PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

DTFT, DFT, ZT, and FFT 493

R.5.104 The properties of DFT are similar to DTFT and ZT. For completeness, the most

important properties are summarized in Table 5.3.

Note that the notation f(n)N = f(n) denotes the interval (0, N − 1), and for any n

outside the interval f(n)N = f(n + rN), for r = 1 , 2 , 3 , .... Recall that the symbol * is

used to denote the complex conjugate.

Note also that the convolution associated with DFT is the circular convolution,

even when in most cases, the linear convolution is the desired end result.

Recall that if f 1 (n) is of length N 1 and f 2 (n) of length N 2 , then the length of the

linear convolution [f 1 (n) ⊗ f 2 (n)] is of length N 1 + N 2 − 1. This would suggest that

the sequences f 1 (n) and f 2 (n) should be appended with N 2 − 1 and N 1 − 1 zeros,

respectively, to make the two sequences of equal length. Then the circular convo-

lution of f 1 (n) with f 2 (n) of the augmented vectors returns the linear convolution

given by f 1 (n) ⊗ f 2 (n).

R.5.105 The DFT and DTFT share many similarities. One of the differences is that DFT

assumes that the operations are circular, which is limited over the range 0 to

N − 1. Recall that DFT is an N-point to N-point transformation.

If the input time sequence is over the range 0 ≤ n ≤ N − 1 , then its transform is

defi ned over the same range 0 ≤ k ≤ N − 1.

The time shift cannot be implemented in the conventional way, since the fi nite

sequence N is defi ned over a given domain and everything else outside that

domain is assumed nonexistent. If f(n) is shifted by n 0 creating f(n − n 0 ), then it

would no longer be in the range 0 ≤ n ≤ N − 1. A new type of shift must then be

defi ned where the shifted resulting sequence is over the range 0 ≤ n ≤ N − 1. This

new shifting operation is called the circular shift.

R.5.106 For example, let the sequence f(n) be given by the sequence f(n) = [1 2 3 4 5 6 7].

Then the circular shift is illustrated by the following two examples, which are

self-explanatory:

a. f(n − 1) = [7 1 2 3 4 5 6]

b. f(n − 3) = [5 6 7 1 2 3 4]

TABLE 5.3

Properties of the DFT

N-Point Time Sequence N-Point DFT

F 1 (n) F 1 (k)

f 2 (n) F 2 (k)

af 1 (n) + bf 2 (n) aF 1 (k) + bF 2 (k)

F(n − no) WFkNKno()

WfnNKn^0 () F(k − k 0 )

[()( 12 )]

0

1

fmfn mN

m

N





∑ F^1 (k).^ F^2 (k)

f 1 (n). f 2 (n) (1/ ) [ ( )]

0

1

NFmFKm

m

N

12 () 



∑

(1/N)F(n) f(−k)

f *(n) F *(−k)N

f *(−n)N F *(k)