Handbook for Sound Engineers

DSP Technology 1165

i.e., the sequence x[n] is a sum of scaled and delayed
impulses. If , i.e., the system
response to the delayed impulse at n=k, then the output
y[n] can be formed as

(31-9)

If the system is also time invariant, then
hk[n]=h[n k], and the output y[n] is given by

(31-10)

This representation is known as the convolution sum
and is commonly written as y[n]=x[n]•h[n]. The
convolution system takes two sequences, x[n] and h[n],
and produces a third sequence y[n]. For each value of
y[n], the computation requires multiplying x[k] by
h[n k] and summing over all valid indices for k where
the signals are non-zero. To compute the output
y[n+ 1], move to the next point, n + 1, and perform the
same computation. The convolution is an LTI system
and is a building block for many larger systems.

As an example, consider the convolution of the
sequences in Fig. 31-6 where h[n] has only three
non-zero sample values and x[n] is a cosine sequence
that has non-zero sample values for nt0.

The computation of

is performed as follows. Values of x[n] for n< 0 are 0.
Only the computation for the first three output samples
are shown.

The result of the convolution is shown in Fig. 31-7

and has the sample values shown in Table 31-2.

31.4 Frequency Domain Representation

Having defined an LTI system, it is possible to look at

the signal from the frequency domain perspective and

understand how a system changes the signals in the fre-

quency domain. The frequency domain represents sig-

nals as a combination of various frequencies from low

frequency to high frequency. Each time-domain signal

has a representation as a collection of frequency compo-

nents where each frequency component can be thought

of as sinusoids or tones. Sinusoids are important

because a sinusoidal input to a linear time-invariant sys-

tem generates an output of the same frequency but with

amplitude and phase determined by the system. This

property makes the representation of signals in terms of

sinusoids very useful.

xn>@ xk>@G>@nk–

k

= ¦

hk>@n =T^`G>@nk–

yn>@=Txn^`>@

Txk>@G>@nk–

k

¦

̄¿

®¾

­½

=

xk>@hk>@n

k

= ¦

yn>@ xk>@hn k>@–

k

= ¦

hk>@xn k>@–

k

= ¦

yn>@ h

k 0=

2

= ¦ >@kxn k>@–

Figure 31-6. A convolution example with two sequences.

x[n] is the same signal from Fig. 31-4 with values shown in

Table 31-1, and h[n] has the values shown above.

(^0) n
1
nx ][ = u[n]
16
n
1
hn][
0.5
0.25
01 2
cos (^216 πn)
y>@ 0 =h>@ 0 x>@ 0 ++h>@ 1 x>@1– h>@ 2 x>@2–
=1.0
y>@ 1 =h>@ 0 x>@ 1 ++h>@ 1 x>@ 0 h>@ 2 x>@1–
=1.4239
y>@ 2 =h>@ 0 x>@ 2 ++h>@ 1 x>@ 1 h>@ 2 x>@ 0
=1.4190