Discrete-time signals and systems [PDF]

Discrete-time signals and systems

x[−3] x[−2] x[−4] x[−1]

See Oppenheim and Schafer, Second Edition pages 8–93, or First Edition pages 8–79.

x[4]

x[0] 1

... −4

−3

−2

... 2

3

−1 0

4 x[1]

1 Discrete-time signals

n

x[3] x[2]

A discrete-time signal is represented as a sequence of numbers: x D fxŒng;

(This can be plotted using the MATLAB function stem.) The value xŒn is undefined for noninteger values of n.

1 < n < 1:

Here n is an integer, and xŒn is the nth sample in the sequence. Discrete-time signals are often obtained by sampling continuous-time signals. In this case the nth sample of the sequence is equal to the value of the analogue signal xa .t / at time t D nT : xŒn D xa .nT /;

Sequences can be manipulated in several ways. The sum and product of two sequences xŒn and yŒn are defined as the sample-by-sample sum and product respectively. Multiplication of xŒn by a is defined as the multiplication of each sample value by a. A sequence yŒn is a delayed or shifted version of xŒn if

1 < n < 1:

yŒn D xŒn

The sampling period is then equal to T , and the sampling frequency is fs D 1=T .

with n0 an integer. The unit sample sequence

xa .1T / ...

n0 ;

1 ... t

x[1]

is defined as

For this reason, although xŒn is strictly the nth number in the sequence, we often refer to it as the nth sample. We also often refer to “the sequence xŒn” when we mean the entire sequence. Discrete-time signals are often depicted graphically as follows:

1

n

0

ıŒn D

8 1 the sequence grows in magnitude as n increases. n

0

is defined as

A sinusoidal sequence ...

uŒn D

8