Ch2(2) Discrete System.pdf

Discrete-Time Systems Discrete-Time Systems:
Examples
• A discrete-time system processes a given
• 2-input, 1-output discrete-time systems -
input sequence x[n] to generates an output
Modulator, adder
sequence y[n] with more desirable
properties • 1-input, 1-output discrete-time systems -
Multiplier, unit delay, unit advance
• In most applications, the discrete-time
system is a single-input, single-output
system:
Discrete ­ time y[n]
x[n] System
Input sequence Output sequence
1 2
Copyright © 2001, S. K. Mitra Copyright © 2001, S. K. Mitra
Discrete-Time Systems: Examples Discrete-Time Systems:Examples
n
• Accumulator -y[n] = å x[l]
l=­¥ • Accumulator - Input-output relation can
n­1 also be written in the form
= å x[l] + x[n] = y[n ­1] + x[n] ­1 n
l=­¥ y[n] = å x[l] + å x[l]
l=­¥ l=0
• The output y[n] at time instant n is the sum n
of the input sample x[n] at time instant n = y[­1] + å x[l], n ³ 0
and the previous output y [ n ­ 1 ] at time l=0
instant n ­ 1 , which is the sum of all • The second form is used for a causal input
previous input sample values from ­ ¥ to n ­1 sequence, in which case y [ ­ 1 ] is called
the initial condition
• The system cumulatively adds, ., it
accumulates all input sample values
3 4
Copyright © 2001, S. K. Mitra Copyright © 2001, S. K. Mitra
Discrete-Time Systems:Examples Discrete-Time Systems:Examples
s[n] = 2[n()n ], d[n] - random signal
• M-point moving-average system - 8
d[n]
s[n]
1 M­1 6 x[n]
y[n] = å x[n ­ k] 4
M k=0
2
Amplitude
• Used in smoothing random variations in 0
-2
data 0 10 20 30 40 50
Time index n
7
• An application: Consider s[n]
6 y[n]
x[n] = s[n] + d[n], 5
4
3
where s[n] is the signal corrupted by a noise Amplitude
2
d[n] 1
5 6 0
0 10 20 30 40 50
Copyright © 2001, S. K. Mitra Time index n Copyright © 2001, S. K. Mitra
1
Discrete-Time Systems:Examples Discrete-Time Systems:
• Linear interpola