Review on Fourier …

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## Review on Fourier …

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**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**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**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)**Demands for Broadband Access**Courtesy of Milos Milosevic (Schlumberger)**DSL Broadband Access Standards**Courtesy of Shawn McCaslin (Cicada Semiconductor, Austin, TX)**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**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**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)**Fourier Series**• General representationof a periodic signal • Fourier seriescoefficients • Compact Fourierseries**Existence of the Fourier Series**• Existence • Convergence for all t • Finite number of maxima and minima in one period of f(t)**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**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**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**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.**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**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**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**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**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**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**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)**F**Fourier Transform Pairs F(w) f(t) (p) (p) t w 0 -w0 w0 0**t**Fourier Transform Pairs sgn(t) 1 -1**Fourier vs. Laplace Transform Pairs**Assuming that Re{a} > 0**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**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**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)**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**Conditions**f(t) 0, when |t| f(t) is differentiable Derivation of property:Given f(t) F(w) Time Differentiation Property**Summary**• Definition of Fourier Transform • Two ways to find Fourier Transform • Use definitions • Use properties