Transcript Chap2_Lect2 - faraday - Eastern Mediterranean University

Chapter 2
Fourier Transform and Spectra
Topics:
 Fourier transform (FT) of a waveform
 Properties of Fourier Transforms
 Parseval’s Theorem and Energy Spectral Density
 Dirac Delta Function and Unit Step Function
 Rectangular and Triangular Pulses
 Convolution
Huseyin Bilgekul
Eeng360 Communication Systems I
Department of Electrical and Electronic Engineering
Eastern Mediterranean University
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Fourier Transform of a Waveform
Definition: Fourier transform
The Fourier transform (FT) of a waveform w(t) is:
where
ℑ[.] denotes the Fourier transform of [.]
f is the frequency parameter with units of Hz (1/s).
 W(f) is also called Two-sided Spectrum of w(t), since
both positive and negative frequency components are
obtained from the definition
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Evaluation Techniques for FT Integral
 One of the following techniques can be used to
evaluate a FT integral:
•
•
•
•
Direct integration.
Tables of Fourier transforms or Laplace transforms.
FT theorems.
Superposition to break the problem into two or more
simple problems.
• Differentiation or integration of w(t).
• Numerical integration of the FT integral on the PC via
MATLAB or MathCAD integration functions.
• Fast Fourier transform (FFT) on the PC via MATLAB
or MathCAD FFT functions.
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Fourier Transform of a Waveform
 Definition: Inverse Fourier transform
The Inverse Fourier transform (FT) of a waveform w(t) is:

w(t ) 
j 2 ft
W
(
f
)
e
df


 The functions w(t) and W(f) constitute a Fourier transform pair.

w(t ) 
j 2 ft
W
(
f
)
e
df


Time Domain Description
(Inverse FT)

W( f ) 

w(t )e  j 2 nft dt

Frequency Domain Description
(FT)
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Fourier Transform - Sufficient Conditions

The waveform w(t) is Fourier transformable if it satisfies both Dirichlet
conditions:
1) Over any time interval of finite length, the function w(t) is single
valued with a finite number of maxima and minima, and the number
of discontinuities (if any) is finite.
2)
•
•
w(t) is absolutely integrable. That is,
Above conditions are sufficient, but not necessary.
A weaker sufficient condition for the existence of the Fourier transform is:
E


2
w(t ) dt  
Finite Energy
•
•
where E is the normalized energy.
This is the finite-energy condition that is satisfied by all physically realizable
waveforms.
•
Conclusion: All physical waveforms encountered in engineering practice
are Fourier transformable.
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Spectrum of an Exponential Pulse
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Spectrum of an Exponential Pulse
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Properties of Fourier Transforms
 Theorem : Spectral symmetry of real signals
If w(t) is real, then
Superscript asterisk is conjugate operation.
• Proof:
Take the conjugate
Substitute -f
=
Since w(t) is real, w*(t) = w(t), and it follows that W(-f) = W*(f).
• If w(t) is real and is an even function of t, W(f) is real.
• If w(t) is real and is an odd function of t, W(f) is imaginary.
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Properties of Fourier Transforms
 Spectral symmetry of real signals. If w(t) is real, then:
W ( f )  W ( f )
•
Magnitude spectrum is even about the origin.
|W(-f)| = |W(f)|
•
(A)
Phase spectrum is odd about the origin.
θ(-f) = - θ(f)
(B)
Corollaries of
Since, W(-f) = W*(f)
We see that corollaries (A) and
(B) are true.
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Properties of Fourier Trans. (Summary)
•
f, called frequency and having units of hertz, is just a
parameter of the FT that specifies what frequency we are
interested in looking for in the waveform w(t).
•
The FT looks for the frequency f in the w(t) over all time, that
is, over -∞ < t < ∞
•
W(f ) can be complex, even though w(t) is real.
•
If w(t) is real, then W(-f) = W*(f).
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Parseval’s Theorem and Energy Spectral Density
 Persaval’s theorem gives an alternative method to evaluate energy
in frequency domain instead of time domain.
 In other words energy is conserved in both domains.
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Parseval’s Theorem and Energy Spectral Density
The total Normalized Energy E is given by the area under the
Energy Spectral Density
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TABIE 2-1: SOME FOURIER TRANSFORM THEOREMS
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Example 2-3: Spectrum of a Damped Sinusoid
http://cnyack.homestead.com/files/afourtr/ftdsoid.htm
 Spectral Peaks of the Magnitude spectrum has moved to f=fo
and
f=-fo due to multiplication with the sinusoidal.
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Example 2-3: Variation of W(f) with f
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Dirac Delta Function
 Definition: The Dirac delta function δ(x) is defined by



w( x)d ( x)dx  w(0)
where w(x) is any function that is continuous at x = 0.
An alternative definition of δ(x) is:



d(t)
t
d ( x)dx  1
, x =0
d ( x)  
0, x  0
The Sifting Property of the δ function is



w( x)d ( x  xo )dx  w( xo )
If δ(x) is an even function the integral of the δ function is given by:
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Unit Step Function
 Definition: The Unit Step function u(t) is:
1,
u (t )  
0,
t>0
t<0
Because δ(λ) is zero, except at λ = 0, the Dirac delta function is related to the unit
step function by
du (t )
 d (t )
dt

t

d ( )d   u (t )
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Spectrum of Sinusoids
 Exponentials become a shifted delta
Ad(f-fc)
Aej2fct 
d(f-fc)
H(f )
fc
H(fc) ej2fct
 Sinusoids become two shifted deltas
2Acos(2fct) 
H(fc)d(f-fc)
Ad(f+fc)
Ad(f-fc)
-fc
fc
 The Fourier Transform of a periodic signal is a weighted train of
deltas
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Spectrum of a Sine Wave
A
V ( f )  d ( f  f o )  d ( f  f o ) 
2
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Spectrum of a Sine Wave
Sine Wave with an Arbitrary Phase
w(t )  A sin(0t  0 )  A sin[0 (t 
0
0 )]
f
A j0 fo
W( f )  j e
d ( f  fo )  d ( f  f o )
2
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Sampling Function
 The Fourier transform of a delta train in time domain is again a delta
train of impulses in the frequency domain.
 Note that the period in the time domain is Ts whereas the period in the
frquency domain is 1/ Ts .
 This function will be used when studying the Sampling Theorem.
nd(t-n/Ts)
Tsnd(t-nTs)
-3Ts
-2Ts
-Ts
0
Ts
2Ts
3Ts
t
-1/Ts
0
1/Ts
f
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