convolution
Example
• ConsidertheDT systemdescribedby
y[n]+ ay[n 1]= bx[n]
• Itsimpulseresponsecanbefoundtobe
( a)n b, n= 0,1,2,
h[n] = 
n= 1, 2, 3,
0,
The Convolution Sum
• Thisparticularsummationis calledthe
convolutionsum

y[n] =  x[i]h[n i]
=
x[n] h[n]

• Equationy[n] x[n] h[n] is called
convolutionrepresentationof thesystem
the
• Remark:aDT LTI systemis completely
describedby itsimpulseresponseh[n]
Block Diagram Representation
of DT LTI Systems
• Sincetheimpulseresponseh[n] provides
thecompletedescriptionof aDT LTI
system,wewrite
x[n]
h[n]
y[n]
The Convolution Sum
for Noncausal Signals
• Supposethatwehavetwosignalsx[n] and
v[n] thatarenotzerofornegativetimes
noncausal signals)
• Then,theirconvolutionis expressedby the
two-sidedseries
y[n] =

 x[i]v[n
i

i]
Properties of the Convolution Sum
• Associativity
x[n](v[n] w[n])= (x[n]v[n]) w[n]
• Commutativity
x[n]v[n] = v[n] x[n]
• Distributivityw.r.t.addition
x[n](v[n]+ w[n])= x[n]v[n]+ x[n] w[n]
Example: Computing Convolution
with Matlab
• ConsidertheDT LTI system
x[n]
h[n]
y[n]
• impulseresponse:
h[n] = sin(0.5n), n 0
• inputsignal:
x[n] = sin(0.2n), n 0
Example: Computing Convolution
with Matlab – Cont’d
h[n] = sin(0.5n), n 0
x[n] = sin(0.2n), n 0
Example: Computing Convolution
with Matlab – Cont’d
• Supposewewanttocomputey[n] for
n= 0,1,
0
• Matlabcode:
n=0:40;
x=sin(0.2*n);
h=sin(0.5*n);
y=conv(x,h);
stem(n,y(1:length(n)))
Example: Computing Convolution
with Matlab – Cont’d
y[n] = x[n] h[n]