Discrete-time Fourier series and transform - Continuous and Discrete Time Signals and Systems - page 472

D n

0.13 0.11 0.13 0.11

0.09 0.07 0.06 0.050.050.050.050.050.05 0.06 0.07 0.09

0.13 0.11

0.11 0.11 0.11

0.05 0.06 0.07 0.09

0.09 0.07 0.06 0.050.050.050.050.050.05 0.06 0.07 0.09 0.09 0.07 0.06 0.05

n

−20 −18 −16 −14 −12 −10 −8 −6 −4 −2 20

0 2468 0 2 4 16 18

(a)

< D n

0.51 0.51

0.36

0.51 0.51

0.36

0.51
0.44

0.51
0.36

0.44
0.33

0.44

0.33 0.33

0.07 0.21 0.07 0.21

−14 −12 −10 −8 0 24 6 16 18 20

n

−20 −18 −16 −6 −4 −2 8 0 2 4

−0.2 −0.07 −0.2 −0.07

−0.33 −0.33 −0.33

−0.36 −0.44 −0.36 −0.44 −0.36 −0.44

−0.51 −0.51 −0.51 −0.51 −0.51 −0.51

(b)

Fig. 11.3. (a) Magnitude
spectrum and (b) phase
spectrum of the DTFS
coefficients in Example 11.3.

where the greatest common divisor between the fundamental period N and the
integer constant m is one.

Solution
We first show that the DT sequence x [ k ] is periodic and determine its fundamen-
tal period. It was mentioned in Proposition 1.1 that a DT complex exponential
sequence x [ k ] = exp(j( Ω 0 k + θ )) is periodic if 2 π/ Ω 0 is a rational number. In
this case, 2 π/ Ω 0 = N / m , which is a rational number as m , K and N are all

integers. In other words, the sequence x [ k ] is periodic. Using Eq. (1.8), the
fundamental period of x [ k ] is calculated to be

K 0 = (2 π/ Ω 0 ) p = pN / m ,

where p is the smallest integer that results in an integer value for K 0 . Note
that the fraction N / m represents a rational number, which cannot be reduced
further since the greatest common divisor between m and N is given to be one.
Selecting p = m , the fundamental period is obtained as K 0 = N .
To compute the DTFS coefficients, we express x [ k ] as follows:

A e j θ e j 0 mk x [ k ] =

and compare this expression with Eq. (11.4). For 0 ≤ n ≤ K 0 − 1, we observe

that

A e j θ

= m

if n

D n = (11.15)

= m .

i f n

As a special case, we consider A = 2 , K 0

= 6 , m = 5, and θ = π/ 4. The
magnitude and phase spectra for the selected values are shown in Figs.
11.4(a) and (b), where we have used the periodicity property of the DTFS