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378

 

 

 

 

APPENDIX A: THE IMPULSE FUNCTION

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure A.1. Finite pulse of unit area.

The impulse function can also be written as the derivative of the unit step function: dðtÞ ¼ ddt uðtÞ ðA:1-5Þ

The impulse function can be obtained by limiting operations on a number of functions whose integral has the value 1. Some examples are given below.

 

 

8 a!1½1

a t

ð

Þ&

 

 

 

 

 

 

 

>

lim

 

ae

atu t

 

 

 

 

 

 

 

 

 

 

!1

2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

>

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

>

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

>

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

> alim

 

 

e

j j

 

 

 

 

 

 

 

 

ð

Þ ¼

>

 

 

 

 

 

 

 

 

 

e

 

 

 

ð

 

Þ

> lim

 

 

 

 

 

 

 

 

 

 

d t

 

>

 

 

 

 

 

1

 

 

 

 

 

2

=2a

2

 

A:1-6

 

 

 

>

 

 

 

 

 

 

 

 

 

 

t

 

 

 

 

 

 

>

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ð

 

 

Þ

 

 

 

 

 

 

 

 

 

 

> a

 

0þ p2pa

 

 

 

 

 

 

 

 

 

 

< a

 

 

 

 

 

 

 

 

 

 

 

>

 

!1

 

 

pt

 

 

 

 

 

 

 

 

 

 

 

 

>

 

sin at

 

 

 

 

 

 

 

 

 

 

 

>

 

 

 

 

 

 

 

 

 

 

 

 

 

 

>

 

 

 

 

 

 

 

 

 

 

 

 

 

 

>

 

 

 

 

 

 

 

 

 

 

 

 

 

 

> !

 

 

 

 

 

 

 

 

 

 

 

 

:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

> lim

 

 

 

 

 

 

 

 

 

 

 

 

 

 

It is observed that these functions in the limit approach 0 at t 0 and 1 at t ¼ 0. In the last case, the limit at t 0 does not approach 0, but the function oscillates so fast that Eq. (A.1-6) remains valid.

Another way to construct the impulse function is by using the triangular pulse shown in Figure A.2. Its area is 1. As a shrinks towards 0, the area of 1 remains

(a)

(b)

Figure A.2. (a) A triangular pulse, and (b) its derivative.

APPENDIX A: THE IMPULSE FUNCTION

379

constant, and the base length approaches 0 as the height grows towards infinity. Thus,

dðtÞ ¼ a!0 a

a

;

;

 

lim

1

tri

2t

1

 

1

 

 

 

The derivative of the impulse function can be defined as

d0ðtÞ ¼ dt a!0 a

a

;

;

 

 

d

lim

1

tri

 

2t

1

 

1

 

 

 

 

 

 

 

lim

1

 

d

 

 

2t

; 1; 1

 

 

 

 

¼ a!0 a dt tri a

ðA:1-7Þ

ðA:1-8Þ

The derivative of the triangular pulse is also shown in Figure A.2. d0ðtÞ is referred to as the doublet.

The derivatives of the impulse function can be defined with respect to the following integral:

ðt2

f ðtÞdkðt t0Þdt ¼ ð 1Þkf kðt0Þ

ðA:1-9Þ

t1

where t1 < t0 < t2, dkðtÞ and f kðtÞ denote the kth derivative of dðtÞ and f ðtÞ, respectively.

Some useful properties of the impulse function are the following:

Property 1. Time scaling

1

 

 

t0

 

dðat t0Þ ¼

 

d

t

 

jaj

a

Property 2. Multiplication by a function

f ðtÞdðt t0Þ ¼ f ðt0Þdðt t0Þ; f ðtÞ continuous at t ¼ t0

Property 3.

d½ f ðtÞ&

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ð

d½ f ðtÞ&dt ¼

 

 

 

 

 

1

 

; f 0

ðtkÞ 6¼ 0

 

 

 

 

k

 

 

 

f 0

tk

 

 

 

 

 

1

 

Xj

 

ð

Þj

 

 

ð

Þ ¼

d t

 

 

ð

 

 

Þ ¼

 

 

 

where f 0

t

 

 

d f ðtÞ

, and tk

occurs when f

 

tk

 

 

0.

 

ðA:1-10Þ

ðA:1-11Þ

ðA:1-12Þ

380 APPENDIX A: THE IMPULSE FUNCTION

Property 4. Sum of shifted impulse functions

 

 

 

 

X

 

 

 

X

 

 

k

 

X

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

jTj k

1

d t

¼ k

 

1

 

e j2pkt=T

 

 

 

 

 

 

 

 

1

cosð2pkt=TÞ

 

¼ 1

T

¼ 1

¼ 1 þ 2 k

¼

1

ðA:1-13Þ

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

EXAMPLE A.1 Evaluate the integral

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

 

 

 

t

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ð

 

d

3

1

 

cosð10tÞdt:

 

 

Solution:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

d

 

 

t

1

¼ d

1

ðt

 

3Þ

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3

 

3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

¼ 3dðt 3Þ

 

 

 

 

Hence,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

 

 

 

 

 

 

 

 

 

 

 

ð d

t

1 cosð10tÞdt ¼ 3

ð

 

dðt 3Þ cosð10tÞdt

 

 

 

3

 

 

 

 

 

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

¼ 3 cosð30Þ

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

¼

 

3

p

3

 

 

 

2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

=

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

EXAMPLE A.2 Evaluate

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

lim

 

1

 

e t2=32a2

 

 

 

 

in terms of the impulse function.!1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a

 

 

p2pa

 

 

 

 

 

 

 

 

 

 

 

 

 

Solution:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

p1

 

e t2

=32a2 ¼ p

1

 

 

e ðt=4Þ2=2a2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2pa

 

 

 

 

 

 

 

 

 

 

2pa

 

 

 

 

 

 

 

 

Since

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

lim

 

 

1

 

e t2=2a2

 

 

 

 

 

t

 

 

 

 

 

 

 

 

 

 

!1

 

 

 

 

 

 

 

 

¼ dð Þ

 

 

 

 

 

 

 

 

a

 

 

 

 

p

2pa

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

it follows that

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

 

 

 

 

t2

=32a2

 

 

 

 

 

 

 

t

 

 

 

 

 

 

 

 

 

a!1 p2pa e

 

 

 

 

¼ d

 

 

 

¼ 4dðtÞ

 

 

 

 

 

 

 

 

 

 

 

4

 

 

 

 

 

 

 

lim

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

APPENDIX A: THE IMPULSE FUNCTION

381

EXAMPLE A.3 Evaluate

 

 

 

 

 

 

 

a

 

 

 

 

 

 

 

 

 

 

 

 

 

lim

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a!1 p ð0

cosðtxÞdx

 

 

in terms of the impulse function.

 

 

 

 

 

 

 

 

 

 

 

 

Solution: It is known that

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

 

 

 

 

 

 

 

sin at

 

 

 

 

 

ð0

cosðtxÞdx ¼

 

 

ð Þ

 

 

a > 0

 

 

p

 

 

pt

 

 

Since

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

lim

sinðatÞ

 

 

t

Þ

;

 

 

 

 

a!1

 

 

pt

 

 

 

¼ dð

 

 

it follows that

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a

 

 

 

 

 

 

 

 

 

t

 

lim

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

dð

Þ ¼ a!1 p ð0

 

cosðtxÞdx

ðA:1-14Þ

This result is often written as

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a

 

 

 

 

 

 

 

 

 

lim

 

1

 

e jtxdx

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

dðtÞ ¼ a!1 2p ð

 

 

 

a

Appendix B

Linear Vector Spaces

B.1 INTRODUCTION

In a large variety of applications, Fourier-related series and discrete transforms are used for the representation of signals by a set of basis functions. More generally, this is a subject of linear vector spaces in which basis functions are called vectors. Representation of signals by discrete Fourier-related transforms can be considered to be the same subject as representing signals by basis vectors in infinite and finitedimensional Hilbert spaces, respectively.

The most obvious example of a vector space is the familiar 3-D space R3. In general, the concept of a vector space is much more encompassing. Associated with every vector space is a set of scalars which belong to a field F. The elements of F can be added and multiplied to generate new elements in F. Some examples of fields are the field R of real numbers, the field C of complex numbers, the field of binary numbers given by [0,1], and the field of rational polynomials.

A vector space S over a field F is a set of elements called vectors, with which two operations called addition (þ) and multiplication ( ) are carried out. Let u, v, w be elements (vectors) of S, and a; b be scalars, which are the elements of the field F. S satisfies the following axioms:

1.u þ v ¼ v þ u 2 S

2.a ðu þ vÞ ¼ a u þ a v 2 S

3.ða þ bÞ u ¼ a u þ b u

4.ðu þ vÞ þ w ¼ u þ ðv þ wÞ

5.ða bÞ u ¼ a ðb uÞ

6.There exists a null vector denoted as h such that h þ u ¼ u. h is often written simply as 0.

7.For scalars 0 and 1, 0 u ¼ 0 and 1 u ¼ u.

8.There exists an additive inverse element for each u, denoted by u, such that

u þ ð uÞ ¼ h

u also equals 1 u.

Diffraction, Fourier Optics and Imaging, by Okan K. Ersoy

Copyright # 2007 John Wiley & Sons, Inc.

382

PROPERTIES OF VECTOR SPACES

383

Some important vector spaces are the following:

Fn is the space of column vectors with n components from a field F. Two special cases are R3 whose components are from R, the vector space of real numbers, and Cn whose components are from C, the vector space of complex numbers.

Fm n is the space of all m n matrices whose components belong to the field F. Some other vector spaces are discussed in the examples below.

EXAMPLE B.1 Let V and W be vector spaces over the same field F. The Cartesian product V W consists of the set of ordered pairs fv; wg with v 2 V and w 2 W. V W is a vector space. Vector addition and scalar multiplication on V W are defined as

fv; wg þ fp; qg ¼ fv þ p; w þ qg

afv; wg ¼ fav; awg

h ¼ fhv; hwg

where a 2 F, v and p 2 V, w and q 2 W, hv and hw are the null elements of V and W, respectively.

The Cartesian product vector space can be extended to include any number of vector spaces.

EXAMPLE B.2 The set of all complex-valued continuous functions of the variable t over the interval [a,b] of the real line forms a vector space, denoted by C[a,b]. Let u and v be vectors in this space, and a 2 F. The vector addition and the scalar multiplication are given by

ðu þ vÞðtÞ ¼ uðtÞ þ vðtÞ ðauÞðtÞ ¼ auðtÞ

The null vector h is the function identically equal to 0 over [a,b].

B.2 PROPERTIES OF VECTOR SPACES

In this section, we discuss properties of vector spaces which are valid in general, without being specific to a particular vector space.

Subspace

A nonempty vector space L is a subspace of a space S if the elements of L are also the elements of S, and S has possibly more number of elements.

384

APPENDIX B: LINEAR VECTOR SPACES

Let M and N be subspaces of S. They satisfy the following two properties:

1. The intersection M \ N is a subspace of S.

2. The direct sum M N is a subspace of S. The direct sum is described below.

Direct Sum

A set S is the direct sum of the subsets S1 and S2 if, for each s 2 S, there exists unique s1 2 S1 and s2 2 S2 such that s ¼ s1 þ s2. This is written as

S ¼ S1 S2 ðB:2-1Þ

EXAMPLE B.3 Let ða; bÞ ¼ ð 1; 1Þ in Example B.2. Consider the odd and even functions given by

ueðtÞ ¼ ueð tÞ

u0ðtÞ ¼ u0ð tÞ

The odd and even functions form subspaces So and Se, respectively. Any function xðtÞ in the total space S can be decomposed into even and odd functions as

u

 

t

Þ ¼

uðtÞ þ uð tÞ

ð

B:2-2

Þ

 

e

ð

2

 

u

 

t

Þ ¼

uðtÞ uðtÞ

ð

B:2-3

Þ

 

0

ð

2

 

Then,

 

 

 

 

 

 

 

uðtÞ ¼ ueðtÞ þ u0ðtÞ

ðB:2-4Þ

Consequently, the direct sum of So and Se equals S.

Convexity

A subspace Sc of a vector space S is convex if, for each vector s0 and s1 2 Sc, the vector s2 given by

s2 ¼ ls0 þ ð1 lÞs1; 0 l 1

ðB:2-5Þ

also belongs to Sc.

In a convex subspace, the line segment between any two points (vectors) in the subspace also belongs to the same subspace.

PROPERTIES OF VECTOR SPACES

385

Linear Independence

A vector u is said to be linearly dependent upon a set S of vectors vi expressed as a linear combination of the vectors in S:

X

u ¼ ckvk

k

if u can be

ðB:2-6Þ

where ck 2 F.

x is linearly independent of the vectors in S if Eq. (B.2-6) is invalid. A set of vectors is called a linearly independent set if each vector in the set is linearly independent of the other vectors in the set.

The following two properties of a set of linearly independent vectors are stated without proof:

1.A set of vectors u0; u1 . . .

ck ¼ 0 for all k.

2.If u0; u1 . . . are linearly ck ¼ bk for all k.

P

are linearly independent iff k ckuk ¼ 0 means

PP

independent, then,

k ckuk ¼ k bkuk means

Span

A vector u belongs to the subspace spanned by a subset S if u is a linear combination of vectors in S, as in Eq. (B.2-6). The subspace spanned by the vectors in S is denoted by span (S).

Bases and Dimension

A basis (or a coordinate system) in a vector space S is a set B of linearly independent vectors such that every vector in S is a linear combination of the elements of B.

The dimension M of the vector space S equals the number of elements of B. If M is finite, S is a finite-dimensional vector space. Otherwise, it is infinitedimensional.

If the elements of B are b0; b1; b2 . . . bM 1, then any vector x in S can be written

as

MX1

x ¼ wkbk

ðB:2-7Þ

k¼0

 

where wk s are scalars from the field F.

The number of elements in a basis of a finite-dimensional vector space S is the same as in any other basis of the space S.

386

APPENDIX B: LINEAR VECTOR SPACES

In an orthogonal basis, the vectors bm are orthogonal to each other. If B is an orthogonal basis, taking the inner product of x with bm in Eq. (B.2-7) yields

MX1

ðu; bmÞ ¼ wkðbk; bmÞ ¼ wmðbm; bmÞ

k¼0

 

 

 

 

so that

 

 

 

 

w

ðx; bmÞ

ð

B:2-8

Þ

m ¼

ðbm; bmÞ

 

EXAMPLE B.4 If the space is Fn, the columns of the n n identity matrix I are linearly independent and span Fn. Hence, they form a basis of Fn. This is called the standard basis.

B.3 INNER-PRODUCT VECTOR SPACES

The vector spaces of interest in practice are usually structured such that there are a norm indicating the length or the size of a vector, a measure of orientation between two vectors called the inner-product, and a distance measure (metric) between any two vectors. Such spaces are called inner-product vector spaces. The rest of the chapter is restricted to such spaces. Their properties are discussed below.

An inner product of two vectors u and v in an inner-product vector space S is written as (u,v) and is a mapping S S ! D, satisfying the following:

1.ðu; vÞ ¼ ðv; uÞ

2.ðau; vÞ ¼ aðu; vÞ; a being a scalar

3.ðu þ v; wÞ ¼ ðu; wÞ þ ðv; wÞ

4.ðu; uÞ > 0 when u ¼6 0, and ðu; uÞ ¼ 0 if u ¼ 0

When u and v are N-tuples,

u¼ ½u0 u1 . . . uN 1&t

v¼ ½v0v1 . . . vN 1&t

ðu; vÞ can be defined as

XN 1

ð

u; v

Þ ¼

u

v

ð

B:3-1

Þ

 

k

k

 

k¼0

INNER-PRODUCT VECTOR SPACES

387

When f ðtÞ and gðtÞ are continuous functions in the vector space C½a; b& with pointwise addition and scalar multiplication, the inner product ð f ; gÞ is given by

ðb

ðB:3-2Þ

ð f ; gÞ ¼ f ðtÞg ðtÞdt

a

p

The Euclidian norm of a vector u is ðu; uÞ, and is denoted by juj. The vectors u and v are orthogonal if ðu; vÞ ¼ 0. In the space Fn, a more general norm is defined by

 

 

X

#

1=p

jujp

¼

ðB:3-3Þ

"k 0 jukjp

 

 

n 1

 

 

 

 

¼

 

 

where 1 p < 1. p ¼ 2 gives the Euclidian norm.

Below we discuss important properties of inner-product vector spaces in general, and some inner-product vector spaces in particular.

Distance Measure (Metric)

The distance dðu; vÞ between two vectors u and v shows how similar the two vectors are. Inner-product vector spaces are also metric spaces, the metric being dðu; vÞ. The distance dðu; vÞ can be defined in many ways provided that it satisfies the following:

1.dðu; vÞ 0

2.dðu; uÞ ¼ 0

3.dðu; vÞ ¼ dðv; uÞ

4.d2ðu; vÞ d2ðu; wÞ þ d2ðw; vÞ

The last relation is usually called the Schwarz inequality or the triangular inequality. The Euclidian distance between two vectors x and y is given by

dðu; vÞ ¼ ju vj ¼ ½ðu v; u vÞ&1=2

ðB:3-4Þ

It is observed that the norm of a vector u is simply dðu; hÞ, h being the null vector. Even though the Euclidian distance shows the similarity between u and v, its magnitude depends on the norms of u and v. In order to remove this dependence, it

can be normalized by dividing it by jujjvj.

Examples of Inner-Product Vector Spaces

Two most common inner product spaces are the set R of real numbers and the set C of complex numbers, with their natural metric dðu; vÞ being the Euclidian distance between u and v.

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