Discrete Fourier transform - Continuous and Discrete Time Signals and Systems - page 528

N

A

x w [ k ] X w ( W )

2 p T 1

k W

012 … N − 1 −4 p −2 p 0 2 p 4 p

(k) (l)

S 2 ( W ) s 2 [ k ]

2 p

1

k W

0 −4 p −2 p 2 p 4 p

− M M 0

2 p

M

(m) (n)

N

x 2 [ k ]

X w ( W )

A

MT 1

k

W

012 … 0

− M N − 1 M

−4 p −2 p 0 2 p 4 p

2 p

M

(o) (p)

N

x 2 [ k ] X 2 [ r ]

MT 1

A

r

k

0012 … − M N − 1 M −2 M − M 0 M 2 M

(q) (r)

derive the DTFT of the DT sequence x 1 [ k ] as follows:

Fig. 12.1. ( cont. )

x 1 [ k ] = ∞

x ( mT 1 ) δ ( t − mT 1 ) . (12.4)

m =−∞

Calculating the CTFT of both sides of Eq. (12.4) yields

∞

x ( mT 1 )e − j ω mT 1 .

X 1 ( ω ) =

(12.5)

m =−∞

Substituting x 1 [ m ] = x ( mT 1 ) and Ω = ω T 1 in Eq. (12.5) leads to

∞

x 1 [ m ]e − j m Ω ,

X 1 ( Ω ) = X 1 ( ω ) ω= Ω / T 1 =

m =−∞