Handbook for Sound Engineers

1168 Chapter 31

31.6.1 Continuous to Discrete Conversion

The most common method for converting a continu-
ous-time signal, xc(t ̧ into a discrete-time signal, x[n], is
to uniformly sample the signal every T seconds with the
equation

(31-15)

This generates a sequence of samples, x[n], where
the value of x[n] is the same as the value of xc(twhen-
ever t=nT—i.e., at each sampling interval T. 1 eT is
known as the sampling frequency and is usually
expressed in Hertz or cycles per second.
Mathematically, when a continuous-time signal is
sampled, the resulting signal has a frequency response
that is related to the underlying continuous-time signal
frequency response and the sampling rate. As shown
next, this has significant ramifications for how often the
signal must be sampled in order for the digital sequence
to be reconstructed into an analog signal that accurately
represents the original signal.

The sampling process will be analyzed in the
frequency domain where it will be assumed that a band
limited signal, xc(t , is to be sampled periodically with
sample period T. A band-limited signal is one that has
no signal energy higher than a particular frequency, :N,

as shown in Fig. 31-10, where : represents the

frequency axis of the signal. The reason the signal is

assumed to be band limited is to prevent frequency

aliasing, as will be evident shortly. The assumption of

being band limited is significant although generally

easily realizable in real-world systems.

The sampling of the continuous-time signal, xc(t ,

generates a signal, xs(t , from equation

(31-16)

xs(t is the collection of values of xc(t at the sampling

interval of T. A convenient representation of this signal

is as a collection of delayed and weighted impulse func-

tions. The amplitude is the value at the sampling instant

and the samples are spaced out by the sampling period

T. The process can be analyzed in the frequency domain

by first representing the Fourier transform of the

impulse sequence as a sequence of impulses in the fre-

quency domain^6. This means that a sequence of equally

spaced impulses in the time domain have a frequency

representation that is a sequence of equally spaced

impulses in the frequency domain, spaced by the sam-

pling frequency 2Se T. This is shown as

(31-17)

where,

:s=2SeT is the sampling frequency in radians/second.

The Fourier transform of the sampled signal, xs(t ,

becomes

(31-18)

Figure 31-8. A block diagram of an FIR system where the input x[n] is fed into a system that multiplies the delayed input

signal with the filter coefficients bk and sums the results together to form the output y[n].

x[n] z^1 z z^1

x[n 1] x[n 2] x[n  M]

z^1

x[n  M +1]

y[n]

b 0 b 1 bM 1 bM

1

Figure 31-9. Analog-to-digital conversion can be thought

of as a two-step process: converting a continuous-time

signal to a discrete-time signal, x[n], followed by quan-

tizing the sample to create the digital sequence.

Sample

Quantize

T

xc(t)

x[n]

ˆx[n]

xn>@=xc nT, –fn f

xs t xcGnT tnT–

n –= f

f

= ¦

S j:^2 S

T

------ G: – k:s

k –= f

f

= ¦

Xs j:^1

T

--- Xc j :–k:s

k –= f

f

= ¦