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 2π/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 2π/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 = 2π/ 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(w o t), f(t) is always periodic, with period T = 2π/w o , for any w o .
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) = e
jwt
= cos(wt) + j sin(wt) (Euler’s identity)
R.1.46 Let f n (t) be the family of exponential signals of the form
f n (t) = e
jwnt
where
wn = nw 0 , for n = 0, ±1, ±2, …, ±∞
where w 0 is called the fundamental frequency, wn’s are called its harmonic frequencies (see Chapter 4, where w 0 = 2π/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 2π/w 0
f t f
t dt
w
n m
n m
n
w
w
m
o
o
o
Ϫ
π
π
/
/
( ) .
( )
/
∫
*
ϭ
ϭ
2
0
for
for
where f m (t)* denotes the complex conjugate of f m (t). For example, if f m (t) = e
jwnt
, then
f m (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 sinc n (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.
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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 2π/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 2π/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 = 2π/ 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(w o t), f(t) is always periodic, with period T = 2π/w o , for any w o .
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) = e
jwt
= cos(wt) + j sin(wt) (Euler’s identity)
R.1.46 Let f n (t) be the family of exponential signals of the form
f n (t) = e
jwnt
where
wn = nw 0 , for n = 0, ±1, ±2, …, ±∞
where w 0 is called the fundamental frequency, wn’s are called its harmonic frequencies (see Chapter 4, where w 0 = 2π/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 2π/w 0
f t f
t dt
w
n m
n m
n
w
w
m
o
o
o
Ϫ
π
π
/
/
( ) .
( )
/
∫
*
ϭ
ϭ
2
0
for
for
where f m (t)* denotes the complex conjugate of f m (t). For example, if f m (t) = e
jwnt
, then
f m (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 sinc n (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.
CRC_47760_CH001.indd 17
CRC_47760_CH001.indd 17
7/25/2008 4:16:13 PM
7/25/2008 4:16:13 PM