Introduction to Acoustics

524 Part D Hearing and Signal Processing
Part D 14.12
limits,
X(ω)
TD/2 �
dt
TD
−TD/2
′ e −iωt′
= 1
×
(N−1)/2
�
n=−(N−1)/2
14.12 The z-Transform
inωot ′
X(nωo) e e −iωTD/2 inωoTD/2
e ,
(14.138)
Like the discrete Fourier transform, the z-transform is
well suited to describing sampled signals. We consider
x(t) to be a sampled signal so that it is defined at discrete
time points t = tk = kTs,whereTs is the sampling period.
Then the time dependence of x can be described by an
index, xk = x(tk). The z-transform of x is
X(z) =
∞�
k=−∞
xkz −k . (14.140)
The quantity z is complex, with amplitude A and
phase ϕ,
z = A e iϕ = A e iωTs , (14.141)
where ϕ is the phase advance in radians per sample.
In the special case where A = 1, all values of z lie
on a circle of radius 1 (the unit circle) inthecomplex
z plane. In that case the z-transform is equivalent to the
discrete Fourier transform. An often-overlooked alternative
view is that the z-transform is an extension of
the Fourier transform wherein the angular frequency ω
becomes complex,
so that
ω = ωR + iωI , (14.142)
z = e −ωITs e iωRTs . (14.143)
The extended Fourier transform will not be pursued
further in this chapter.
A well-defined z-transform naturally includes a function
of variable z, but the function itself is not enough.
In order for the inverse transform to be unique, the definition
also requires that the region of the complex plane
in which the transform converges must also be specified.
which reduces to
X(ω) =
(N−1)/2
�
sin[(ω − nωo)TD/2]
X(nωo)
(ω − nωo)TD/2
n=−(N−1)/2
×e −iωTD/2 iπn
e . (14.139)
Table 14.2 z-Transform pairs
xk X(z) Radius of
convergence
δk,k 0 z −k 0 all z
a k uk z/(z − a) |z| > a
ka k uk az/(z − a) 2 |z| > a
a k cos(ωoTsk)uk
a k sin(ωoTsk)uk
z 2 −az cos(ωoTs)
z 2 −2az cos(ωoTs)+a 2
az sin(ωoTs)
z 2 −2az cos(ωoTs)+a 2
|z| > a
|z| > a
To illustrate that point, one can consider two different
functions xk that have the same z-transform function,
but different regions of convergence.
Consider first the function
xk = 2 k
xk = 0 for k < 0 .
for k ≥ 0 , (14.144)
This two-line function can be written as a single line by
using the discrete Heaviside function uk. The function
uk is defined as zero when k is a negative integer and as
+1whenk is any other integer, including zero. Then xk
above becomes
xk = 2 k uk . (14.145)
The z-transform of xk is
X(z) =
∞�
(2/z) k . (14.146)
k=0
The sum is a geometric series, which converges to
X(z) =
1
= z/(z − 2) (14.147)
1 − 2/z