Tutorial Sheet No.7
1. Use the analysis equation to evaluate the numerical values of one period of the
Fourier series coefficients of the periodic signal
∞
x[n] = ∑ (4.δ (n − 4m) +8.δ (n − 1 − 4m) )
m = −∞
2. Each of the two sequences x1 (n) and x2 (n) has a period N=4, and the corresponding
Fourier series coefficients are specified as
DTFS
x1 (n) ←⎯
⎯→ ak
DTFS
x2 (n) ←⎯
⎯→ bk
where a0 = a3 =
1
1
a1 = a2 = 1 and b0 = b1 = b2 = b3 = 1
2
2
Use the multiplication property; determine the Fourier series coefficients ck form the
signal g (n) = x1 (n).x2 (n) .
3. If the periodic sequence x(n) =
∑c e ω
j
k =< N >
0 kn
k
; ω0 =
2π
. Find the values of c k in
N
terms of x(n) and ω0 .
[Nov. 08]
4. Prove that c k = c k + N .
[Dec.06]
5. Calculate the D.T.F.S. of x(n) = cos
π .n
N
.
[Dec.06]
6. Let x1 (n) and x2 (n) be the periodic sequence with period N and their D.T.F.S. given
N −1
N −1
k =0
k =0
by x1 (n) = ∑ d k e jω0 kn , and x2 (n) = ∑ ek e jω0 kn . Show that the sequence x(n) = x1 (n).x2 (n)
N −1
∑
ck e
can be expressed c = ∑ d m .ek − m ,where x(n) =
k m =0
k =< N >
k = +∞
7. Consider the sequence x(n) = ∑ δ (n − 4k ) .
k = −∞
jω0 kn
.
[Dec.08]
(a) Sketch x(n)
(b) Find the Fourier coefficients c of x(n) .
k
8. Determine the D.T.F.S. representation for each of the following sequence.
(a) cos
π
3
(b) cos 2
n + sin
π
8
π
4
n
n
9. Let x(n) be a real and odd periodic signal with period N = 7 and Fourier coefficients
ak .Given that a15 = j , a16 = 2 j , a17 = 3 j . Determine the values of a0 , a−1 , a− 2 and a−3 .
10. Consider the following three discrete –time signals with the fundamental period of
N = 6.
x(n) = 1 + cos
π
3
n
π⎞
⎛π
y (n) = sin ⎜ n + ⎟
4⎠
⎝3
x(n). y (n)
(a) Determine the Fourier coefficients of x(n) .
(b) Determine the Fourier coefficients of y (n) .
(c) Use the results of parts (a) and (b) , along with the multiplication property of the
discrete –time Fourier series, to determine the Fourier series coefficients of
z (n) = x(n). y (n) .