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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