PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

DTFT, DFT, ZT, and FFT 471

f. Convolution in time

f 1 (n) ⊗ f 2 (n) ↔ F 1 (ejW). F 2 (ejW)

g. Convolution in frequency

fnfn 12 Fe 1 jW Fe 2 jW

1

2

() ()↔⋅{ ( )⊗( )} Fe Fejjd



^ 



(^1) W
2

12 () (())

nn

n

∫

h. Parseval’s theorem is given by











fn Fe dW

n

()^221 ( )jW


 ^2







∑ 
 ∫

i. The initial time value is given by

fFedW() ( )0jW

1

2





 



∫

j. The initial frequency value is given by

Fej f n

n

()^0
()

 



∑

k. Subtraction of shifted sequences

f(n) − f(n − 1) ↔ [1 − ejW]. F(ejW)

R.5.39 A partial list of the most frequently used transform pairs are summarized in

Table 5.1.

TABLE 5.1

DTFT Pairs

Time Frequency

f(n) F(e jW)

δ(n) 1

δ(n − n 0 )e−jn^0 W

e−jw 0 n 2 πδ(W − w 0 )

1 2 πδ(W)

cos(w 0 n) π[δ(W + w 0 ) + δ(W − w 0 )]

sin(w 0 n) jπ[δ(W + w 0 ) − δ(W − w 0 )]

u(n) [1/1 − e−jw] + πδ(W)

anu(n) 1/(1 − ae−jW), (|a|< 1)

(a + 1)anu(n) 1/(1 − e−jw)^2 , (|a|< 1)

___________(nn^ !(+k^ k −^ − 1)!1)! anu(n) __________^1

(1 − ae−jW)k

, (|a| < 1)