Page 77 - DSP Integrated Circuits

62 Chapter 3 Digital Signal Processing

used to encode the information and to exploit different types of signal carriers and
circuit devices. The simplest type of modulation is to let the signal carrier (voltage)
vary according to the continuous time signal, e.g., a speech signal. Such a signal,
which varies continuously both over a range of signal values and in time, is called
an analog signal. An analog signal is denoted

The signal may be a complex function in the complex domain, but usually both
y and t are real quantities. Note that we usually refer to the independent vari-
able(s) as "time" even if in certain applications the variable may represent, for
example, the spatial points (pixels) in an image. It is not a trivial matter to extend
the theory for one-dimensional systems to two- or multidimensional systems, since
several basic properties of one-dimensional systems do not have a direct corre-
spondence in higher-dimensional systems.
In many cases, the signal does not vary continuously over time. Instead, the
signal is represented by a sequence of values. Often, these signals are obtained
from measurements (sampling) of an analog quantity at equidistant time
instances. A sampled signal with continuously varying signal values is called a
discrete-time signal:

where T is an associated positive constant. If the sequence is obtained by sampling
an analog signal, then T is called the sample period.
If signal values are restricted to a countable set of values, the corresponding
signal is referred to as a digital signal or sequence:

Unfortunately, it is common practice not to distinguish between discrete-time
and digital signals. Digital signals are, in principle, a subset of discrete-time signals.
Digital signals are usually obtained by measurements of some physical quantity
using an A/D converter with finite resolution.
We will frequently make use of the following special sequences:

The impulse sequence is sometimes called the unit sample sequence. When-
ever convenient, we choose to drop the T and simply write x(ri) instead of x(nT).

3.4 THE FOURIER TRANSFORM

Many signal processing systems exploit the fact that different signals and/or
unwanted noise occupy different frequency bands or have in some other way dif-