MATLAB在数字信号处理中的应用指南

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Sequence in

Example 2.2a

Sequence

Example

22b

FIGURE 2.2

Sequences in Example

2.2

This example shows that the four

sig*

functions developed

this

section provide a convenient approach for sequence manipulations.

EXAMPLE

2.3

Generate the complex-valued signal

e(-0.1+~0.3)n

-105

and plot its magnitude, phase, the real part, and the imaginary part in four

separate subplots.

The plot of the sequence is shown in Figure

2.3.

Chapter 2 DISCRETE-TIME SIGNALS AND SYSTEMS

real

part

FIGURE

2.3

Complez-valued sequence plots in Example

2.3

SOME USEFUL

There are several important results in discretetime signal theory. We will

RESULTS

discuss some that are useful in digital signal processing.

Unit sample synthesis

Any arbitrary sequence x(n) can be synthe

sized

a weighted sum of delayed and scaled unit sample sequences, such

x(n)

k=-m

x(k)6(n- k)

We will use this result in the next section.

Even and

odd

synthesis

real-valued sequence xe(n)

called even

(symmetric) if

Similarly, a real-valued sequence

x,(n)

called odd (antisymmetric)

Discrete-time Signals

We will also need an expression for the sum of any finite number of terms

of the series given by

These two results will be used throughout this book.

Correlations of sequences Correlation is

operation used in many

applications in digital signal processing. It is a measure of the degree to

which two sequences are similar. Given two real-valued sequences x(n) and

y(n) of finite energy, the crosswmlation of x(n) and y(n) is a sequence

~,,(e) defined as

The indeq

is called the shift or lag parameter. The special case of (2.6)

when y(n)

x(n)

called autocorrelation and is defined by

It provides a measure of self-similarity between different alignments of the

sequence.

MATLAB

functions to compute auto- and crosscorrelations are

discussed later in the chapter.

DISCRETE SYSTEMS

Mathematically, a discrete-time system (or discrete system for short) is

described

an operator T(.] that takes a sequence x(n) (called excitation)

and transforms it into another sequence y(n) (called response). That is,

y(n)

T[x(n)l

In DSP we will say that the system processes an input signal into

output

signal. Discrete systems are broadly classified into linear and nonlinear

systems. We will deal mostly with linear systems.

LINEAR

discrete system T[.] is a linear operator I.[.] if and only

L[.]

satisfies

SYSTEMS

the principle of superposition, namely,

Chapter

DISCRETE-TIME SIGNALS AND SYSTEMS

Using (2.1) and (2.8), the output y (n) of a linear system to an arbitrary

input

(n) is given by

The response

[6 (n

k)] can be interpreted

the response of a linear

system at time

due to a unit sample (a well-known sequence) at time k.

It is called an impulse response and

denoted by h (n, k). The output

then

given by the superposition summation

The computation of (2.9) requires the time-varying impulse response

h (n,

k),

which in practice

not very convenient. Therefore time-invariant

systems are widely

used

DSP.

Linear

time-invariant

(LTI)

system

linear system in which an

input-output pair, x(n) and y(n),

invariant to a shift

in time is called

a linear timeinvariant system. For an LT1 system the

L[.]

and the shifting

operators are reversible

shown below.

We will denote an LT1 system by the operator

LTI

[.l.

Let x(n) and y(n)

be the input-output pair of an LT1 system. Then the time-varying function

h (n,

becomes a time-invariant function h (n

k), and the output from

(2.9) is given by

y(n)

LTI

[x(n)]

x(k)h(n

(2.10)

k=-m

The impulse response of an LT1 system is given by h(n). The mathemat-

ical operation in

(2.10)

is called a linear convolution sum and is denoted

Discrete Systems

Hence an LT1 system is completely characterized in the time domain by

the impulse response

h(n)

as shown below.

We will explore several properties of the convolution in Problem

2.12.

Stability

This is a

very

important concept in linear system theory. The

primary reason for considering stability is to avoid building harmful sys-

tems

or to avoid burnout or saturation in the system operation. A system

said to be

bounded-input bounded-output

(BZBO)

stable

if every bounded

input produces a bounded output.

An LT1 system is BIBO stable if and only if its impulse response is

abso-

lutely summable.

BIB0 Stability

Ih(n)l

(2.12)

-m

CawaZity

This important concept

necessary to make sure that sys-

tems

can

be built. A system is said to be causal if the output at index

depends only on the input up to and including the index

that is, the

output does not depend on the future values of the input. An LT1 system

causal if and only if the impulse response

Such a sequence

termed a

causal

sequence.

In signal processing, unless

otherwise stated, we will always assume that the system

causal.

CONVOLUTION

We introduced the convolution operation

(2.11)

to describe the response

of an LT1 system. In DSP it

important operation and has many other

uses that we will see throughout this book. Convolution can be evaluated

in many different ways. If the sequences are mathematical functions (of

finite or infinite duration), then we can analytically evaluate

(2.11)

for all

to obtain a functional form of

y(n).

Chapter

DISCRETE-TIME SIGNALS AND SYSTEMS

EXAMPLE

2.5

Let the rectangular pulse

x(n)

u(n)

u(n

10)

of Example

2.4

an input

LT1 system with impulse response

Determine the output

y(n).

The input

x(n)

and the impulse response

h(n)

are shown in Figure

2.5.

Fkom

(2.11)

The sum

2.14

is almost

geometric series sum except that the term

u(n

takes diierent values depending on

and

There are three dierent conditions

under which

u(n

can

evaluated.

CASE

Then

u(n

0, 0

Hence from

(2.14)

In this

case

the nonzero values of

x(n)

and

h(n)

not

overlop.

Impulse

Response

FIGURE

2.5

The input sequence and the impulse response in Ezample

2.5

Convolution

CASE

Then

u(n

Hence from

(2.14)

In this

case

the impulse response

h(n) partially overlaps

the input

x(n).

CASE

iii

Then

u(n

and from

(2.14)

In this last case

h(n) completely overlaps z(n).

The complete response

given by

(2.15), (2.16),

and

(2.17).

It is shown in

Figure

2.6

which depicts the distortion of the input pulse.

The above example can also be done using a method called graphical

convolution, in which

(2.11)

given a graphical interpretation. In this

method

h(n

interpreted

folded-and-shafied

version of

h(k).

The output

y(n)

obtained

a sample sum under the overlap of

x(k)

and

h(n

k).

use

an example to illustrate this.

EXAMPLE

2.6

Given the following two sequences

determine the convolution

y(n)

x(n)

h(n).

Solution

In Figure

2.7

we show four plots. The topleft plot shows

x(k)

and

h(k),

the

original sequences. The togright plot shows

s(k)

and

h(-k),

the folded version

Ourwt

Sequence

FIGURE

2.6

The output sequenm in Example

2.5

Chapter

DISCRETE-TIME SIGNALS AND SYSTEMS

~(k)

and

h(k)

x(k) and h(-k)

1.1

"'""":'i

sdM:

dashed.

x(k)

and

h(-l-kl

x(k) and h(2-k)

solid:

dashed:

solid

dashed:

n=-l

n=2

FIGURE

2.7

Gmphical convolution in Example 2.6

h(k).

The bottom-left plot shows

x(k)

and

h(-l

k),

the folded-and-shifted-

by--l version of

h(k).

Then

The bottom-right plot shows

x(k)

and

h(2

k),

the folded-and-shifted-by-2

version of

h(k),

which gives

Thus we have obtained two values of

y(n).

Similar graphical calculations

can

be done for other remaining values of

y(n).

Note that the beginning point

(first

nonzero sample) of

y(n)

is given by

-3

(-1)

-4,

while the end point

(the last nonzero sample) is given by

The complete output

given by

Students are strongly encouraged to verify the above result. Note that the result-

ing sequence

y (n)

has a

longer length

than both the

(n)

and

h (n)

sequences.

MATLAB

If arbitrary sequences are of infinite duration, then

MATLAB

cannot be

IMPLEMEN-

used

directly

to compute the convolution.

MATLAB

does provide a built-in

TATION

function called conv that computes the convolution between two finite-

Convolution

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