Time Domain Representation of Continuous and Discrete Signals 17
where A is a real number and represents its amplitude, N the period given by an
integer, α the phase angle in radians or degrees, and 2 π/N its angular frequency in
radians.
R.1.42 Clearly, a discrete time sequence may or may not be periodic. A discrete sequence
is periodic if f(n) = f(n + N), for any integer n, or if 2 π/N can be expressed as rπ,
where r is a rational number.
R.1.43 For example, cos(3n) is not a periodic sequence since 3 = rπ, and clearly r cannot
be a rational number. On the other hand, consider the sequence cos(0.2π n), that is
periodic since 0.2π = rπ or r = 0.2 = 2/10, where r is clearly a rational number, then
the period is given by N = 2 π/ 0.2π or N = 10.
R.1.44 Observe that for the case of a continuous time sinusoidal function of the form f(t) =
A cos(wot), f(t) is always periodic, with period T = 2 π/wo, for any wo.
R.1.45 The most important signal, among the standard signals used in circuit analysis,
electrical networks, and linear systems, in general, is the sinusoidal wave, in either
of the following forms:
f(t) = sin(wt)
f(t) = cos(wt)
or most effective as a complex wave
f(t) = ejwt = cos(wt) + j sin(wt) (Euler’s identity)
R.1.46 Let fn(t) be the family of exponential signals of the form
fn(t) = ejwnt
where
wn = nw 0 , for n = 0, ±1, ±2, ..., ±∞
where w 0 is called the fundamental frequency, wn’s are called its harmonic fre-
quencies (see Chapter 4, where w 0 = 2 π/T). This family possesses the property
called orthogonal, which means that the following integral over a period shown
for the products of any two members of the family is either zero or a constant
given by 2 π/w 0ft f tdtwnmnmn
ww
moooπ π//
(). ()/
∫
*
2
0
forforwhere fm(t)* denotes the complex conjugate of fm(t). For example, if fm(t) = ejwnt, then
fm(t)* = e−jwnt. For the special case in which the orthogonal constant is one, the family
is called orthonormal.
R.1.47 There are a number of orthonormal families. Some of the most frequently used
orthonormal families in system analysis are
a. Hermite
b. Laguerre
c. sinc (where sincn(t) = sin(t − nπ)/[π(t − nπ)])
R.1.48 The Hermitian orthonormal family of signals are generated starting from the
Gaussian signal
Her 0 = e−[t^2/4]
and all other members are generated by successive differentiations with respect to t.