Handbook for Sound Engineers

DSP Technology 1163

together on a sample-by-sample basis. Other systems,
such as an MPEG audio compression system may oper-
ate on frames of data that have 1152 samples in each
frame. The choice of whether to operate sample-
by-sample or frame-by-frame is made by the system
designer and algorithm developer.

A fundamental system is the ideal delay. The ideal
delay system delays or advances a sequence by the
delay amount. This system is defined by the equation

(31-5)

where,

nd is an integer that is the delay of the signal.

The ideal delay system creates an output y[n] by

shifting the input signal, x, by nd samples to the right

when nd is positive. This means that the value of the

output signal y[n] at a particular index n is the value of

the input signal at index n–nd. For example if the

signal is delayed by three samples, then nd=3 and the

output value y[7] is equal to the value of x[4]—i.e., the

value of x[k] at k= 4 now appears at y[ j], j=7. The

system shifted the input signal three samples to the right

as shown in Fig. 31-5.

The moving average system takes an average of the

input signal over some window and then moves to the

next sample and takes an average over the new window,

etc. The general moving average system is defined by

the equation below, where M 1 and M 2 are positive inte-

gers. It is called a moving average because to compute

each output, y[n], the filter must be moved to the next

index and the average recomputed.

(31-6)

Figure 31-3. A cosine sequence of period 16. This particular cosine sequence is an infinite sequence of values that repeat

with a period of 16 samples.

0 n

1

cos ( 162 )

16

πn

Figure 31-4. The product of the cosine sequence with the
unit step sequence, u[n]. Notice that all the signal values
for n < 0 are set to 0.

Table 31-1. The Values of the Signal x[n] in Figure

31-4

x[0] 1.0000 x[6] 0.7071 x[12] 0.0000

x[1] 0.9239 x[7] 0.9239 x[13] 0.3827

x[2] 0.7071 x[8] 1.0000 x[14] 0.7071

x[3] 0.3827 x[9] 0.9239 x[15] 0.9239

x[4] 0.0000 x[10] 0.7071 ...

x[5] 0.3827 x[11] 0.3827

0 n

1

u[n]

16

cos ( 162 πn)

yn>@=xn n>@f– d– n f

Figure 31-5. The cosine signal from Fig. 31-4 delayed by

nd= 3 samples. This delay shifts the sequence to the right

by three samples.

0 n

1 16 (^3 )] [^3 ]

cos [^2 πn un

16

yn>@

1

M 1 ++M 2 1

------------------------------ xn k>@–

kM=– 1

M 2

= ¦