1. 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. Frequency Domain Description (FT) Time Domain Description (Inverse FT)

5. 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 asterisk denotes conjugate operation. • Theorem : Spectral symmetry of real signals If w(t) is real, then • 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. Corollaries of 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) 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

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

22. d(x) x Dirac Delta Function • Definition: The Dirac deltafunctionδ(x) is defined by where w(x) is any function that is continuous at x = 0. An alternative definition of δ(x) is: The Sifting Propertyof 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. Ad(f-fc) Ad(f-fc) H(fc)d(f-fc) Aej2pfct d(f-fc) H(f) fc fc H(fc) ej2pfct Ad(f+fc) 2Acos(2pfct) -fc 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

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 t -3T -2T -T 0 T 2T 3T 0 4/T -2/T 2/T

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