Discrete-time Fourier series and transform - Continuous and Discrete Time Signals and Systems - page 474

Based on Eq. (11.5), and noting that Ω 0 = 2 π/ K 0 , the DTFS coefficients are

given by

1
K 0

− jn Ω 0 k

D n = B sin( m Ω 0 k + θ ) e

k =< K 0 >

e j( m Ω 0 k +θ ) − e j( m Ω 0 k +θ )
2 j

1
K 0

− j n Ω 0 k

= B e

k =< K 0 >

k = K 0 e j( m − n ) Ω 0 k k = K 0 e

B
2 K 0

B
2 K 0 e

e j θ

− j θ − j( m + n ) Ω 0 k

=− j + j .

summation I summation II

In proving Proposition 11.2, we used the following summation:

K 0 − 1

K 0 if k = m

e j n Ω 0 ( k − m )

=

= m .

0i f k

n = 0

Therefore, summations I and II are given by

k = K 0 e j( m − n ) Ω 0 k

K 0 if n = m

I = =

0i f n = m ;

k = K 0 e

K 0 if n =− m

− j( m + n ) Ω 0 k

II = =

0i f n =− m ,

which results in the following values for the DTFS coefficients:

− j B

2 e j θ

for n = m

j B

D n = (11.18)

− j θ

2 e

for n =− m

0 elsewhere,

within one period ( − m − m − 1)).

As a special case, let us consider the DTFS for the following discrete sinu-
soidal sequence:

≤ n ≤ ( K 0

2 π
7

k + π
4

y [ k ] = 3 sin ,

which has a fundamental period of K 0 = 7. Substituting B = 3 , m = 1, and

θ = π /4 into Eq. (11.18), we obtain

− j 3

2 e j 4

for n = 1

j 3

− j π
4

D n = (11.19)

2 e

for n =− 1

0 elsewhere,

for − 1 ≤ n ≤ 5. The magnitude and phase spectra for the sinusoidal sequence
are shown in Figs. 11.5(a) and (b).