U. Glaeser

frequency components in excess B Hz. Another early enabler of the DSP revolution was the Cooley–Tukey
fast Fourier transform (FFT) algorithm. The FFT made many signal-processing tasks practical for the first
time using, in many instances, only software. Another defining DSP moment occurred when the first
DSP microprocessors (DSP µ p) made a marketplace appearance beginning in the late 1970s. These devices
provided an affordable and tangible means of developing hardware and embedded solutions with a
minimum risk and effort. Regardless of the origins, today’s DSP objects and systems have become part
of a pervasive technology, appearing in a myriad of applications, and supported with a rich and deep
technological infrastructure. DSP is now a discipline unto itself, with its own professional societies,
academic programs, trained practitioners, and industrial infrastructure.
24.2 Digital Signals and Systems
Digital systems process digital signals in the time or frequency-domain. Systems can be analyzed and
characterized by the system’s response to a pure impulse (i.e., δ[
k]),
called the impulse response denoted
h[
k]
= { h[0],
h[1],
h[2],…}.
The sequence of sample values h[
k],
called a time-series,
can also be mathematically
represented using a z-transform.
The z-transform
of an arbitrary time-series x[
k],
consisting of
∞ – k
sample values x[
k]
= { x[0],
x[1],
x[2],…},
is given by X(
z)
= Σk=0 x [ k]z.
The z operator is defined in
terms of the Laplace transform delay operator, namely z = e , where Ts
is the sample period. The z-
transforms of common signals are reported in standard table of z-transforms,
such as those shown in
Table 24.1. The common signals shown in Table 24.1 can be manipulated and combined, using the
property list shown in Table 24.2, to synthesize higher-order and more complex signals. In addition to
the properties listed in Table 24.2, the initial value theorem x[0]
= and the final value theorem
x[
∞]
=
provide a convenient means of evaluating two end points of a time-series. The
mapping of a z-transformed
signal X(
z)
back into the time-domain is performed in a piecemeal manner.
Specifically, the inverse z-transform
X(
z)
is normally expressed in partial fraction, or Heaviside expansion
having the form X(
z)
=
, where Xi(
z)
is an element of Table 24.1 corresponding to a discretetime
signal xi[
k],
with Ai
is a Heaviside coefficient associated with the term Xi(
z).
The inverse z-transform
of X(
z)
is given by x[
k]
=
.
sT – s
lim X( z)
z→∞
lim ( z – 1)X(
z)
z→∞
M
Σi=1 AiX i( z)
M
Aix i[ k]
© 2001 by CRC Press LLC
TABLE 24.1
Σ i=1
z-Transforms
of Primitive Time Functions
Discrete-time signal x[
k]
z-transform
X(
z)
δ[
k]
(impuise) 1
u[
k]
(unit step)
z/
( z − 1)
k
a u[
k]
(exponential)
z/
( z − a)
sin[ bkTs]
u[
kTs]
(sine wave)
cos[ bkTs]
u[
kTs]
(cosine wave)
k
a sin(bkTs)u[kTs] (damped sine)
2
sin( bTs)
z/
( z − 2z
cos( bTs)
+ 1)
2
( z − cos( bTs))
z/
( z − 2z
cos( bTs)
+ 1)
a sin(bTs)z/(z 2 − 2az cos(bTs) + a 2 a
)
k cos(bkTs)u[kTs] (damped cosine) (z − a cos(bTs))z/(z 2 − 2az cos(bTs) + a 2 )
TABLE 24.2 Properties of z-Transforms
Property Time-Series z-Transform
Linearity x1[k] + x2[k] X1(z) + X2(z) Real scaling ax[k] aX(z)
Complex scaling w k x[k] X(z/w)
Time reversal x[−k] X(1/z)
Modulation e −ak x[k] X(e a z)
Summation k
x[ n]
zX(z)/(z − 1)
∑
n= – ∞
Shift delay x[k − 1] z −1 X(z) − zx[0]