Page 72 - Fundamentals of Communications Systems

2.24 Chapter Two

Examining Eq. (2.81) shows that if the sampling rate is higher than twice
the highest significant frequency component of the continuous time signal then
1 f
X(e j 2π f ) = X (2.82)
T s T s
and

X(f ) = T s X(e j 2π( fT s ) ) (2.83)
Note the highest significant frequency component is often quantified through
the definition of the bandwidth (see Section 2.2.3). Consequently the rule of
thumb for sampling is that the sampling rate should be at least twice the band-
width of the signal. A sampling rate of exactly twice the bandwidth of the signal
is known as Nyquist’s sampling rate.

2.4.2 Integration
Many characteristics of continuous time signals are defined by an integral. For
example, the energy of a signal is given in Eq. (2.1) as an integral. The Fourier
transform, the correlation function, convolution, and the power are other exam-
ples of signal characteristics defined through integrals. Matlab does not have
the ability to evaluate integrals but the values of the integral can be approx-
imated to any level of accuracy desired. The simplest method of computing
an approximation solution to an integral is given by the Riemann sum first
introduced in calculus.

Definition 2.17 A Riemann sum approximation to an integral is
b − a k(b − a)
b N N
x(t)dt ≈ x a − + = h x(k) (2.84)
a N N
k=1 k=1
b−a
where ∈ [−(b − a)/N , 0] and h = is the step size for the sum.
N
A Riemann sum will converge to the true value of the finite integral as the
number of points in the sum goes to infinity. Note that the DTFT for sampled
signals can actually be viewed as a Riemann sum approximation to the Fourier
transform.
2.4.3 Commonly Used Functions
This section details some Matlab functions that can be used to implement the
signals and systems that are discussed in this chapter. Help with the details of
these functions is available in Matlab.

Standard Stuff
■ cos — cosine function
■ sin — sine function
■ sinc — sinc function