Introduction - Introduction to Digital Signal Processing and Filter Design - page 14

When σ 0 = 0,

the real and imaginary parts of this complex exponential

sequence are | A | cos (ω 0 n + φ) and | A |

sin (ω 0 n

+ φ) , respectively, and are real

sinusoidal sequences with an amplitude equal to | A | .When σ 0 > 0, the two

sequences increase as n →∞ and decrease when σ 0 < 0as n →∞ .When

e σ 0 n .

ω 0 = φ = 0, the sequence reduces to the real exponential sequence | A |

1.3.7 Properties of cos( ω 0 n )

e jω 0 n

When A j sin (ω 0 n) .

This function has some interesting properties, when compared with the
continuous-time function e jω 0 t and they are described below.
First we point out that ω 0 in x(n)

= 1, and σ 0 = φ = 0, we get x(n) = = cos (ω 0 n) +

e jω 0 n

=
1 /T ,where f s is the sampling frequency in hertz and T is the sampling period
in seconds, specifically, ω 0 = 2 πf 0 /f s

= is a frequency normalized by f s

ω 0 T ,where ω 0 =2 πf 0 is the actual real
frequency in radians per second and f 0 is the actual frequency in hertz. Therefore
the unit of the normalized frequency ω 0 is radians. It is common practice in
the literature on discrete-time systems to choose ω as the normalized frequency
variable, and we follow that notation in the following chapters; here we denote
ω 0 as a constant in radians. We will discuss this normalized frequency again in
a later chapter.

=

e jω 0 n , two frequen-
cies separated by an integer multiple of 2 π are indistinguishable from each
other. In other words, it is easily seen that e jω 0 n

Property 1.1 In the complex exponential function x(n) =

e j(ω 0 n + 2 πr) . The real part and =

e jω 0 n , which are sinusoidal functions,
also exhibit this property. As an example, we have plotted x 1 (n)

the imaginary part of the function x(n) =

= cos ( 0 . 3 πn)

and x 2 (n) = cos ( 0 . 3 π + 4 π)n in Figure 1.8. In contrast, we know that two

e jω 2 t or their real and imag-
inary parts are different if ω 1 and ω 2 are different. They are different even if they
are separated by integer multiples of 2 π . From the property e jω 0 n

e jω 1 t and x 2 (t) continuous-time functions x 1 (t) = =

e j(ω 0 n + 2 πr)
above, we arrive at another important result, namely, that the output of a discrete-
time system has the same value when these two functions are excited by the
complex exponential functions e jω 0 n or e j(ω 0 n + 2 πr) . We will show in Chapter 3
that this is true for all frequencies separated by integer multiples of 2 π ,and
therefore the frequency response of a DT system is periodic in ω .

=

Property 1.2 Another important property of the sequence e jω 0 n is that it is
periodic in n . A discrete-time function x(n) is defined to be periodic if there
exists an integer N such that x(n x(n) ,where r is any arbitrary integer

and N is the period of the periodic sequence. To find the value for N such that
e jω 0 n

+ rN) =

is periodic, we equate e jω 0 n to e jω 0 (n + rN) . Therefore e jω 0 n e jω 0 n e jω 0 rN , =

which condition is satisfied when e jω 0 rN = 1, that is, when ω 0 N = 2 πK ,where