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Understanding Fourier Transform: Properties and Applications

Explore the Fourier transform of waveforms, Parseval’s theorem, Dirac Delta function, convolution, and more. Learn about frequency and time domain descriptions, energy spectral density, and spectral symmetry.

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Understanding Fourier Transform: Properties and Applications

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  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 Huseyin Bilgekul 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 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

  7. Spectrum of an Exponential Pulse

  8. Properties of Fourier Transforms Superscript asterisk is 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.

  9. 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.

  10. 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).

  11. 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.

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

  13. TABIE 2-1: SOME FOURIER TRANSFORM THEOREMS

  14. Example 2-3: Spectrum of a Damped Sinusoid • Spectral Peaks of the Magnitude spectrum has moved to f = foand f = -fodue to multiplication with the sinusoidal.

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

  16. 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 Property of the δ function is If δ(x) is an even function the integral of the δ function is given by:

  17. 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

  18. 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

  19. Spectrum of a Sine Wave

  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 Tswhereas the period in the frquency domain is 1/ Ts . • This function will be used when studying the Sampling Theorem. t -3Ts -2Ts -Ts 0 Ts 2Ts 3Ts 0 -1/Ts 1/Ts f

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