Digital Signal Processing Reference
In-Depth Information
(v)
(vi)
(a)
Find the Fourier coefficients of the exponential form for each signal.
(b)
Find the Fourier coefficients of the combined trigonometric form for each signal.
x(t) = cos
7t
x(t) = 4(cos
t)
( sin 4t)
4.3.
(
a)
Determine whether the following functions can be represented by a Fourier series:
(i)
(ii)
(iii)
(iv)
where and
(b)
For those signals in part (a) that can be represented by a Fourier series, find the
coefficients of all harmonics, expressed in exponential form.
x(t) = cos(3t) + sin (5t)
x(t) = cos(6t) + sin (8t) + e
j2t
x(t) = cos(t) + sin (pt)
x
1
(t) = sin (
p
6
)
x
2
(t) = sin (
p
9
).
x
3
(t) = x
1
(t) + x
2
(3t)
4.4.
A periodic signal
x
(
t
) is expressed as an exponential Fourier series:
q
k=-
q
C
k
e
jkv
o
t
.
x(t) =
N
(t) = x(t - t
o
)
Show that the Fourier series for
is given by
q
k=-
q
N
k
e
jkv
o
t
,
N (t) =
in which
ƒ
N
k
ƒ = ƒC
k
ƒ
N
k
= ∠
and
∠
C
k
- kv
o
t
o
.
4.5.
For a real periodic signal
x
(
t
), the
trigonometric form
of its Fourier series is given by
x(t) = A
0
+
q
k= 1
[A
k
cos
kv
o
t + B
k
sin kv
o
t].
Express the exponential form Fourier coefficients
C
k
in terms of
A
k
and
B
k
.
4.6.
This problem will help illustrate the orthogonality of exponentials. Calculate the fol-
lowing integrals:
2p
sin
2
(t)dt
(a)
L
0
2p
sin
2
(2t)dt
(b)
L
0
2p
(c)
sin(t)sin(2t)dt
L
0
(d)
Explain how the results of parts (a), (b), and (c) illustrate the orthogonality of
exponentials.
