Handbook for Sound Engineers

1166 Chapter 31

As an example, assume an input signal x[n] is
defined as x[n]=ejZn —i.e., a complex exponential
(Euler’s relationship from complex number theory that
states that e jZn=cos(Zn)+jsin(Zn), where Z is the
radian frequency that ranges from 0 dZdS), then
using the convolution sum of

generates

(31-11)

(31-12)

By defining

we have

where,

H(e jZ represents the phase and amplitude determined

by the system.

This shows that a sinusoidal (or, in this case, the

complex exponential) input to a linear time invariant

system will generate an output that has the same

frequency but with an amplitude and phase determined

by the system.

H(e jZ is known as the frequency response of the

system and describes how the LTI system will modify

the frequency components of an input signal. The trans-

formation

is known as the Fourier transform of the impulse

response, h[n]. If H(e jZ is a low-pass filter, then it has a

frequency response that attenuates high frequencies but

not low frequencies—hence it passes low frequencies. If

H(e jZ is a high-pass filter, then it has a frequency

response that attenuates low frequencies but not high

frequencies.

In many instances it is more useful to process a

signal or analyze a signal from the frequency domain

than in the time domain either because the phenomenon

of interest is frequency based or our perception of the

phenomenon is frequency based.

An example of this is the family of MPEG audio

compression standards that exploits the frequency prop-

erties of the human auditory system to dramatically

reduce the number of bits required to represent the

signal without significantly reducing the audio quality.

31.5 The Z-Transform

The Z-transform is a generalization of the Fourier trans-

form that permits the analysis of a larger class of sys-

Figure 31-7. The output, y[n], from the convolution of x[n]
and h[n] in Fig. 31-6.

Table 31-2. The Result of the Convolution in Fig. 31-7

y[0] 1.0000 y[11] 0.9672 y[22] 0.8984

y[1] 1.4239 y[12] 0.3681 y[23] 1.3731

y[2] 1.4190 y[13] 0.2870 y[24] 1.6387

y[3] 0.9672 y[14] 0.8984 y[25] 1.6548

y[4] 0.3681 y[15] 1.3731 y[26] 1.4190

y[5] 0.2870 y[16] 1.6387 y[27] 0.9672

y[6] 0.8984 y[17] 1.6548 y[28] 0.3681

y[7] 1.3731 y[18] 1.4190 y[29] 0.2870

y[8] 1.6387 y[19] 0.9672 y[30] 0.8984

y[9] 1.6548 y[20] 0.3681 ...

y[10] 1.4190 y[21] 0.2870

0 n

1

y[n]=cos ( 162 n)u[n]•h[n]

16

0 k

1

x[k] =cos (^216 k)u[k]

16

1

h[n ]

0.5

0.25

k

k π

π

yn>@ hk>@xn k>@–

k

=¦

yn>@ hk>@e

jZ nk–

k

=¦

yn>@e

jZn

hk>@e

• jZk

k

©¹ ̈ ̧¦

§·

=

He jZ hk>@e–jZk

k

=¦

yn>@=He jZe^ jZn

He jZ hk>@e–jZk

k

=¦