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Appendix 2

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Title: Appendix 2


1
Appendix 2
  • Fourier Transform and Spectra
  • Topics
  • Fourier transform (FT) of a waveform
  • Properties of Fourier Transforms
  • Parsevals Theorem and Energy Spectral Density
  • Dirac Delta Function and Unit Step Function
  • Rectangular and Triangular Pulses
  • Convolution

2
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


3
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.


4
Fourier Transform of a Waveform
  • Definition Inverse Fourier transform
  • The Inverse Fourier transform (FT) of a
    waveform w(t) is
  • The functions w(t) and W(f) constitute a Fourier
    transform pair.

Frequency Domain Description (FT)
Time Domain Description (Inverse FT)
5
Fourier Transform - Sufficient Conditions
  • The waveform w(t) is Fourier transformable if it
    satisfies both Dirichlet conditions
  • 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.
  • 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

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.

6
Spectrum of an Exponential Pulse
W(f), we have
7
Spectrum of an Exponential Pulse
8
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.

9
Properties of Fourier Transforms
  • Spectral symmetry of real signals. If w(t) is
    real, then
  • Magnitude spectrum is even about the origin.
  • W(-f) W(f) (A)
  • Phase spectrum is odd about the origin.
  • ?(-f) - ?(f) (B)

Since, W(-f) W(f) We see that corollaries
(A) and (B) are true.
10
Properties of Fourier Transform(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 -8 lt t lt 8
  • W(f ) can be complex, even though w(t) is real.
  • If w(t) is real, then W(-f) W(f).

11
Parsevals Theorem and Energy Spectral Density
  • Persavals theorem gives an alternative method to
    evaluate energy in frequency domain instead of
    time domain.
  • In other words energy is conserved in both
    domains.

12
Parsevals Theorem and Energy Spectral Density
Comparing,
13
TABLE 2-1 SOME FOURIER TRANSFORM THEOREMS
14
Example Spectrum of a Damped Sinusoid
From already found results,
  • Spectral Peaks of the Magnitude spectrum has
    moved to ffo and f-fo due to multiplication
    with the sinusoidal.

15
Example 2-3 Variation of W(f) with f
16
Dirac Delta Function
  • Definition The Dirac delta function d(x) is
    defined by

where w(x) is any function that is continuous at
x 0. An alternative definition of d(x) is
The Sifting Property of the d function is
If d(x) is an even function the integral of the d
function is given by
17
Unit Step Function
  • Definition The Unit Step function u(t) is

Because d(?) is zero, except at ? 0, the Dirac
delta function is related to the unit step
function by
18
Spectrum of Sinusoids
  • Exponentials become a shifted delta
  • Sinusoids become two shifted deltas
  • The Fourier Transform of a periodic signal is a
    weighted train of deltas

19
Spectrum of a Sine Wave
before
20
Spectrum of a Sine Wave
21
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.

Ts?nd(t-nTs)
?nd(t-n/Ts)
t
0
Ts
2Ts
3Ts
-Ts
-2Ts
-3Ts
0
1/Ts
-1/Ts
f
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