1. 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

2. A sum of sines and cosines=
sin(x) A A
3 sin(x)
+ 1 sin(3x) B
A+B
+ 0.8 sin(5x) C
A+B+C
Accept without proof that every function is a sum of
sines/cosines
As frequency increases – more details are added
Low frequency – main details
Hight frequency – fine details
Coef decreases with the frequency
+ 0.4 sin(7x) D
A+B+C+D

3. 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

4. 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 (i.e., 1/s).
W(f) is also called a two-sided spectrum of w(t), since both positive and negative frequency components are obtained from the definition

5. 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.

6. 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.
Time Domain description
(Inverse FT)
Frequency Domain description
(FT)

7. 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.

8. Spectrum of an exponential pulse

9. Plots of functions X(f) and Y(f)

10. Properties of Fourier Transforms
Theorem : Spectral symmetry of real signals
If w(t) is real, then
Superscript asterisk denotes the 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.

11. Properties of Fourier Transforms
Corollaries of
If w(t) is real,
Magnitude spectrum is even about the origin (i.e., f = 0),
|W(-f)| = |W(f)| ………………(A)
Phase spectrum is odd about the origin.
θ(-f) = - θ(f) ……………….(B)
Since, W(-f) = W*(f)
We see that corollaries (A) and (B) are true.

12. Properties of Fourier Transforms - 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).

13. Parseval’s Theorem and Energy Spectral Density

14. 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.
The total Normalized Energy E is given by the area under the Energy Spectral Density

15. TABIE 2-1: SOME FOURIER TRANSFORM THEOREMS

16. Example 2-3: Spectrum of a Damped Sinusoid
Spectral Peaks of the Magnitude spectrum has moved to f=fo and f=-fo due to multiplication with the sinusoidal.

17. Example 2-3: Variation of W(f) with f

18. Dirac Delta Function & Unit Step Function
d(t)
Definition:
The Dirac delta function δ(x) is defined by
where w(x) is any function that is continuous at x = 0.
An alternative definition of δ(x) is:
From (2-45), the Sifting Property of the δ function is
If δ(x) is an even function the integral of the δ function is given by:

19. Definition: The Unit Step function u(t) is:
Because δ(λ) is zero, except at λ = 0, the Dirac delta function is related to the unit step function by
and consequently,

20. Example 2-4: Spectrum of a Sine Wave

21. Example 2-4: Spectrum of a Sine Wave (contd..)

22. Fourier Transform and Spectra
Chapter 2
Fourier Transform and Spectra
Topics:
Rectangular and Triangular Pulses
Spectrum of Rectangular, Triangular Pulses
Convolution
Spectrum by Convolution
Huseyin Bilgekul
EEE360 Communication Systems I
Department of Electrical and Electronic Engineering
Eastern Mediterranean University

23. Rectangular and Triangular Pulses

24. Example 2-5: Spectrum of a Rectangular Pulse
= T Sa( Tf)
T Sa(Tf )
Note the inverse relationship between the pulse width T and the zero crossing 1/T

25. Example 2-5: Spectrum of a Rectangular Pulse
To find the spectrum of a Sa function we can use duality theorem From Table 2.1
Duality: W(t)  w(-f)
T Sa( Tf)
Because Π is an even and real function
2WSa( 2WT)

26. Example 2-4: Spectrum of a 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
Time Delay Theorem: w(t-Td)  W(f) e-jωTd
When we apply this to:
We get:

27. Example 2-6: Spectrum of a Triangular Pulse

28. Spectrum of Rectangular, Sa and Triangular Pulses

29. Table 2.2 Some FT pairs

30. Evaluation involves 3 steps….
Convolution
Definition:
The convolution of a waveform w1(t) with a waveform w2(t) to produce a third waveform w3(t) 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 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)).

31. Example: Convolution of rectangular pulse with exponential
For 0< t < T
For t > T