Ersoy O.K. Diffraction, Fourier optics, and imaging (Wiley, 2006)(ISBN 0471238163)(427s) PEo

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.

It is observed that these functions in the limit approach 0 at t 6¼ 0 and 1 at t ¼ 0. In the last case, the limit at t 6¼ 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

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

380 APPENDIX A: THE IMPULSE FUNCTION

EXAMPLE A.3 Evaluate

a

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.

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

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

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.

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

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

k¼0

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

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

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

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.