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