1. Fourier Transform and Spectra
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
Erhan A. Ince
Eeng360 Communication Systems I
Department of Electrical and Electronic Engineering
Eastern Mediterranean University

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 waveform w(t) can be obtained from the spectrum via the inverse Fourier transform (IFT) :
The functions w(t) and W(f) constitute a Fourier transform pair.
Time Domain Description
(Inverse FT)
Frequency Domain Description
(FT)

5. Quadrature components
Quadrature components
Phasor components

6. Fourier Transform - Sufficient Conditions
The waveform w(t) is Fourier transformable if it satisfies the 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.

7. Spectrum of an Exponential Pulse

8. Phasor components

10. Spectrum of an Exponential Pulse

11. 1 -1

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

14. 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.
θ(-f) = - θ(f) (B)
Corollaries of
Since, W(-f) = W*(f)
We see that corollaries (A) and (B) are true.

15. Properties of Fourier Transform
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).

16. 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.
(2-41)

17. Parseval’s Theorem and Energy Spectral Density
The total Normalized Energy E is given by the area under the Energy Spectral Density

19. TABIE 2-1: Properties in Spectrum Domain

20. Example 2-3: Spectrum of a Damped Sinusoid
the Magnitude spectrum for w(t) has moved to f = fo and f = -fo due to multiplication with the sinusoidal.

21. Example 2-3: Spectrum of a Damped Sinusoid
Variation of W(f) with f

22. Dirac Delta Function
Definition: The Dirac delta function δ(x) is defined by x
d(x)
where w(x) is any function that is continuous at x = 0.
An alternative definition of δ(x) is:
The Sifting Property of the δ function is
If δ(x) is an even function the integral of the δ function is given by:

23. Unit Step Function 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

24. 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 fc
Ad(f-fc)
Aej2pfct 
H(f )
d(f-fc)
H(fc)d(f-fc)
H(fc) ej2pfct
2Acos(2pfct)  fc
Ad(f-fc)
-fc
Ad(f+fc)

25. Spectrum of a Sine Wave +

26. Spectrum of a Sine Wave

27. Fourier Transform of an Impulse Train
FT of any periodic signal can be represented as:
An impulse train in time domain can be represented as:
This signal is periodic hence (1) can be used to get its FT
2𝜋/𝑇 f
-3T
-2T -T T 2T 3T t
-2/T
2/T
4/T

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