傅里叶变换：工程师入门指南

数字信号处理

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"Fourier Transforms: An Introduction for Engineers" 是一本由Robert M. Gray和Joseph W. Goodman合著的书籍，主要针对工程师介绍傅里叶变换的概念和应用，旨在帮助读者深入理解数字信号处理。傅里叶变换是信号处理领域中的核心概念，它在数学、物理、工程和计算机科学中具有广泛的应用。这本书可能涵盖了以下知识点： 1. **傅里叶变换基础**：书中可能首先介绍了傅里叶变换的基本理论，包括离散傅里叶变换（DFT）和连续傅里叶变换（CTFT）。傅里叶变换能够将信号从时域转换到频域，揭示信号的频率成分。 2. **傅里叶逆变换**：书中可能会讲解如何通过傅里叶逆变换将频域表示恢复到时域表示，这对于信号的分析和滤波至关重要。 3. **快速傅里叶变换（FFT）**：由于DFT计算复杂度高，实际应用中通常使用快速傅里叶变换算法，这可能是书中重点讨论的内容之一，因为FFT极大地提高了计算效率。 4. **信号分析**：傅里叶变换在信号分析中的应用，如检测信号中的周期性成分、噪声过滤、谱分析等，可能是书中讨论的实际案例。 5. **滤波器设计**：傅里叶变换在设计滤波器中的作用，如低通、高通、带通和带阻滤波器的设计，以及滤波器性能的分析和优化。 6. **图像处理**：在图像处理领域，二维傅里叶变换用于分析图像的频率特性，进行图像压缩、增强或降噪。 7. **其他应用**：除了信号处理，傅里叶变换还应用于通信系统（如调制解调）、频谱分析、声学、光学、热力学等多个工程领域，书中可能也会涉及这些领域的实例。 8. **数学背景**：虽然面向工程师，但书可能也提供了一定的数学背景，帮助读者理解变换背后的数学原理。 9. **教学方法**：鉴于提到向Ron Bracewell致敬，可能书中采用了启发式和实践性的教学方式，结合实例和问题解决策略来教授傅里叶变换。《Fourier Transforms: An Introduction for Engineers》是为工程专业人士提供的一本深入浅出的傅里叶变换教程，它不仅解释了理论，还强调了其在数字信号处理中的实际应用。

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PREFACE xix

Imaginary

Real

Figure 0.1: Complex Numbers

the

complex number is denoted by

the

vector z, with magnitude A

and

phase

has a real component x

and

imaginary component y.

The

complex conjugate

defined as x - iy.

The

real

and

imaginary

parts

are

easily found by

x =

!R(z)

~(z

+ z*), y =

~(z)

~(z

- z*)

Sinusoids

and

complex exponentials play a basic role throughout this

book.

They

are related by Euler's formulas: For all real

()

cos(}

+ isin(}

ilJ

cos(}

= 2

ilJ

sin 8 = 2i

A complex-valued function g = {get); t E

assignment of

complex-valued numbers

get}

to every t E T.

A signal

{get};

t E

is said

be even if

-t)

get};

t E

PREFACE

said

be odd if

-t)

-g(t);

tEn.

A slight variation

this definition

common: strictly speaking, an odd

signal must satisfy

g(O)

= 0 since -g(O) =

g(O).

This condition

some-

times dropped so

that

the

definition becomes

-t)

-g(t)

for all t

For example, the usual definition of

the

sign function meets

the

strict def-

inition,

but

the

alternative definition (which is +1 for t

0 and

-1

for

t < 0) does not.

The

alternative definition is, however,

odd function if

one ignores

the

behavior

t =

A signal is Hermitian if

g(-t)

= g*(t);

tEn.

For example, a complex exponential g(t) = ei21r/ot is Hermitian. A signal

anti-Hermitian if

-t)

-g*(t);

tEn.

examples, sin t and te-'>'Itl are odd functions of

tEn,

while cos t

and e-'>'Itl are even.

shall have occasion

deal with modular arithmetic. Given a posi-

tive real number T

any real number a can be written uniquely in the

form

a =

+ r where the "remainder" term is in

the

interval

[0,

T).

This

formula defines

a mod T = r,

that

is, a mod T is what

left of a when the

largest possible number of integer multiples of

T is subtracted from

This

is often

stated

as "a modulo T."

The

definition can be summarized by

a mod T = r if a =

+ r where r E

[0,

and k E

(0.1)

More generally

can define modular arithmetic on any interval

[a,

b),

a by

a mod

[a,

= r if a =

+ r where r E

[a,

and k E

(0.2)

Thus

the

important

special case a mod T

an abbreviation for a mod

[0,

T). By "modular arithmetic" is meant doing addition and subtraction

within an interval. For example, (0.5

+ 0.9) mod 1 = 1.14 mod 1 = .14 mod

1 and

(-0.3)

mod 1 = 0.7.

Acknowledgements

gratefully acknowledge our debt

the many students who suffered

through early versions of this book and who considerably improved

their corrections, comments, suggestions, and questions.

also acknowl-

ege

the

Industrial Affiliates Program of the Information Systems Labora-

tory, Stanford University, whose continued generous support provided the

computer facilities used

write

and

design this book.

CHAPTER

SIGNALS

AND

SYSTEMS

while

the

signal

{h(t)j

t E

(-oo,oo)}

defined by

h(t)

{sint

t E

[0,211')

tEn,

t ¢

[0,211')

infinite

duration

but

time-limited. Clearly

the

properties of these two

signals will strongly resemble each other,

but

there will

subtle and im-

portant

differences in their analysis.

When

the

set T

continuous as in these examples,

say

that

the

signal

continuous

time

signal,

continuous

parameter

signal, or simply

a waveform.

The

shorter

notation get)

often used for

the

signal

{g(t)j

t E

T},

but

this can cause confusion as

g(t)

could either represent

the

value

the

signal

a specific time t

the

entire waveform

g(t)

for all t E T. To

lessen this confusion

common

denote

the

entire signal by simply

the

name of

the

function gj

that

is,

9 = {g(t)j t E

when

the

index set T

clear from context.

also fairly common practice

use boldface

denote

the

entire signalj

that

is, g =

{g(t)j

t E

T}.

Signals can also

sequences, such as sampledsinusoids

{sin(nT)j

n E

Z},

where Z is

the

set

all integers {

...

-2,

-1,0,

...

a geometric

progression {rnj n = 0,

1,2,

...

a sequence of binary

data

{unj n E

Z},

where all

the

are either 1 or

Analogous

the

waveform case

can denote such a sequence as

{g(t)j

t E

with

the

index set T now

being a set

integers.

is more common, however,

use subscripts

rather

than

functional notation and

use indexes like k, I, n, m instead

of t for

the

index for sequences. Thus a sequence will often

denoted

by {gnj

nET}.

still use

the

generic notation 9 for a signal

this

type.

The

only difference between

the

first and second types is

the

nature

the

index set T. When T is a discrete set such as

the

integers or

the

nonnegative integers,

the

signal 9 is called a discrete

time

signal, discrete

parameter

signal, sequence,

time

series.

the

waveform case, T may

have infinite

duration

(e.g., all integers)

finite duration (e.g.,

the

integers

from °

N - 1).

the

above examples

the

index set T

one-dimensional,

that

is,

con-

sists

some collection of real numbers. Some signals are best modeled as

having multidimensional index sets. For example, a two-dimensional square

sampled image intensity

raster

could be written as {gn,kj n = 1,

...

k =

...

,K},

where each gn,k represents

the

intensity

nonnegative number)

of a single picture element

pixel in

the

image,

the

pixel located in

the

nth

column

and

kth

row

the

square image. Note

that

in this case

the

1.1.

WAVEFORMS

AND

SEQUENCES

dummy arguments correspond

space rather

than

time. There is nothing

magic

about

the

name of a dummy variable, however, and

could still

use

t here if

wished.

For example, a typical magnetic resonance (MR) image consists of a

256

x 256 square array of pixel intensities, each represented by an integer

from

° (black, no light)

1 =

511

(white, fully illuminated).

shall wish

display such images on screens which support only 8 and not

9 bits in order

generate examples, however,

shall consider MR images

consist of 256 x 256 square array of pixel intensities, each represented by

integer from °

-1

255.

note

that,

in fact, the raw

data

used

to generate

images constitute (approximately) the Fourier transform of

the

image. Thus when MR images are rendered for display, the

basic"

operation is

inverse Fourier transform.

A square continuous image raster might be represented by a wave-

form depending on two arguments

{g(x,

y)j

x E

[0,

al, y E

[0,

a]}. A

three-dimensional sequence of image rasters could be expressed by a sig-

nal

{gn,k,lj

n =

0,1,2,

...

, k = 1,2,

...

,K,

1 = 1,2,

...

,K},

where now

the

time index (any nonnegative integer) and k and 1 are

the

spatial

indices. Here

the

index set T

three-dimensional and includes both time

and space.

In all

these different signal types, the signal has

the

general form

9 = {g(t)j t E

T},

where T is

the

domain

definition or index

set

of the signal, and where g(t)

denotes

the

value of

the

signal

"time" or parameter or dummy variable

In general, T can be finite, infinite, continuous, discrete, or even vector

valued. Similarly

g(t)

can take on vector values,

that

is, values in Euclidean

space.

shall, however, usually focus on signals

that

are real or complex

valued,

that

is, signals for which g(t)

either a real number or a complex

number for all

t E T.

mentioned before, when T is discrete

will often

write

or something similar instead of g(t).

In summary, a signal is

just

a function whose domain

definition is T

and whose range is the space

real

complex numbers.

The

nature of T

determines whether

the

signal

continuous time or discrete time and finite

duration

infinite duration. The signal

real-valued or complex-valued

depending on

the

possible values of g(t).

Although T appears

be quite general,

will need

impose some

structure

get useful results and

will focus on a

few

special cases

that

are

the

most important examples for engineering applications.

The

most common index sets for the four basic types of signals are listed in

Table 1.1.

The

subscripts of

the

domains inherit their meaning from their

place in

the

table,

that

is,

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