傅里叶变换在雷达与信号处理中的应用解析

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"《Fourier Transforms in Radar and Signal Processing》是David Brandwood撰写的一本专业书籍，由Artech House出版。这本书详细探讨了在雷达和信号处理领域中傅里叶变换的应用。" 傅里叶变换是数学和工程领域的一个核心工具，特别是在信号处理和雷达技术中具有极其重要的地位。在雷达系统中，傅里叶变换被用来分析和理解接收到的信号，从而识别目标的特性、速度、距离等关键信息。通过傅里叶变换，复杂的时域信号可以转化为频域表示，这使得我们可以更容易地检测和分离不同的频率成分。信号处理方面，傅里叶变换是滤波、压缩、解码和通信信号分析的基础。它允许工程师将信号分解为不同频率的简单成分，这对于去除噪声、提取有用信息以及实现各种数字信号处理算法至关重要。例如，它可以用于频谱分析，确定信号的频谱分布，或者在通信系统中用于调制和解调。书中可能涵盖了以下主题： 1. 基本傅里叶变换理论：包括连续傅里叶变换、离散傅里叶变换（DFT）以及快速傅里叶变换（FFT）的定义、性质和计算方法。 2. 傅里叶变换在雷达信号分析中的应用：如脉冲压缩、匹配滤波、距离-多普勒处理等，这些都是雷达系统中用于提高分辨率和探测能力的关键技术。 3. 信号采样理论：根据奈奎斯特定理，理解如何正确采样信号以避免信息损失。 4. 频域滤波：介绍如何使用傅里叶变换设计和实现滤波器，以去除噪声或选择性增强特定频率成分。 5. 射频(RF)和微波信号处理：讨论这些频率范围内信号的傅里叶变换特性。 6. 实际应用案例：可能包括雷达系统设计的实际示例，以及信号处理问题的解决方案。 7. 数字信号处理(DSP)算法：介绍如何使用傅里叶变换实现现代数字信号处理器和软件定义无线电中的算法。 8. 最新进展和挑战：可能讨论傅里叶变换在新兴技术如MIMO雷达、认知无线电、以及信号处理中的新应用和挑战。《Fourier Transforms in Radar and Signal Processing》是一本深入探讨傅里叶变换在雷达和信号处理领域应用的专业书籍，对理解和应用这一关键技术具有极高的价值。

Introduction

and this is the subject of his Chapter 2, in which his rules and pairs are

introduced. Waveforms and spectra are connected, of course, by a Fourier

transform relationship, and this technique is the principal tool of the ‘‘time-

and-frequency analysis’’ that Woodward implies is the basis of much of the

mathematical study of information theory, radio, and radar.

It is not claimed that Woodward’s rules and pairs themselves are

particularly original. The pairs are well known and, as Woodward says, the

rules are ‘‘known by heart by most circuit mathematicians’’ [1]. What is

perhaps new is the careful specification of the set of rules and the committed

and consistent use of them to obtain transforms very directly and concisely.

In addition to the results required for the mathematical study of radar,

Woodward uses this approach to derive important general results very neatly

(again already known) such as Parseval’s theorem, sampling theorems, and

Poisson’s formula. What is more clearly new and valuable is Woodward’s

contribution to notation, including the rect and sinc functions, the comb

function and the rep operator. The term sinc function has since become more

widely accepted and used, although, regrettably, some writers use sinc x to

mean sin x/x instead of the correct sin

␲

x. (The form sin x /x is unlikely

to arise using Woodward’s methods, so these writers are mixing the standard

integration method with Woodward’s misrepresented notation, leading to

inconsistency.) The comb function and the rep operator are used for describ-

ing sampled or repetitive waveforms and their spectra, and hence enable the

whole field of Fourier series to be incorporated into the field of Fourier

transforms. Thus, the Fourier series can now be seen as a particular form

of the Fourier transform, rather than the Fourier transform seen just as a

limiting case of the Fourier series. For suitable waveforms, this enables the

Fourier series coefficients to be obtained without explicit integration.

1.3 Outline of the Rules and Pairs Method

To use the rules and pairs, the function to be transformed must first be

expressed carefully in the notation in which they are expressed, that is, in

terms of the basic functions included in the table of Fourier transform pairs.

This table gives the transforms of these functions and the table of rules

provides the relationships between these transforms (sums, products, convolu-

tions, and appropriate scaling factors, for example) as determined by the

relationships between the input basic functions.

The notation is specific and specialized, but is reasonably natural and

quickly absorbed, and it is given in Chapter 2 with the tables of the rules

4 Fourier Transforms in Radar and Signal Processing

and pairs. When the transform has been obtained using the rules and pairs,

the resulting expression requires interpretation and may benefit from rear-

ranging and simplifying. Sketches of the functions can be useful to bring

to life a mathematical expression, and illustrations of functions and their

transforms have been provided fairly generously, particularly in Chapters 2

and 3.

A feature of the notation is that a given function (or waveform) can

sometimes be correctly described in more than one way, leading to more

than one expression for its transform (or spectrum), though of course these

expressions must be equivalent. One of these expressions will be more appro-

priate and convenient for one particular case under study than another, so

there is an art, based on experience and imagination (as required for solving

certain differential equations or some problems in integration), in choosing

the description of the waveform that will yield the spectrum represented in the

required form. This alternative description approach is particularly effective in

producing general results or theorems, as Woodward [1] shows. A good

example is in Woodward’s proof of the sampling theorem in the time domain,

given in Chapter 4, where, by expressing the spectrum of a waveform in

two different ways, the equivalence of a continuous waveform and its sampled

form is established.

1.4 The Fourier Transform and Generalized Functions

The concept of the Fourier series seems intuitively very reasonable: that any

periodic function can be represented by a sum of elementary periodic func-

tions, either sine and cosine functions or, equivalently, complex exponentials

(cisoids). The frequencies of the elementary functions are integer multiples

(including zero, giving a constant function) of the repetition frequency of

the periodic function. The sum may turn out to be infinite, but users of

this mathematical tool are generally content to let mathematicians justify

such a sum, determining the conditions under which it converges; however,

for problems arising in practice, in physics or engineering, for example, it

is ‘‘obvious’’ that such a sum does converge. Thus, we can put for f a real

or complex function of a real variable x, with period X [so that f (x + X )

= f (x)],

f (x) =

∑

∞

cos (2

␲

nx/X ) +

∑

∞

sin (2

␲

nx/X ) =

∑

∞

=−∞

exp (2

␲

inx/X )

(1.1)

6 Fourier Transforms in Radar and Signal Processing

An approach to finding the Fourier transform of a constant function,

say f (x) = 1 for all real x, is to find a sequence of functions that do have

transforms as given by (1.4) and that approach f in the limit of some

parameter. For example, we could choose f

(x) to be the function

exp (−

␲

). Putting this into (1.4), we find that its transform is g

( y )

= n exp (−

␲

). We see that in the limit as n →∞, f

(x) → f (x ) = 1,

in that however small we choose the positive number

⑀

, for any x we can

find a value of n such that f

(x) > 1 −

⑀

. Also, as n →∞we have g

(0)

→∞, and for nonzero values of y, g

( y ) → 0, as we can always find a

value of n such that g

( y ) <

⑀

, for any positive

⑀

. The limiting function g

is the Dirac

␦

-function, which plays an important part in this theory. It is

not strictly a function in the ordinary sense, but is a generalized function

in Lighthill’s terminology [2].

The fact that there is some difficulty in using (1.4) where f (x)isa

constant function (and also if it is periodic) does not mean that such functions

do not have Fourier transforms. This problem has been tackled formally

and the subject of Fourier transforms put on a rigorous basis by Laurent

Schwarz. However, to do this it was necessary to generalize the idea of a

function to include the

␦

-function, and indeed the term generalized function

has been introduced by Temple and presented clearly and accessibly by

Lighthill [2]. As it is shown in this text that, in general, ordinary functions

satisfy the definition of generalized functions (as a limiting sequence of

suitable functions), the practical result is that we can confidently accept and

include

␦

-functions (and rows of

␦

-functions, in the case of a line spectrum)

with ordinary functions for the purpose of Fourier transform operations and

analysis.

The

␦

-function has been obtained as the limit of a series of Gaussian

functions above (also in Woodward [1, p. 28]), the limit of a series of

triangular functions [1, p. 15], and the limit of a series of rectangular

functions (Figure 2.3). The fact that different sequences can be used is

included in Lighthill’s definition, though he adds the condition that the

functions should be differentiable everywhere, which actually rules out the

series of triangular and rectangular functions.

There is no reference to generalized function theory by Woodward;

Schwarz’s work was published in 1950–1951, only shortly before Wood-

ward’s book (1953), and the further spreading of these ideas by Temple

(1955) and Lighthill (1958) followed later. The Dirac

␦

-function is an

example of the not uncommon case where physics and engineering have

required a new mathematical tool. This has been devised and given a very

reasonable justification (for example, as the limit of a series of triangular

functions), only later to be given a more rigorous mathematical definition.

Introduction

1.5 Complex Waveforms and Spectra in Signal Processing

The approach to Fourier transforms presented here uses the complex Fourier

transform, by which a waveform, real or complex, is expressed as a sum or

integral of complex exponentials [see (1.3)], which are elementary complex

waveforms. The idea of a complex waveform should not be seen as just a

mathematical convenience, with the ‘‘real-world’’ waveform just the real

part. The elementary complex waveform exp (2

␲

ift) can be represented as

the pair of real waveforms cos (2

␲

ft) and sin (2

␲

ft) in two channels, which

must be handled appropriately—that is, according to the rules of complex

arithmetic. We recall that a complex number can be represented as an ordered

pair of real numbers; that is, z = x + iy can be written as (x, y), satisfying

the rules (x

, y

) + (x

, y

) = (x

+ x

, y

+ y

) and (x

, y

) ⭈ (x

, y

)

= (x

− y

, x

+ x

). This avoids the explicit use of the imaginary

constant i, if this worries the practical user, but we need only consider that

i acts as a form of switch that moves a waveform from channel 1 (real) to

channel 2 (imaginary), or from channel 2 to channel 1, with a sign change

in this case. We note that by using complex waveforms, meaning is given

to the idea of negative frequency. Compared with the positive frequency

form, this corresponds to an inversion of the waveform in the second channel.

In signal processing, it is convenient to use the analytic signal, which

is the complex waveform corresponding to the real waveform that is present,

as received, for example, from a radio or radar antenna or sonar sensor. Thus

if the waveform is expressed as a(t) cos [2

␲

t +

␾

(t)], that is, a carrier at

intermediate frequency (IF) or radio frequency (RF) f

, with amplitude a

and phase

␾

, which may be time-varying, in the general case, then we form

the complex form a(t) exp i [2

␲

t +

␾

(t)], which is the pair {a(t) cos [2

␲

␾

(t)], a(t) sin [2

␲

t +

␾

(t)}. The second member of this pair is obtained

from the first by a Hilbert transform, which in effect performs a wideband

−90-degree phase shift. (Thus all cosine components in the signal, whatever

their frequencies, become sines, and sines become −cosines.) In practice,

this can be achieved, with a high degree of fidelity (for moderate fractional

bandwidths) by a 3-dB hybrid directional coupler. The two (real) outputs

of this coupler can be considered as the required (complex) waveform pair.

The advantage of using the analytic signal is that (at least when on a carrier)

the spectrum is ‘‘one-sided,’’ with only positive frequency components, in

contrast to the two-sided spectrum of the real waveform. If the complex

waveform is then mixed down to complex baseband, using the complex

local oscillator (LO) exp (−2

␲

t) we obtain the complex waveform

a(t ) exp [i

␾

(t)], which is the part of the waveform, the modulation, con-

taining the information of interest. At baseband the spectrum will contain

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