Discrete-time Fourier series and transform - Continuous and Discrete Time Signals and Systems - page 496

Solution
Using Table 11.2, the DTFT of cos( Ω 0 k )isgivenby

m =−∞ [ δ ( Ω

∞

DTFT
←−−→

cos( Ω 0 k ) π +

Ω 0 − 2 m π ) + δ ( Ω

−

Ω 0 − 2 m π )] .

Using the frequency-shifting property,

m =−∞ [ δ ( Ω

∞

DTFT
←−−→ cos ( Ω 0 k )e j Ω 1 k π +

Ω 0 − Ω 1 − 2 m π )

+ δ ( Ω − − − 2 m π )]

Ω 0 Ω 1

and

m =−∞ [ δ ( Ω

∞

DTFT
←−−→

− j Ω 1 k

cos( Ω 0 k )e π + + − 2 m π )

Ω 0 Ω 1

+ δ ( Ω − + − 2 m π )] .

Ω 0 Ω 1

Adding the two DTFT pairs and noting that [exp( j Ω 1 k ) + exp( − j Ω 1 k )] =
2 cos( Ω 1 k ), we obtain

m =−∞ [ δ ( Ω

∞

←−−→ π
2

DTFT

cos( Ω 0 k ) cos( Ω 1 k ) +

Ω 0 − Ω 1 − 2 m π )

+ δ ( Ω − − − 2 m π )

Ω 0 Ω 1

+ δ ( Ω + + − 2 m π )

Ω 0 Ω 1

+ δ ( Ω − + − 2 m π )] .

Ω 0 Ω 1

The above DTFT can also be obtained by expressing

2 cos( Ω 0 k ) cos( Ω 1 k ) = cos[( Ω 0 +

Ω 1 ) k ] + cos[( Ω 0 − Ω 1 ) k ]

and calculating the DTFT of the right-hand side of the above expression.

11.5.7 Time differencing

The time differencing in the DT domain is the counterpart of differentiation in
the CT domain. The time-differencing property is stated as follows.
If

DTFT
←−−→ x [ k ] X ( Ω )

then

DTFT
←−−→

− j Ω ] X ( Ω ) .

x [ k ] − x [ k − 1] [1 − e (11.44)

The proof of Eq. (11.44) follows directly from the application of the time-
shifting property.