Revisiting the Basics - Digital Signal Processing: A Practitioner's Approach - page 24

1

0.5

0

−0.5

− 1

0 5 10 15 20 25 30

Time

Figure 2.3 Impulse response h k

2.1.5.1 Convolution: Response to a Sequence

Consider a system defined by its unit sample response h k , and let the input sequence
be u k of length N. Then the response due to u 1 can be obtained using the scaling
property as u 1 h k , and the response due to u 2 ;

u 3 ; u 4 ; ...; u N þ 1 can be obtained using

the scaling and shifting properties as

u 2 h k 1 ; u 3 h k 2 ; u 4 h k 3 ; ...; u N þ 1 h k N :

Now, using superposition we obtain the response due to the sequence u k by
summing all the terms as

u 1 h k þ u 2 h k 1 þ u 3 h k 2 þ u 4 h k 3 þþ u N þ 1 h k N : ð 2 : 5 Þ

We need to remember that the term f u 1 h k g is a sequence obtained by multiplying a
scalar u 1 with another sequence h k . This mental picture is a must for better
understanding the operations below. We obtain the response y k through a familiar
operation known as convolution: 2

y k ¼ X

n ¼ N

u n þ 1 h k n : ð 2 : 6 Þ

n ¼ 0

While deriving (2.6) we have used shift invariance properties. The above convolu-
tion operation is represented symbolically as y k ¼ u k h k . The binary operator
which represents convolution is linear and follows all the associative and commu-
tative properties. In the frequency domain, convolution takes the form of multi-
plication, Y ð j

!Þ¼ U ð j !Þ H ð j !Þ; where all the j ! functions are corresponding

frequency domain functions.

2 y ð t Þ¼ R 1 1 u ðÞ h ð t Þ d :