PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

DTFT, DFT, ZT, and FFT 515

Example 5.4

Given the discrete sequence f(n) = [1 2 3],

a. Evaluate by hand the DTF coeffi cients of f(n)

b. Create the script fi le DFT_IDFT that returns the DTF of f(n)

c. Recover the sequence f(n) by evaluating the IDTF of part b

ANALYTICAL Solution

Fk f ne

jNnk

n

N

()
 ()



 2

0

1 

∑

Ffn

n

() ()01236

0

2





∑

Ffne e e i

j n

n

() 1 ( ) 1 2 jj 3 1 5000. 0 8660.

2

3

0

2



 
^2343 





∑ 

Ffne ee i

j n

n

() 2 () 1 2 jj 3 1 5000. 0 8660.

4

3

0

2



 
^4383 





∑ 

MATLAB Solution

% Script file: DFT _ IDFT

fn = [1 2 3]

DTF _ fn = fft(fn);

disp(‘**************** R E S U L T S **************** ’)

disp(‘The given time sequence is f(n) = [1 2 3]’)

disp(‘The DTF of f(n)=[1 2 3] using fft is given by :’)

disp(DTF _ fn)

fnn = ifft(DTF _ fn);

disp(‘The IDTF of the DFT of f(n)=[1 2 3] using ifft is given by:’)

disp(fnn)

disp(‘**************************************************** ’)

The script fi le DFT_IDFT is executed and the results are as follows:

******************* R E S U L T S ***********************

The given time sequence is f(n) = [1 2 3]

The DTF of f(n) = [1 2 3] using fft is given by:

6.0000 -1.5000 + 0.8660i -1.5000 - 0.8660i

The IDTF of the DFT of f(n) = [1 2 3] using ifft is given by:

1 2 3

************************************************************

Example 5.5

Create the script fi le DTFT_DFT that returns the magnitude and phase plots of DTFT as

well as DFT of the following discrete-time sequence:

fn

n

N

N

n

k

()

cos

 3

16

0

(^1) 
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
∑  for