Discrete-time Fourier series and transform - Continuous and Discrete Time Signals and Systems - page 476

Fig. 11.6. (a) Time-limited
sequence x [ k ]; (b) its periodic
extension.

x [ k ]

k

M 1 0 M 2

(a)

˜

x K 0 [ k ]

k

K 0

− K 0 M 1 0 M 2

(b)

Since x K 0 [ k ] is periodic with fundamental period K 0 (or fundamental frequency
Ω 0 ), we can express it using the DTFS as follows:

n = K 0 D n e j n Ω 0 k ,

x K 0 [ k ] = (11.22)

where the DTFS coefficients D n are given by

k = K 0 x K 0 [ k ]e

1
K 0

− j n Ω 0 k ,

D n =

for 1 ≤ n ≤ K 0 . Using Eq. (11.21), the above equation can be expressed as

follows:

k =−∞ x [ k ]e

∞

1
K 0

− j n Ω 0 k

D n = lim

K 0 →∞

(11.23)

for 1 ≤ n ≤ K 0 . Let us now define a new function X ( Ω ), which is continuous
with respect to the independent variable Ω :

k =−∞ x [ k ]e

∞

− j Ω k .

X ( Ω ) = (11.24)

In Eq. (11.24), the independent variable Ω is continuous in the range −∞ ≤
Ω

≤∞ . In terms of X ( Ω ), Eq. (11.23) can be expressed as follows:

1
K 0

D n = lim

K 0 →∞

X ( n Ω 0 ) . (11.25)

= n Ω 0 .
Given the DTFS coefficients D n of x K 0 [ k ] , the aperiodic sequence x [ k ] can
be obtained by substituting the values of D n in Eq. (11.22) and solving for
M 1

The function X ( n Ω 0 ) is obtained by sampling X ( Ω ) at discrete points Ω

≤ k ≤ M 2 . The resulting expression is given by

1
K 0 X ( n Ω 0 )e j n Ω 0 k .

x [ k ] =

K 0 →∞ x K 0 [ k ] =

lim lim

K 0 →∞

(11.26a)

n = K 0