Signals and Systems - Electrical Engineering

500 C H A P T E R 8: Discrete-Time Signals and Systems

For the given case wherey[n]=δ[n] andx[n]=u[n], we get, according to the above,

h[0]=

1

x[0]

y[0]= 1

h[1]=

1

x[0]

(

y[1]−h[0]x[1]

)

= 0 − 1 =− 1

h[2]=

1

x[0]

(

y[2]−h[0]x[2]−h[1]x[1]

)

= 0 − 1 + 1 = 0

h[3]=

1

x[0]

(

y[3]−h[0]x[3]−h[1]x[2]−h[2]x[3]

)

= 0 − 1 + 1 − 0 = 0

..

.

and, in general,h[n]=δ[n]−δ[n−1].

The length of the convolutiony[n] is the sum of the lengths of the inputx[n] and of the impulse

responseh[n] minus one. Thus,

length ofh[n]=length ofy[n]−length ofx[n]+ 1

When usingdeconvwe need to make sure that the length ofy[n] is always larger than that ofx[n]. If

x[n] is of infinite length, like whenx[n]=u[n], this would require an even longery[n], which is not

possible. However, MATLAB can only provide a finite-support input, so we make the support of

y[n] larger. In this example we have found analytically that the impulse responseh[n] is of length

2, so if the length ofy[n] is chosen so that lengthy[n] is larger than the length ofx[n] by one, we

get the correct answer (case (a) in the script below); otherwise we do not (case (b)). Run the two

cases to verify this (get rid of % symbol to run case (b)).

%%%%%%%%%%%%%%%%%%%%%%

% Example 8.25 --- Deconvolution

%%%%%%%%%%%%%%%%%%%%%%

clear all

x = ones(1, 100);

y = [1 zeros(1, 100)]; % (a) correct h

% y = [1 zeros(1, 99)]; % (b) incorrect h

[h, r] = deconv(y, x)

n

Bounded-Input Bounded-Output Stability

Stability characterizes useful systems. A stable system provides well-behaved outputs for well-behaved

inputs. Bounded-input bounded-output (BIBO) stability establishes that for a bounded (that is what

is meant by well-behaved) inputx[n] the output of a BIBO-stable systemy[n] is also bounded. This

means that if there is a finite boundM<∞such that|x[n]|<Mfor alln(you can think of it as

an envelope [−M,M] inside which the input is in for all time), the output is also bounded (i.e.,

|y[n]|<LforL<∞and alln).