Transcript Chapter2_Lect3.ppt

Chapter 2
Fourier Transform and Spectra
Topics:
 Rectangular and Triangular Pulses
 Spectrum of Rectangular, Triangular Pulses
 Convolution
 Spectrum by Convolution
Huseyin Bilgekul
Eeng360 Communication Systems I
Department of Electrical and Electronic Engineering
Eastern Mediterranean University
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Rectangular Pulses
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Triangular Pulses
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Spectrum of a Rectangular Pulse
t
w(t )   
T





  W ( f )  T  Sa  Tf 

Rectangular pulse is a time window.
FT is a Sa function, infinite frequency content.
Shrinking time axis causes stretching of frequency axis.
Signals cannot be both time-limited and bandwidth-limited.
Note the inverse relationship between the pulse width T and the zero crossing 1/T
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Spectrum of Sa Function
 To find the spectrum of a Sa function we can use duality theorem.
Duality: W(t)  w(-f)
Because Π is an even and real function
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Spectrum of a Time Shifted Rectangular Pulse
• The spectra shown in previous slides are real because the time domain pulse
(rectangular pulse) is real and even.
• If the pulse is offset in time domain to destroy the even symmetry, the spectra will
be complex.
• Let us now apply the Time delay theorem of Table 2.1 to the Rectangular pulse.
1
 t T 
2
v(t )   
 T 


T
Time Delay Theorem:
w(t-Td)  W(f) e-jωTd
We get:
V( f ) T
sin( fT )
 fT
 ( f )  e j fT  Sa( fT )
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Spectrum of a Triangular Pulse
 The spectrum of a triangular pulse can be obtained by direct evaluation of the FT
integral.
 An easier approach is to evaluate the FT using the second derivative of the
triangular pulse.
 First derivative is composed of two rectangular pulses as shown.
 The second derivative consists of the three impulses.
 We can find the FT of the second derivative easily and then calculate the FT of
the triangular pulse.
dw(t )
dt
d 2 w(t )
dt 2
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Spectrum of a Triangular Pulse
dw(t )
dt
d 2 w(t )
dt 2
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Spectrum of Rectangular and Sa Pulses
Duality Theorem if w(t )  W ( f ) Then W(t )  w(  f )
t 
    TSa  Tf
T 

 f 
Then 2WSa  2 Wt    

 2W 
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Table 2.2 Some FT pairs
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Key FT Properties
 Time Scaling; Contracting the time axis leads to an expansion of the
frequency axis.
 Duality
• Symmetry between time and frequency domains.
• “Reverse the pictures”.
• Eliminates half the transform pairs.
 Frequency Shifting (Modulation); (multiplying a time signal by an
exponential) leads to a frequency shift.
 Multiplication in Time
• Becomes complicated convolution in frequency.
• Mod/Demod often involves multiplication.
• Time windowing becomes frequency convolution with Sa.
 Convolution in Time
• Becomes multiplication in frequency.
• Defines output of LTI filters: easier to analyze with FTs.
x(t)*h(t)
x(t)
h(t)
X(f)
H(f)
X(f)H(f)
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Convolution
 The convolution of a waveform w1(t) with a waveform w2(t) to
produce a third waveform w3(t) which is
where w1(t)∗ w2(t) is a shorthand notation for this integration operation and ∗ is
read “convolved with”.
If discontinuous wave shapes are to be convolved, it is usually easier to evaluate
the equivalent integral
 Evaluation of the convolution integral involves 3 steps.
•
•
•
Time reversal of w2 to obtain w2(-λ),
Time shifting of w2 by t seconds to obtain w2(-(λ-t)), and
Multiplying this result by w1 to form the integrand w1(λ)w2(-(λ-t)).
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Example for Convolution
T

t


2
w1 (t )   
 T

-





t
T
w 2 (t)=e u (t )
For
0< t < T
For t > T
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Convolution
 y(t)=x(t)*z(t)=  x(τ)z(t- τ )d τ
• Flip one signal and drag it across the other
• Area under product at drag offset t is y(t).
x(t)
-1
0
x(t)
1
t
t
-4
z(t-t)
z(t)
t
-1
0
z(-2-t)
z(-6-t)
-6
z(t)
-2
t
1
z(2-t)
z(0-t)
-1
0
t-1
1
t
t
t+1
z(4-t)
t
2
2
y(t)
-6
-4
-2
-1
0
1
2
t
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