信号频谱分析：调制-传输-解调过程与空间方法

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在"Spectral Analysis of Signals"（佩特雷·斯托伊卡和罗纳德·摩西合著，2005年普林塞出版社出版，ISBN: 0-13-113956-8）的下半部分，章节6探讨了空间方法，主要聚焦于信号的频谱分析。本节讨论的核心概念是将带通信号与基带信号的关系及其在通信过程中的作用。首先，假设接收到的信号x(t)是一个带通信号，其频谱如图6.2所示。带通信号的频谱完全由对应的基带信号（或低通信号）决定。基带信号的谱密度IS(w)与图6.2中的信号相对应，如图6.3所示。基带信号set(t)与x(t)之间的转换过程，即调制，是指将信号从基带形式转换到带通形式；而解调则是将带通信号还原回基带信号的过程。 6.2.1 调制-传输-解调过程在这个过程中，由于物理信号x(t)是实值的，其频谱IX(c)是对称的，即关于c=0，这反映在图6.2中。然而，经过解调后的信号set(t)的频谱可能不再是偶函数，这取决于具体的调制技术（例如幅度调制、频率调制等）。在后续的讨论中，虽然这些关于调制和解调过程的评论并不直接针对信号源定位问题，但它们有助于理解和解释文本中的某些陈述，有助于我们理解信号处理中的基本原理。通过这种转换，我们可以看到信号处理中频域操作的重要性，如滤波、混频和解调，这些都是通信系统的基础。在实际应用中，例如无线通信、雷达和信号处理系统设计中，理解这些概念对于信号质量控制、干扰分析和信号恢复至关重要。此外，作者可能还会讲解不同的调制类型（如AM、FM、PM等）对信号频谱的影响，以及如何通过谱分析来区分不同的调制模式。对于多径传播环境中的信号，还会涉及到多径效应和时延扩散如何影响频谱特性，这对于信号的检测和接收机设计具有显著影响。本章节深入探讨了信号从基带到带通，再到信号源定位或通信链路中的其他环节的转换过程，提供了理解信号处理中频域分析和技术的关键洞察。理解这些概念对于从事相关领域研究和实践的工程师来说，是不可或缺的知识基石。

6.4 PARAMETRIC METHODS

In this section,

postulate the array model (6.2.21). Furthermore, the noise e(t) is assumed to

be spatially white with components having identical variance:

293

(6.4.1)E {e(t)e*(t)} =

(j2J

Section 6.4 Parametric Methods

in the cases where such infonnation is available, for example in the form

a covariance

model

the data, a nonparametric approach does not give the performance that one can achieve

with

a parametric (model-based) approach. The parametric approach to

DOA

estimation

the subject

the next section.

In addition, the signal covariance matrix

:-i:

P = E {s(t)s*(t)}

(6.4.2)

',".

is assumed to be nonsingular (but not necessarily diagonal; hence, the signals may be (partially)

correlated). When the signals are fully correlated, so that

P is singular, they are said to be coherent.

Finally, we assume that the signals and the noise are uncorrelated with one another.

Under the previous assumptions, the theoretical covariance matrix

the array output vector

is given

(6.4.3)

There is a direct analogy between the array models (6.2.21) and (6.4.3) and the corresponding

models encountered in our discussion

the

sinusoids-in-n~e

case

Chapter 4. More specif-

ically, the "nonlinear regression" model (6.2.21)

the array is analogous

(4.2.6), and the

array covariance model (6.4.3) is much the same as (4.2.7). The consequence

these analogies

that all methods introduced

Chapter 4 for frequency estimation can also be used for DOA

estimation

without any essential modification. In the following, we briefly review these methods,

with a view to pointing out any differences from the frequency-estimation application. When the

assumed array model is a good representation

reality, the parametric DOA estimation methods

provide highly accurate

DOA

estimates, even

adverse situations (such as low SNR scenarios).

OUf

main thrust in this text has been to understand the basic ideas behind the presented spectral

estimation methodologies, so we do not dwell on the details

the analysis required to establish the

statistical properties

the

DOA

estimators discussed in what follows; see, however, Appendix B

for a discussion on the Cramer-Rao bound and the best accuracy achievable in DOA estima-

tion problems. Such analysis details are available in

[STOICA

AND

NEHORAI

1989A~

STOICA

AND

NEHORAI

1990;

STOICA

AND

SHARMAN

1990;

STOICA

AND

NEHORAI

1991;

VIBERG

AND

OTTER-

STEN

1991; RAo

AND

HARI

1993].

For

reviews

many

the recent advances in spatial-spectral

analysis, the reader can consult

[PILLAI

1989],

[OTTERSTEN,

VIBERG,

STOICA,

ANDNEHORAI

1993],

and [V

TREES

2002].

Section 6.4 Parametric Methods

295

consideration. With this observation, the proof

the fact that the computationally efficient beam-

fonning method provides an approximate solution

(6.4.7) in scenarios with well-separated

DOAs is concluded. •

One difference between (6.4.7) and the corresponding optimization problem in the frequency-

estimation application (see (4.3.8) in Section 4.3) lies in the fact that, in the frequency-estimation

application, only one "snapshot"

data is available, in contrast to the N snapshots available in

the DOA-estimation application. Another, more important difference is that, for non-ULA cases,

the matrix A in (6.4.7) does not have the Vandennonde structure

the corresponding matrix in

(4.3.8). As a consequence, several

the algorithms used

(approximately) solve the frequency-

estimation problem (such

the one in

[KUMARESAN,

SCHARF,

AND

SHAW

1986] and

[BRESLER

AND

MACOVSKI

1986]) are no longer applicable

solving (6.4.7) unless the array is a ULA.

6.4.2 Yule-Walker Method

The matrix

which lies at the basis

the Yule-Walker method (see Section 4.4), can be

constructed from any block

R in (6.4.3) that does not include diagonal elements.

be more

precise, partition the array model (6.2.21) into the following two nonoverlapping parts:

[

y(t)]

[A]

[e(t)]

y(t)

= A

s(t)

e(t)

Because

e(t)

and

e(t)

are uncorrelated (by assumption), we have

r A E {y(t)y*(t)} =

APA*

which is assumed to be

dimension M x L (with M +L = m). For

(6.4.8)

(6.4.9)

>n,

L>n

(6.4.10)

(which cannot hold unless

m > 2n), the rank

r is equal

n (under weak conditions) and the

- n)-dimensional null space

this matrix contains complete information about the DOAs.

see this, let G be an

(L -

matrix whose columns form a basis of

N(r).

(G can be

obtained from the SVD

se~

Result R15 in Appendix A.) Then

have

which

implies (using the fact that

rank(AP) =

that

A*G

This observation can be used,

the manner of Sections 4.4 (YW) and 4.5 (MUSIC), to estimate

the DOAs from a sample estimate

r, such as

1 N

r = N

Ly(t)y*(t)

t=1

(6.4.11)

Chapter 6 Spatial Methods

296

Unlike all the other methods discussed in what follows, the Yule-Walker method does not impose

the rather stringent condition

(6.4.1). The Yule-Walker method requires only that E {e(t)e*(t)} =

0, which is a much weaker assumption. This

a distinct advantage

the Yule-Walker method

(see

[VIBERG,

STOICA,

AND

OTTERSTEN

1995] for details). Its relative drawback is that it cannot be

used unless

m > 2n (all the other methods require only that m > n); in general, it has been found

to provide accurate DOA estimates only in those applications involving large-aperture arrays.

Interestingly enough, whenever the condition (6.4.1) holds (Le., the noise at the array output

is spatially white), we can use a modification

the technique above that does not require that

m > 2n

[FUCHS

1996].

see this, let

e.E{Y(t)y*(t)}=R[~]

xL)

where yet) is

defined in (6.4.8); hence f is made from the last L columns

R. By making

use

the expression (6.4.3) for

we obtain

(6.4.12)

Because the noise terms in

(t)

and y

(t)

are correlated, the noise is still present in f (as can be

seen from (6.4.12»; hence,

f is not really a YW matrix. Nevertheless, f has a property similar

to that

the YW matrix

as we now show.

First, observe that

The matrix

p +PA*AP is readily shown to be nonsingular

and only

P is nonsingular.

f*f

has the same form as R in (6.4.3), we conclude that (for m

L > n) the

- n)

matrix

whose columns are the eigenvectors

f that correspond to the multiple minimum

eigenvalue

, satisfies

(6.4.13)

A*G

instead

the more restrictive conditions

- L > n, L > n} (see (6.4.10» required in the

method based on r. Observe that (6.4.14) can always be satisfied

m > n,

wherea~

(6.4.10) requires that m > 2n. Finally, note that r is made from the first m - L rows

The columns

Gare also equal to the

right singular vectors

f corresponding to the

multiple minimum singular value

For numerical precision reasons, Gshould be computed

from the singular vectors

f rather than from the eigenvectors

f*f

(see Section A.8.2).

Because (6.4.13) has the same form as

A*G

= 0, we can use (6.4.13) for subspace-based

DOA

estimation in exactly the same way as we used

A*G

= a (see equation (4.5.6) and the

discussion following it in Chapter 4). Note that, for the method based on

f to be usable,

require only that

m:;::L>n

(6.4.14)

Section 6.4 Parametric Methods

297

and hence contains "less infonnation" than

this provides a quick intuitive explanation

wh~

the method based on r requires more sensors to be applicable than does the method based

6.4.3 Pisarenko and MUSIC Methods

6.4.5

ESPRIT

Method

6.4.4

Min-Norm

Method

There is no essential difference between the use

the Min-Norm method for frequency estimation

and

for DOA-finding in the noncoherent case. As for MUSIC, in the DOA-estimation application,

the

Min-Norm method should not be used

scenarios with coherent signals, and the Root

Min-Norm algorithm can only be used in the ULA case

[KUMARESAN

AND

TUFfS

1983]. In

addition, the key property that the true DOAs are asymptotically the

unique solutions

the

Min-Norm estimation problem holds in the ULA case (see Complement 6.5.1), but not necessarily

for other array geometries.

(6.4.15)

(6.4.16)

(m.

O]A

Az = [0

Im]A

the ULA case, ESPRIT can be used for DOA estimation exactly as it is for frequency estimation.

(See Section 4.7.) In the non-ULA case, ESPRIT can be used only in certain situations.

precisely, and unlike the other algorithms in this section, ESPRIT can be used for DOA finding

only

the array contains two identical subarrays that are displaced by a known displacement

vector

[RoY

AND

KAILATH

1989;

STOICA

AND

NEHORAI

1991]. Mathematically, this condition can

fonnulated as follows: Let

denote the number

sensors in the two twin subarrays, and let

AI and Az denote the submatrices

A corresponding to these subarrays. The sensors in the array

are

numbered arbitrarily, so it constitutes no restriction to assume that

made from the first

rows in A and

from the last

m.:

The MUSIC algorithm (with Pisarenko

a special case), developed in Section 4.5 for the

frequency-estimation application, can be used without modificauon for DOA estimation

[BIENVENU

1979;

SCHMIDT

1979;

BARABELL

1983]. There are only minor differences

betwe~n

the DOA and

the frequency-estimation applications

MUSIC,

pointed out next.

First, in the spatial application, we can choose between the Spectral and Root MUSIC esti-

mators only

the case

a ULA. For most

the other array geometries, only Spectral MUSIC

applicable.

Second, the standard MUSIC algorithm (4.5.15) breaks down in the case

coherent signals,

because, in that case, the rank condition (4.5.1) no longer holds. (Such a situation cannot occur

the frequency-estimation application, because P is always (diagonal and) nonsingular there.)

However, the

modified MUSIC algorithm (outlined at the end

Section 4.5) can be used when

the

signals are coherent, provided that the array is uniform and linear. This is so because the

property (4.5.23), on which the modified MUSIC algorithm is based, continues to hold even

singular. (See Exercise 6.14.)

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