Page 71 - Fundamentals of Communications Systems

Signals and Systems Review 2.23

2.4 Utilizing Matlab
The signals that are discussed in a course on communications are typically
defined over a continuous time variable, e.g., x(t). Matlab is an excellent package
for visualization and learning in communications engineering and will be used
liberally throughout this text. Unfortunately Matlab uses signals that are de-
fined over a discrete time variable, x(k). Discrete time signal processing basics
are covered in the prerequisite signals and linear systems and comprehensive
treatments are given in [Mit98, OS99, PM88, Por97]. This section provides a
brief discussion of how to transition between the continuous time functions
(communication theory) and the discrete time functions (Matlab). The exam-
ples considered in Matlab will reflect the types of signals you might measure
when testing signals in the lab.

2.4.1 Sampling
The simplest way to convert a continuous time signal, x(t), into a discrete time
signal, x(k), is to sample the continuous time signal, i.e.,

x(k) = x(kT s + )
1
where T s is the time between samples. The sample rate is denoted f s = . This
T s
conversion is an important part of analog–to–digital conversion (ADC) and is
a common operation in practical communication system implementations. The
discrete time version of the signal is a faithful representation of the continuous
time signal if the sampling rate is high enough.
To see that this is true it is useful to introduce some definitions for discrete
time signals.

Definition 2.16 For a discrete time signal x(k), the discrete time Fourier transform
(DTFT) is
∞
j 2π fk
j 2π f
X(e ) = x(k)e (2.80)
k=−∞
For clarity when the frequency domain representation of a continuous time
signal is discussed it will be denoted as a function of f , e.g., X(f ) and when
the frequency domain representation of a discrete time signal is discussed it
will be denoted as a function of e j 2π f , e.g., X(e j 2π f ). This DTFT is a continuous
function of frequency and since no time index is associated with x(k), the range
of where the function can potentially take unique values is f ∈ [−0.5, 0.5].
Matlab has built-in functions to compute the discrete Fourier transform (DFT),
which is simply the DTFT evaluated at uniformly spaced points.
For a sampled signal, the DTFT is related to the Fourier transform of the
continuous time signal via [Mit98, OS99, PM88, Por97].
∞
1 f − n
X(e j 2π f ) = X (2.81)
T s T s
n=−∞