Handbook for Sound Engineers

1162 Chapter 31

software is developed—subroutine by subroutine (one
level of abstraction at a time). This section will intro-
duce some fundamental systems and also introduce the
useful properties that some systems possess.

31.3.1 Sequences

Discrete-time signals, also called sequences, are most
often created by sampling analog, or continuous-time,
signals. By sampling a continuous-time signal, a
sequence of samples, really a sequence of numbers, can
be processed and manipulated in a digital signal proces-
sor. Before going further into the sampling process, an
introduction to signal and system theory will be pre-
sented, starting with discrete-time signals.

Discrete-time signals are represented mathematically
as a sequence of numbers. The notation used will denote
a sequence, x, as x={x[n]} where n is the index of the
nth element in the sequence. In terms of notation, x[n]
represents both the nth sample in the sequence and the
entire sequence that is a function of n. The index, n, can
range over all values from –f to +f.
From a programming perspective, a sequence can be
thought of as an infinitely large array of data indexed by
an integer variable. In reality, an infinitely long array is
not practical, so a sequence is usually represented as a
continuous stream of data. Often it is assumed that the
sequence starts at time = 0 (n= 0) and ends some finite
time later (n=M ).

There are several sequences that are fundamental
building blocks of DSP systems. These are the unit
impulse, the unit step sequence, and the sinusoid (cosine
or sine). The unit impulse is a signal that has a value of
1 at index n = 0 and is 0 everywhere else as shown in
Fig. 31-1. Mathematically this is denoted by

(31-1)

Having defined the unit impulse, it is possible to
represent a sequence x[n] as a sum of delayed impulses
that have a value of x[k] at n=k. Mathematically this is
formulated as

(31-2)

which simply says that the value of x[n] is the collection
of its individual samples at time n=k.

The unit step is a signal that starts at index 0 with

value 1 and has value 1 for all positive indices as shown

in Fig. 31-2. Mathematically, this is denoted by

(31-3)

The cosine signal is a sinusoid of frequency Z and

phase IAn example of the cosine signal is shown in

Fig. 31-3. Mathematically, the cosine signal is denoted

by

(31-4)

All sequences can also be represented by the

numbers that are the sample values x[n]. Table 31-1

shows the sample values for the sequence in Fig. 31-4.

Only the first sixteen sample values are listed because

the sequence repeats itself after the 16th value (x[15]).

31.3.2 Systems

Systems transform input signals into output signals.

Some commonly used systems include the ideal delay

system that delays the output relative to the input and

the moving average system that performs some simple

low-pass filtering. Systems operate on a signal by oper-

ating on each sample individually or groups of samples

at a time. For instance, multiplying a sequence by a con-

stant can be implemented by multiplying each sample of

the sequence by the constant. Similarly, the addition of

two sequences is performed by adding the signals

G>@n

0 nz 0

̄^1 n 0=

®

­

=

xn>@ xk>@G>@nk–

k

=¦

Figure 31-1. Unit impulse sequence has a value of 1 at

n= 0 and is 0 everywhere else.

Figure 31-2. The unit step sequence has a value of 1 for

nt0 and is 0 everywhere else.

0 n

1

Dn][

un>@

0 n 0

̄^1 nt^0

®

­

=

0 n

1

nu ][

cos>@n = cos Zn+I