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3.6 Sampling of Analog Signals 65

EXAMPLE 3.2
Determine the z-transform for the sequence:

There are several ways to find the sequence x(n) from X(z). Essentially X(z)
l
must be expanded into a power series in z~ or z or both. The sequence values cor-
respond to the coefficients of the power series. A formal way to obtain these values
directly is to compute the following integral.
The inverse z-transform is

where C is a contour inside the region of convergence, followed counterclockwise
around the origin. The inverse ^-transform can be determined by using residue cal-
culus or by partial fraction expansions in the same way as for the Laplace trans-
form. Expansion in terms of simple geometrical series is sometimes useful [4,6,17,
32, 40]. In practice, the numerical inversion of a z-transform is often done by filter-
ing an impulse sequence through a filter with the appropriate transfer function.

3.6 SAMPLING OF ANALOG SIGNALS

"Mr. Twain, people say you once had a twin brother. Is it true?"
"Oh, yes. But, when we were very young, I died, fortunately,
he has lived on, aliasing my name"
—an anecdote
Most discrete-time and digital signals are in practice obtained by uniform sampling
of analog signals. The information in the analog signal, x a(t), can under certain con-
ditions be retained and also reconstructed from the discrete-time signal. In this
chapter, we will neglect possible loss of information due to finite resolution in the
sample values. How to choose the necessary data word length is discussed in [6].