Discrete Fourier transform - Continuous and Discrete Time Signals and Systems - page 529

which is the standard definition of the DTFT introduced in Chapter 11. The
DTFT spectrum X 1 ( Ω )of x 1 [ k ] is obtained by changing the frequency axis
ω of the CTFT spectrum X 1 ( ω ) according to the relationship Ω

= ω T 1 . The

DTFT spectrum X 1 ( Ω ) is illustrated in Fig. 12.1(h).

Step 2: Time limitation The discretized signal x 1 [ k ] can possibly be of infi-
nite length. Therefore, it is important to truncate the length of the discretized
signal x 1 [ k ] to a finite number of samples. This is achieved by multiplying the
discretized signal by a rectangular window,

10 ≤ k ≤ ( N − 1)
0

w [ k ] = (12.6)

elsewhere,

of length N . The DTFT X w ( Ω ) of the time-limited signal x w [ k ] = x 1 [ k ] w [ k ]
is obtained by convolving the DTFT X 1 ( Ω ) with the DTFT W ( Ω ) of the rect-
angular window, which is a sinc function. In terms of X 1 ( Ω ), the DTFT X w ( Ω )
of the time-limited signal is given by

1
2 π

sin(0 . 5 N Ω )
sin(0 . 5 Ω )

e − j( N − 1) / 2

X w ( Ω ) = X 1 ( Ω ) ⊗ , (12.7)

which is shown in Fig. 12.1(l) with its time-limited representation x w [ k ] plotted
in Fig. 12.1(k). Symbol ⊗ in Eq. (12.7) denotes the circular convolution.

Step 3: Frequency sampling The DTFT X w ( Ω ) of the time-limited signal
x w [ k ] is a continuous function of Ω and must be discretized to be stored on a
digital computer. This is achieved by multiplying X w ( Ω ) by a frequency-domain
impulse train, whose DTFT is given by

Ω −

∞

2 π
M

2 π m
M

S 2 ( Ω ) = δ . (12.8)

m =−∞

The discretized version of the DTFT X w ( Ω ) is therefore expressed as follows:

1
M

sin(0 . 5 N Ω )
sin(0 . 5 Ω )

e − j( N − 1) / 2 X 2 ( Ω ) = X w ( Ω ) S 2 ( Ω ) = X 1 ( Ω ) ⊗

∞

2 π m
M

δ

Ω −

. (12.9)

m =−∞

The DTFT X 2 ( Ω ) is shown in Fig. 12.1(p), where the number M of frequency
samples within one period ( −π ≤ Ω ≤ π )of X 2 ( Ω ) depends upon the funda-
mental frequency Ω 2 = 2 π/ M of the impulse train S 2 ( Ω ). Taking the inverse
DTFT of Eq. (12.9), the time-domain representation x 2 [ k ] of the frequency-
sampled signal X 2 ( Ω )isgivenby

∞

x 2 [ k ] = [ x w [ k ] ∗ s 2 [ k ]] = [ x 1 [ k ] w [ k ]] ∗ δ ( k − mM ) , (12.10)

m =−∞

and is shown in Fig. 12.1(o).