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Understanding Fourier Analysis: Demystifying Periodic Signals and Signal Processing Techniques

Delve into the world of Fourier analysis, learn about periodic signals and advanced signal processing methods. Explore Fourier series, spectrogram representation, and modulation techniques. Gain insights into signal properties and frequencies through this comprehensive review.

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Understanding Fourier Analysis: Demystifying Periodic Signals and Signal Processing Techniques

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  1. Review on Fourier …

  2. Slides edited from: • Prof. Brian L. Evans and Mr. Dogu Arifler Dept. of Electrical and Computer Engineering The University of Texas at Austin course: EE 313 Linear Systems and Signals Fall 2003

  3. Fourier Series

  4. Spectrogram Demo (DSP First) • Sound clips • Sinusoid with frequency of 660 Hz (no harmonics) • Square wave with fundamental frequency of 660 Hz • Sawtooth wave with fundamental frequency of 660 Hz • Beat frequencies at 660 Hz +/- 12 Hz • Spectrogram representation • Time on the horizontal axis • Frequency on the vertical axis

  5. Transmitter Receiver Channel Receiver Transmitter upstream Home ServiceProvider downstream Frequency Content Matters • FM radio • Single carrier at radio station frequency (e.g. 94.7 MHz) • Bandwidth of 165 kHz: left audio channel, left – right audio channels, pilot tone, and 1200 baud modem • Station spacing of 200 kHz • Modulator/Demodulator (Modem)

  6. Demands for Broadband Access Courtesy of Milos Milosevic (Schlumberger)

  7. DSL Broadband Access Standards Courtesy of Shawn McCaslin (Cicada Semiconductor, Austin, TX)

  8. Multicarrier Modulation • Discrete Multitone (DMT) modulation ADSL (ANSI 1.413) and proposed for VDSL • Orthogonal Freq. Division Multiplexing (OFDM) Digital audio/video broadcasting (ETSI DAB-T/DVB-T) Courtesy of Güner Arslan (Cicada Semiconductor) channel frequency response magnitude a carrier a subchannel frequency Harmonically related carriers

  9. Periodic Signals • f(t) is periodic if, for some positive constant T0 For all values of t, f(t) = f(t + T0) Smallest value of T0 is the period of f(t). sin(2pfot) = sin(2pf0t + 2p) = sin(2pf0t + 4p): period 2p. • A periodic signal f(t) Unchanged when time-shifted by one period Two-sided: extent is t (-, ) May be generated by periodically extending one period Area under f(t) over any interval of duration equal to the period is the same; e.g., integrating from 0 to T0 would give the same value as integrating from –T0/2 to T0 /2

  10. Sinusoids • f0(t) = C0 cos(2 p f0 t + q0) • fn(t) = Cn cos(2 p n f0 t + qn) • The frequency, n f0, is the nth harmonic of f0 • Fundamental frequency in Hertz is f0 • Fundamental frequency in rad/s is w = 2 p f0 Cn cos(n w0 t + qn) =Cn cos(qn) cos(n w0 t) - Cn sin(qn) sin(n w0 t) =an cos(n w0 t) + bn sin(n w0 t)

  11. Fourier Series • General representationof a periodic signal • Fourier seriescoefficients • Compact Fourierseries

  12. Existence of the Fourier Series • Existence • Convergence for all t • Finite number of maxima and minima in one period of f(t)

  13. Fundamental period T0 = p Fundamental frequency f0 = 1/T0 = 1/p Hz w0 = 2p/T0 = 2 rad/s f(t) 1 e-t/2 -p 0 p Example #1

  14. Fundamental period T0 = 2 Fundamental frequency f0 = 1/T0 = 1/2 Hz w0 = 2p/T0 = p rad/s f(t) A -1 0 1 -A Example #2

  15. Fundamental period T0 = 2p Fundamental frequency f0 = 1/T0 = 1/2p Hz w0 = 2p/T0 = 1 rad/s f(t) 1 -p/2 -p -2p p/2 p 2p Example #3

  16. Fourier Analysis

  17. Periodic Signals • For all t,x(t + T) = x(t) x(t) is a period signal • Periodic signals havea Fourier seriesrepresentation • Cncomputes the projection (components) of x(t) having a frequency that is a multiple of the fundamental frequency 1/T.

  18. Fourier Integral • Conditions for the Fourier transform of g(t) to exist (Dirichlet conditions): x(t) is single-valued with finite maxima and minima in any finite time interval x(t) is piecewise continuous; i.e., it has a finite number of discontinuities in any finite time interval x(t) is absolutely integrable

  19. Laplace Transform • Generalized frequency variable s = s + j w • Laplace transform consists of an algebraic expression and a region of convergence (ROC) • For the substitution s = j w or s = j 2 p f to be valid, the ROC must contain the imaginary axis

  20. Fourier Transform • What system properties does it possess? • Memoryless • Causal • Linear • Time-invariant • What does it tell you about a signal? • Answer: Measures frequency content • What doesn’t it tell you about a signal? • Answer: When those frequencies occurred in time

  21. Useful Functions • Unit gate function (a.k.a. unit pulse function) • What does rect(x / a) look like? • Unit triangle function rect(x) 1 x -1/2 0 1/2 D(x) 1 x -1/2 0 1/2

  22. sinc(x) 1 x -3p -2p -p 0 p 2p 3p Useful Functions • Sinc function • Even function • Zero crossings at • Amplitude decreases proportionally to 1/x

  23. F(w) f(t) t F 1 w t -t/2 0 t/2 -6p -4p -2p 2p 4p 6p 0 t t t t t t Fourier Transform Pairs

  24. f(t) = 1 1 t 0 F Fourier Transform Pairs F(w) = 2 p d(w) (2p) w 0 (2p) means that the area under the spike is (2p)

  25. F Fourier Transform Pairs F(w) f(t) (p) (p) t w 0 -w0 w0 0

  26. t Fourier Transform Pairs sgn(t) 1 -1

  27. Fourier Transform Properties

  28. Fourier vs. Laplace Transform Pairs Assuming that Re{a} > 0

  29. t F(w) w -6p -4p -2p 2p 4p 6p t t t t t t 0 Duality • Forward/inverse transforms are similar • Example: rect(t/t)  t sinc(wt / 2) • Apply duality t sinc(t t/2)  2 p rect(-w/t) • rect(·) is even t sinc(t t /2)  2 p rect(w/t) f(t) 1 t -t/2 0 t/2

  30. Scaling • Same as Laplacetransform scaling property |a| > 1: compress time axis, expand frequency axis |a| < 1: expand time axis, compress frequency axis • Effective extent in the time domain is inversely proportional to extent in the frequency domain (a.k.a bandwidth). f(t) is wider  spectrum is narrower f(t) is narrower  spectrum is wider

  31. Time-shifting Property • Shift in time • Does not change magnitude of the Fourier transform • Does shift the phase of the Fourier transform by -wt0 (so t0 is the slope of the linear phase)

  32. Frequency-shifting Property

  33. Modulation

  34. F(w) 1 w -w1 w1 0 Modulation • Example: y(t) = f(t) cos(w0 t) f(t) is an ideal lowpass signal Assume w1 << w0 • Demodulation is modulation followed by lowpass filtering • Similar derivation for modulation with sin(w0 t) Y(w) 1/2 F(w+w0) 1/2 F(w-w0) 1/2 w -w0 - w1 -w0 + w1 w0 - w1 w0 + w1 0 -w0 w0

  35. Conditions f(t)  0, when |t|  f(t) is differentiable Derivation of property:Given f(t) F(w) Time Differentiation Property

  36. Time Integration Property

  37. Summary • Definition of Fourier Transform • Two ways to find Fourier Transform • Use definitions • Use properties

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