907

240.561

247.532

252.499

188

141.810 145.850 151.923 157.282 163.617 174.583 187.334

200.690

213.239

220.991

227.863

236.027

241.694

248.681

253.659

189

142.678 146.730 152.822 158.197 164.551 175.549 188.334

201.725

214.305

222.076

228.964

237.147

242.827

249.829

254.818

190

143.545 147.610 153.721 159.113 165.485 176.514 189.334

202.760

215.371

223.160

230.064

238.266

243.959

250.977

255.976

191

144.413 148.491 154.621 160.028 166.419 177.479 190.334

203.795

216.437

224.245

231.165

239.386

245.091

252.124

257.135

192

145.282 149.372 155.521 160.944 167.354 178.445 191.334

204.829

217.502

225.329

232.265

240.505

246.223

253.271

258.292

193

146.150 150.254 156.421 161.860 168.288 179.410 192.334

205.864

218.568

226.413

233.365

241.623

247.354

254.418

259.450

194

147.020 151.135 157.321 162.776 169.223 180.376 193.334

206.898

219.633

227.496

234.465

242.742

248.485

255.564

260.607

195

147.889 152.017 158.221 163.693 170.158 181.342 194.334

207.932

220.698

228.580

235.564

243.860

249.616

256.710

261.763

196

148.759 152.900 159.122 164.610 171.093 182.308 195.334

208.966

221.763

229.663

236.664

244.977

250.746

257.855

262.920

197

149.629 153.782 160.023 165.527 172.029 183.273 196.334

210.000

222.828

230.746

237.763

246.095

251.876

259.001

264.075

198

150.499 154.665 160.925 166.444 172.964 184.239 197.334

211.034

223.892

231.829

238.861

247.212

253.006

260.145

265.231

199

151.370 155.548 161.826 167.361 173.900 185.205 198.334

212.068

224.957

232.912

239.960

248.329

254.135

261.290

266.386

200

152.241 156.432 162.728 168.279 174.835 186.172 199.334

213.102

226.021

233.994

241.058

249.445

255.264

262.434

267.541

201

153.112 157.316 163.630 169.196 175.771 187.138 200.334

214.136

227.085

235.077

242.156

250.561

256.393

263.578

268.695

202

153.984 158.200 164.532 170.114 176.707 188.104 201.334

215.170

228.149

236.159

243.254

251.677

257.521

264.721

269.849

203

154.856 159.084 165.435 171.032 177.643 189.071 202.334

216.203

229.213

237.240

244.351

252.793

258.649

265.864

271.002

204

155.728 159.969 166.338 171.951 178.580 190.037 203.334

217.237

230.276

238.322

245.448

253.908

259.777

267.007

272.155

205

156.601 160.854 167.241 172.869 179.516 191.004 204.334

218.270

231.340

239.403

246.545

255.023

260.904

268.149

273.308

206

157.474 161.739 168.144 173.788 180.453 191.970 205.334

219.303

232.403

240.485

247.642

256.138

262.031

269.291

274.460

207

158.347 162.624 169.047 174.707 181.390 192.937 206.334

220.337

233.466

241.566

248.739

257.253

263.158

270.432

275.612

208

159.221 163.510 169.951 175.626 182.327 193.904 207.334

221.370

234.529

242.647

249.835

258.367

264.285

271.574

276.764

209

160.095 164.396 170.855 176.546 183.264 194.871 208.334

222.403

235.592

243.727

250.931

259.481

265.411

272.715

277.915

210

160.969 165.283 171.759 177.465 184.201 195.838 209.334

223.436

236.655

244.808

252.027

260.595

266.537

273.855

279.066

211

161.843 166.169 172.664 178.385 185.139 196.805 210.334

224.469

237.717

245.888

253.122

261.708

267.662

274.995

280.217

212

162.718 167.056 173.568 179.305 186.076 197.772 211.334

225.502

238.780

246.968

254.218

262.821

268.788

276.135

281.367

213

163.593 167.943 174.473 180.225 187.014 198.739 212.334

226.534

239.842

248.048

255.313

263.934

269.912

277.275

282.517

214

164.469 168.831 175.378 181.145 187.952 199.707 213.334

227.567

240.904

249.128

256.408

265.047

271.037

278.414

283.666

215

165.344 169.718 176.283 182.066 188.890 200.674 214.334

228.600

241.966

250.207

257.503

266.159

272.162

279.553

284.815

216

166.220 170.606 177.189 182.987 189.828 201.642 215.334

229.632

243.028

251.286

258.597

267.271

273.286

280.692

285.964

217

167.096 171.494 178.095 183.907 190.767 202.609 216.334

230.665

244.090

252.365

259.691

268.383

274.409

281.830

287.112

218

167.973 172.383 179.001 184.828 191.705 203.577 217.334

231.697

245.151

253.444

260.785

269.495

275.533

282.968

288.261

219

168.850 173.271 179.907 185.750 192.644 204.544 218.334

232.729

246.213

254.523

261.879

270.606

276.656

284.106

289.408

220

169.727 174.160 180.813 186.671 193.582 205.512 219.334

233.762

247.274

255.602

262.973

271.717

277.779

285.243

290.556

221

170.604 175.050 181.720 187.593 194.521 206.480 220.334

234.794

248.335

256.680

264.066

272.828

278.902

286.380

291.703

222

171.482 175.939 182.627 188.514 195.460 207.448 221.334

235.826

249.396

257.758

265.159

273.939

280.024

287.517

292.850

223

172.360 176.829 183.534 189.436 196.400 208.416 222.334

236.858

250.457

258.837

266.252

275.049

281.146

288.653

293.996

224

173.238 177.719 184.441 190.359 197.339 209.384 223.334

237.890

251.517

259.914

267.345

276.159

282.268

289.789

295.142

225

174.116 178.609 185.348 191.281 198.278 210.352 224.334

238.922

252.578

260.992

268.438

277.269

283.390

290.925

296.288

226

174.995 179.499 186.256 192.203 199.218 211.320 225.334

239.954

253.638

262.070

269.530

278.379

284.511

292.061

297.433

227

175.874 180.390 187.164 193.126 200.158 212.288 226.334

240.985

254.699

263.147

270.622

279.488

285.632

293.196

298.579

228

176.753 181.281 188.072 194.049 201.097 213.257 227.334

242.017

255.759

264.224

271.714

280.597

286.753

294.331

299.723

229

177.633 182.172 188.980 194.972 202.037 214.225 228.334

243.049

256.819

265.301

272.806

281.706

287.874

295.465

300.868

230

178.512 183.063 189.889 195.895 202.978 215.194 229.334

244.080

257.879

266.378

273.898

282.814

288.994

296.600

302.012

231

179.392 183.955 190.797 196.818 203.918 216.162 230.334

245.112

258.939

267.455

274.989

283.923

290.114

297.734

303.156

232

180.273 184.847 191.706 197.742 204.858 217.131 231.334

246.143

259.998

268.531

276.080

285.031

291.234

298.867

304.299

233

181.153 185.739 192.615 198.665 205.799 218.099 232.334

247.174

261.058

269.608

277.171

286.139

292.353

300.001

305.443

234

182.034 186.631 193.524 199.589 206.739 219.068 233.334

248.206

262.117

270.684

278.262

287.247

293.472

301.134

306.586

235

182.915 187.524 194.434 200.513 207.680 220.037 234.334

249.237

263.176

271.760

279.352

288.354

294.591

302.267

307.728

236

183.796 188.417 195.343 201.437 208.621 221.006 235.334

250.268

264.235

272.836

280.443

289.461

295.710

303.400

308.871

237

184.678 189.310 196.253 202.362 209.562 221.975 236.334

251.299

265.294

273.911

281.533

290.568

296.828

304.532

310.013

238

185.560 190.203 197.163 203.286 210.503 222.944 237.334

252.330

266.353

274.987

282.623

291.675

297.947

305.664

311.154

239

186.442 191.096 198.073 204.211 211.444 223.913 238.334

253.361

267.412

276.062

283.713

292.782

299.065

306.796

312.296

240

187.324 191.990 198.984 205.135 212.386 224.882 239.334

254.392

268.471

277.138

284.802

293.888

300.182

307.927

313.437

241

188.207 192.884 199.894 206.060 213.327 225.851 240.334

255.423

269.529

278.213

285.892

294.994

301.300

309.058

314.578

242

189.090 193.778 200.805 206.985 214.269 226.820 241.334

256.453

270.588

279.288

286.981

296.100

302.417

310.189

315.718

243

189.973 194.672 201.716 207.911 215.210 227.790 242.334

257.484

271.646

280.362

288.070

297.206

303.534

311.320

316.859

244

190.856 195.567 202.627 208.836 216.152 228.759 243.334

258.515

272.704

281.437

289.159

298.311

304.651

312.450

317.999

245

191.739 196.462 203.539 209.762 217.094 229.729 244.334

259.545

273.762

282.511

290.248

299.417

305.767

313.580

319.138

246

192.623 197.357 204.450 210.687 218.036 230.698 245.334

260.576

274.820

283.586

291.336

300.522

306.883

314.710

320.278

247

193.507 198.252 205.362 211.613 218.979 231.668 246.334

261.606

275.878

284.660

292.425

301.626

307.999

315.840

321.417

248

194.391 199.147 206.274 212.539 219.921 232.637 247.334

262.636

276.935

285.734

293.513

302.731

309.115

316.969

322.556

249

195.276 200.043 207.186 213.465 220.863 233.607 248.334

263.667

277.993

286.808

294.601

303.835

310.231

318.098

323.694

250

196.161 200.939 208.098 214.392 221.806 234.577 249.334

264.697

279.050

287.882

295.689

304.940

311.346

319.227

324.832

300

240.663 245.972 253.912 260.878 269.068 283.135 299.334

316.138

331.789

341.395

349.874

359.906

366.844

375.369

381.425

350

285.608 291.406 300.064 307.648 316.550 331.810 349.334

367.464

384.306

394.626

403.723

414.474

421.900

431.017

437.488

400

330.903 337.155 346.482 354.641 364.207 380.577 399.334

418.697

436.649

447.632

457.305

468.724

476.606

486.274

493.132

450

376.483 383.163 393.118 401.817 412.007 429.418 449.334

469.855

488.849

500.456

510.670

522.717

531.026

541.212

548.432

500

422.303 429.388 439.936 449.147 459.926 478.323 499.333

520.950

540.930

553.127

563.852

576.493

585.207

595.882

603.446

550

468.328 475.796 486.910 496.607 507.947 527.281 549.333

571.992

592.909

605.667

616.878

630.084

639.183

650.324

658.215

600

514.529 522.365 534.019 544.180 556.056 576.286 599.333

622.988

644.800

658.094

669.769

683.516

692.982

704.568

712.771

650

560.885 569.074 581.245 591.853 604.242 625.331 649.333

673.942

696.614

710.421

722.542

736.807

746.625

758.639

767.141

700

607.380 615.907 628.577 639.613 652.497 674.413 699.333

724.861

748.359

762.661

775.211

789.974

800.131

812.556

821.347

750

653.997 662.852 676.003 687.452 700.814 723.526 749.333

775.747

800.043

814.822

827.785

843.029

853.514

866.336

875.404

800

700.725 709.897 723.513 735.362 749.185 772.669 799.333

826.604

851.671

866.911

880.275

895.984

906.786

919.991

929.329

850

747.554 757.033 771.099 783.337 797.607 821.839 849.333

877.435

903.249

918.937

932.689

948.848

959.957

973.534

983.133

900

794.475 804.252 818.756 831.370 846.075 871.032 899.333

928.241

954.782

970.904

985.032 1001.630 1013.036 1026.974 1036.826

950

841.480 851.547 866.477 879.457 894.584 920.248 949.333

979.026 1006.272 1022.816 1037.311 1054.334 1066.031 1080.320 1090.418

1000 888.564 898.912 914.257 927.594 943.133 969.484 999.333 1029.790 1057.724 1074.679 1089.531 1106.969 1118.948 1133.579 1143.917