Assignment # 04
1) Solution to Midterm Exam (already
discussed) [For undergrad only]
2) Example 3.6, 3.8, 3.12, 3.17
3) Problems 3.3. 3.4, 3.13, 3.14, 3.15,
3.22(a), 3.30, 3.38
Due Date (29th Nov & 1st Dec for DL students)
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Fourier Transforms-Communication Systems-Lecture Slides, Slides of Data Communication Systems and Computer Networks
Sir Chiranjeev Mehta delivered this lecture at Alagappa University for Communication Systems course. It includes: Midterm, Average, Paper, Weight, Probably, Bridge, Fourier, Aperiodic, Domain, Periodic, Impulse
Typology: Slides
2011/2012
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Assignment # 04 1) Solution to Midterm Exam (already
discussed) [For undergrad only]
2) Example 3.6, 3.8, 3.12, 3.17 3) Problems 3.3. 3.4, 3.13, 3.14, 3.15,
3.22(a), 3.30, 3.
Due Date (
th
Nov
st
Dec
for
DL
students
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Midterm Papers Checked Finally
Minimum marks
Î
08 / 70
Maximum
Î
55 / 70
(79%)
Average (approx)
Î
31 / 70
(45 %)
Therefore, I will not change the weight-age most probably
Bridge to Fourier Transform
So far only Periodic Signals have been discussed andtheir corresponding frequency domain
Æ
Fourier Series
What about non-periodic Signals
Æ
Fourier Transform
Bridging Fourier Series and Transform
Consider a periodic signal below:
We know that the Fourier coefficients will be:
sin(
1 0
1
k k
k
T
k
T T
a
k
ω π
x
T
(t)
t
-T
1
T
1
T
0
-T
0
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What happens in frequency domain !?!?
As the period
Æ
, the fundamental frequency
ω
0
Æ
- So,
the distance between the two consecutive a
k s becomes
zero, and the sketch of a
k
becomes continuous, what is
called as Fourier Transform.
At the other side, the signal x(t) becomes non-periodicand takes the form:
This means the Fourier Transform can represent a non-periodic signal on the frequency-domain.
x(t)
t
-T
1
0 T
1
Center for Advanced Studies in
Engineering
9
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We have represented a non-periodic signal by a Fourier Integral rather than a Fourier series
We call X(w) the direct Fourier transform of x(t), and x(t) the Inverse Fourier transform of X(w)
x(t) and X(w) are a Fourier transform pair “
)
(
) (
w
X
t x
⇔
∫
∞ ∞−
dw
e
w
X
t
x
jwt
π
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What does it tell you about the signal?
What it doesn’t tell you about the signal?
Given a signal x(t) in time-domain, its Fouriercoefficients (a
k ) or its Fourier Transform (X(
ω
)) are
called as its “
frequency (or line) spectrum
”.
If a
k
or X(
ω
) is complex, then frequency spectrum
is observed by their
magnitude
(|a
k
| or |X(
ω
)|) and
phase
( ∠
a
k
or
∠
X(
ω
)) plots, e.g.:
θ
ω
ω ω
θ
=
∠
= =
) (
) (
) (
X
A
X
Ae
X
j
Examples
t
t
x
)
(
) (
t^0
t
t x
−
=
δ
Fourier Transform of a PeriodicSignal
∑
∑
∞
∞ −
+∞
∞ −
−
=
) ( 2 ) ( ,
) (
,
0
0
ω
k
a
X
Then
e a
t x
if
t k
jk
k
t
j
IFFT
e
0
)
(
2 ,
0
ω
ω
ω
πδ
⎯ ⎯ →
←
−
∵
FT of a sinusoidal function
FT of a Pulse Train
train
pulse
Periodic
t
x
sin(
1 0
1
k k
k
T
k
T T
a
k
FT Property:
Differentiation in Time:
)
(
/ ) (
ω
ω
X
j
dt
t
dx
↔