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Discrete-time Signals 33
The quantity T s is called the sampling interval, and Ω 0 = ω 0 /T s is called
the analog frequency, measured in radians per second.
The fact that n is a discrete variable, whereas t is a continuous
variable, leads to some important differences between discrete-time and
continuous-time sinusoidal signals.
Periodicity in time From our definition of periodicity, the sinusoidal
sequence is periodic if
x[n + N] =A cos(ω 0 n + ω 0 N + θ) =A cos(ω 0 n + θ 0 )=x[n] (2.1)
This is possible if and only if ω 0 N =2πk, where k is an integer. This
leads to the following important result (see Problem P2.5):
The sequence x(n) =A cos(ω 0 n + θ 0 )isperiodic if and only if f 0
△
=
ω 0 /2π = k/N, that is, f 0 is a rational number. If k and N are a
pair of prime numbers, then N is the fundamental period of x(n) and
k represents an integer number of periods kT s of the corresponding
continuous-time sinusoid.
Periodicity in frequency From the definition of the discrete-time sinusoid,
we can easily see that
A cos[(ω 0 + k2π)n + θ 0 ]=A cos(ω 0 n + kn2π + θ 0 )
= A cos(ω 0 n + θ 0 )
since (kn)2π is always an integer multiple of 2π. Therefore, we have the
following property:
The sequence x(n) =A cos(ω 0 n + θ) isperiodic in ω 0 with fundamental
period 2π and periodic in f 0 with fundamental period one.
This property has a number of very important implications:
1. Sinusoidal sequences with radian frequencies separated by integer multiples
of 2π are identical.
2. All distinct sinusoidal sequences have frequencies within an interval of
2π radians. We shall use the so-called fundamental frequency ranges
−π