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# TUTORIAL QAF OR DFS DFT by 2m18Zj

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```									 TUTORIAL QUESTIONS FOR DFS & DFT                                 DIGITAL SIGNAL PROCESSING

1. Find the DFS expansion of the sequence
~(n)  A cos(n )
x
2

2. Compute the N-point DFT of each of the following sequences:
(a) x1 (n)   (n)

(b) x 2 (n)   (n  n0 ),         where 0  n0  N

(c) x3 (n)   n             0n N

3. Consider the sequence
x(n)   (n)  2 (n  2)   (n  3)

(a) Find the four-point DFT, X (k ) , of          x(n) .

(b) Find the finite-length sequence q(n) that has a four-point DFT

Q(k )  W42 k X (k )

(c) If    y(n) is the four-point circular convolution of x(n) with itself, find

y(n) and the four-point DFT Y (k ) .

(d) With h(n)   (n)   (n  1)  2 (n  3) , find the four-point circular

convolution of x(n) with h(n) .
TUTORIAL QUESTIONS FOR DFS & DFT                                DIGITAL SIGNAL PROCESSING

SOLUTION:
3
(a) X (k )   x(n)W4nk  1  2W42 k  W43k
n 0

(b) The sequence q(n) is formed by multiplying the DFT of x(n) by the complex
exponential W42 k . Because this corresponds to a circular shift of x(n) by 2,

q(n)  x(( n  2)) 4

it follows that
q(n)   (n  2)  2 (n)   (n  1)

(c) Y (k )  X (k ) 2  (1  2W42 k  W43k )(1  2W42 k  W43k )

 1  4W42 k  2W43k  4W44 k  4W45k  W46 k
 5  4W4k  5W42 k  2W43k
Therefore,
y(n)  5 (n)  4 (n  1)  5 (n  2)  2 (n  3)

(d) With h(n)   (n)   (n  1)  2 (n  3) , the four-point circular convolution of

x(n) with h(n) may be found using the tabular method. Because, the linear
convolution of x(n) with h(n) is
y(n)  x(n)  h(n)  [1, 1, 2, 5, 1, 4, 2]

then

n                     0      1      2      3         4        5   6    7   8
y(n)                    1      1      2      5         1        4   2    0   0
y(n+4)                   1      4      2      0         0        0   0    0   0
z(n)                    2      5      4      5        -         -    -   -       -

or     z (n)  2 (n)  5 (n  1)  4 (n  2)  5 (n  3)

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