信号与系统解析：MIT6.011春季课程讲义

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"MIT6_011S10_notes.pdf - 这是MIT春季2010年开设的6.011课程（Introduction to Communication, Control and Signal Processing）的课堂笔记，由Alan V. Oppenheim和George C. Verghese撰写。这是一本关于信号、系统和推理的电子书，涵盖了信号与系统的理论及其在通信和控制中的应用。" 在这些笔记中，主要探讨了以下几个关键知识点： 1. **介绍**： - 课程可能涵盖了信号处理的基础概念，包括其在通信和控制系统中的重要性。 2. **信号与系统**： - **信号和模型的属性**：这部分可能讲解了信号的基本类型（如连续时间、离散时间、确定性或随机信号），以及如何用模型来描述信号和系统的行为。 - **线性时不变系统（LTI）**：这是信号处理领域的一个核心概念，LTI系统对所有输入信号保持线性和时不变性质。 - **脉冲响应表示**：LTI系统的输出可以由其脉冲响应和输入信号的卷积来表示。 - **特征函数和变换表示**：LTI系统的特性可以通过特征函数（如阶跃响应）或通过特定变换（如傅里叶变换）来研究。 - **傅里叶变换**：这部分深入讨论了傅里叶变换，它是分析系统频率响应和信号频谱的关键工具。 3. **确定性信号及其傅里叶变换**： - **信号类别**：可能包括不同类型的信号（如阶跃、脉冲、正弦波等）及其对应的傅里叶变换。 - **帕塞瓦尔恒等式、能量谱密度和确定性自相关**：这些是评估信号能量和频率内容的重要概念。 4. **双边拉普拉斯变换和Z变换**： - **双边Z变换**：适用于离散时间信号分析的工具，特别是在逆变换中使用。 - **逆Z变换**：如何从Z域回到时域，这对于理解系统行为至关重要。 - **双边拉普拉斯变换**：在连续时间信号处理中，拉普拉斯变换是傅里叶变换的扩展，用于分析不稳定系统。 5. **离散时间处理连续时间信号**： - **基本结构**：可能讲述了如何设计和实现数字系统来处理模拟信号，包括采样、量化和滤波等步骤。这些笔记提供了深入理解信号处理理论的坚实基础，对于学习通信工程、自动化和控制系统的学生来说，是一份宝贵的资源。通过这些概念，学生能够分析和设计用于信息传输和控制的复杂系统。

28 Chapter 2 Signals and Systems

provided the above sum has a ﬁnite value when z = z

. Note that this sum is

precisely the bilateral Z-transform of the impulse response, and the (interior of

the) set of values of z

for

which the sum takes a ﬁnite value constitutes the ROC

of the Z-transform. As in the CT case, the one-sided exponential z

u[n] is not in

general an eigenfunction.

Again, an important case

is when x[n] = (e

jΩ

)

= e

jΩn

, corresponding to z

(2.13) ha

ving unit magnitude and taking the value e

jΩ

, where Ω — the (real)

“frequency” — denotes

the angular position (in radians) around the unit circle in

the z-plane. Such an

x[n] is bounded for all positive and negative time. Although

we use

a diﬀerent symbol, Ω, for frequency in the DT case, to distinguish it from

the frequency ω in

the CT case, it is not unusual in the literature to ﬁnd ω used in

both CT and DT cases for notational convenience. The corresponding output is

y[n] = H(e

jΩ

jΩn

(2.16)

where

∞

H(e

jΩ

) =

h[n]e

−jΩn

(2.17)

n=−∞

The function H(e

jΩ

) in (2.17) is

the frequency response of the DT system, and is

also the discrete-time Fourier transform (DTFT) of the impulse response. The sum

that deﬁnes

the

DTFT has a ﬁnite value (and can be shown to be a continuous

function of Ω) if

h[n] is absolutely summable, i.e., provided

∞

| h[n] | < ∞ (2.18)

n=−∞

We noted that this condition is equivalent to the system being BIBO stable. As with

the CTFT, the DTFT can be deﬁned

for signals that are not absolutely summable;

we wi

ll elaborate on this later.

Note from (2.17) that

the frequency response for DT systems is always periodic,

with period 2π. The

“high-frequency” response is found in the vicinity of Ω = ±π,

which is

consistent with the fact that the input signal e

±jπn

= (−1)

is the most

rapidly varying DT signal that one can have.

When

the input of an LTI system can be expressed

as a linear combination of

bounded eigenfunctions, for instance (in the CT case),

jω

ℓ

x(t) =

ℓ

e (2.19)

ℓ

then, by linearity, the output is the same linear combination of the responses to

the individual exponentials. By the eigenfunction property of

exponentials in LTI

systems, the response to each exp

onential involves only scaling by the system’s

frequency response. Thus

jω

ℓ

y(t) =

ℓ

H(jω

ℓ

)e (2.20)

ℓ

Similar expressions can be written for the DT case.

°Alan V. Oppenheim and George C. Verghese, 2010

Section 2.2 Linear, Time-Invariant Systems 29

2.2.3 Fourier Transforms

A broad class of input signals can be represen

ted as linear combinations of bounded

exponentials, through the Fourier transform. The synthesis/analysis formulas for

the CTFT are

∞

jωt

dω x(t) = X(j

ω) e (syn

thesis) (2.21)

2π

−∞

∞

x(t) e

−jωt

dt X(jω) =

(analysis) (2.22)

−∞

Note that (2.21) expresses

x(t) as a linear combination of exponentials — but this

weighted combination involves a

continuum of exponentials, rather than a ﬁnite or

countable number. If this signal x(t) is the input to an LTI system with frequency

response H(jω), then by

nearity and the eigenfunction property of exponentials

the output is the same weighted combination of the responses to these exponentials,

∞

jωt

dω y(t) = H(jω)X(jω) e (2.23)

2π

−∞

By viewing this equation as a CTFT synthesis equation, it follows that the CTFT

of y(t) is

Y (jω) = H(j

ω)X(jω) (2.24)

Correspondingly, the

convolution relationship (2.2) in

the time domain becomes

multiplication in the transform domain. Thus, to ﬁnd the

response Y at a particular

frequency point, we only

need to know the input X at that single frequency, and

the frequency response of the system at that frequency. This

simple fact serves, in

large measure, to explain why the

frequency domain is virtually indispensable in

the analysis of LTI systems.

The corresponding DTFT synthesis/analysis pair is deﬁned b

[n] = X(e

jΩ

) e

jΩn

dΩ (synthesis) (2.25)

2π

<2π>

∞

X(e

jΩ

) =

x[n] e

−jΩn

(analysis) (2.26)

n=−∞

where the notation < 2π > on the integral in the synthesis formula denotes integra-

tion over an

y contiguous interval of length 2π, since the DTFT is always periodic in

Ω with period 2π, a

simple consequence of the fact that e

jΩ

is periodic with period

2π. Note

that (2.25) expresses

x[n] as a weighted combination of a continuum of

exponentials.

As in the CT case, it is straightforward to show that

if x[n] is the input to an LTI

mapping, then the output y[n] has DTFT

Y (e

jΩ

) = H(e

jΩ

)X(e

jΩ

) (2.27)

°Alan V. Oppenheim and George C. Verghese, 2010

30 Chapter 2 Signals and Systems

2.3 DETERMINISTIC SIGNALS AND THEIR FOURIER TRANSFORMS

In this section we review

the DTFT of deterministic DT signals in more detail, and

highlight the

classes of signals that can be guaranteed to have well-deﬁned DTFTs.

We shall

also devote some attention to the energy density spectrum of signals that

have DTFTs.

The section will bring out aspects of the DTFT that may not have

been emphasized

in your earlier signals and systems course. A similar development

can be carried

out for CTFTs.

2.3.1 Signal Classes and their Fourier Transforms

The DTFT synthesis and analysis pair in (2.25)

and

(2.26) hold for at least the

three large classes of DT signals described b

elow.

Finite-Action Signals. Finite-action signals, which are

also called absolutely

summable signals or ℓ

(“ell-one”) signals, are deﬁned by the condition

∞

x[k]

< ∞ (2.28)

k=−∞

The sum on the left is called the ‘action’ of the signal. For these ℓ

signals, the

inﬁnite sum that deﬁnes the

DTFT is well behaved and the DTFT can be shown

to be a

continuous function for all Ω (so, in particular, the values at Ω = +π and

Ω = −

π are well-deﬁned and equal to each other — which is often not the case

when signals are not ℓ

Finite-Energy Signals. Finite-energy signals, which are

also called square summable

or ℓ

(“ell-two”) signals, are deﬁned by the condition

∞

x[k]

< ∞ (2.29)

k=−∞

The sum on the left is called the ‘energy’ of the signal.

In discrete-time, an absolutely summable (i.e., ℓ

) signal is always square summable

(i.e., ℓ

). (In continuous-time, the story is more complicated: an absolutely inte-

grable signal need not

be square integrable, e.g., consider x(t) = 1/

√

t for

0 <

t ≤ 1

and x(t) = 0

elsewhere; the source of the problem here is that the signal is not

bounded.) However, the reverse is not true. For example, consider the signal

(sin Ω

n)/πn for 0 < Ω

< π, with the value at n = 0 taken to be Ω

/π, or consider

the signal (1/n)u[n −

1], both of which are ℓ

but not ℓ

. If x[n] is such a signal,

its DTFT X(e

jΩ

) can be thought of as the limit for N → ∞ of the quantity

jΩ

) =

x[k]e

−jΩk

(2.30)

k=−N

and the resulting limit will typically have discontinuities at some values of Ω. For

instance, the transform of (sin Ω

n)/πn has discontinuities at Ω = ±Ω

°Alan V. Oppenheim and George C. Verghese, 2010

Section 2.3 Deterministic Signals and their Fourier Transforms 31

Signals of Slow Growth. Signals of ‘slow’ growth are signals whose magnitude

grows no faster

than polynomially with the time index, e.g., x[n] = n for all n. In

this case X

jΩ

) in (2.30) do

es not converge in the usual sense, but the DTFT

still exists as a general

ized (or singularity) function; e.g., if x[n] = 1 for all n, then

X(e

jΩ

) = 2πδ(Ω) for |Ω| ≤ π.

Within the class of signals of slow gro

wth, those of most interest to us are bounded

(or ℓ ) signa

ls:

∞

x[k]

≤ M

< ∞

(2.31)

i.e., signals whose amplitude has a ﬁxed

and ﬁnite bound for all time. Bounded

everlasting exponentials of the form e

jΩ

, for instance, play a key role in Fourier

transform theory. Suc

h signals need not have ﬁnite energy, but will have ﬁnite

average power o

ver any time interval, where average power is deﬁned as total energy

over to

tal time.

Similar classes of signals are deﬁned in

continuous-time. Speciﬁcally, ﬁnite-action

(or L

) signals comprise those that are absolutely integrable, i.e.,

∞

x(t)

dt < ∞

(2.32)

−∞

Finite-energy (or L

) signals comprise those that are square summable, i.e.,

∞

x(t)

< ∞ (2.33)

−∞

And signals of slow growth are ones for which the magnitude grows no faster than

polynomially with time. Bounded (or L ) con

tinuous-time signals are those for

∞

which the magnitude never exceeds a ﬁnite bound M (so these are slow-growth

signals as well). These may ag

ain not have ﬁnite energy, but will have ﬁnite average

power o

ver any time interval.

In both continuous-time and discrete-time there are many im

portant Fourier trans-

form pairs and Fourier transform properties developed and

tabulated in basic texts

on signals and systems (see, for example, Chapters 4 and

5 of Oppenheim and Will-

sky). For convenience, we include

here a brief table of DTFT pairs. Other pairs

are easily derived from

these by applying various DTFT properties. (Note that the

δ’s in the left column denote unit samples, while those in the right col

umn are unit

impulses!)

°Alan V. Oppenheim and George C. Verghese, 2010

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