Review on Fourier …

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

Fourier Series

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

Frequency Content Matters FM radio –Single carrier at radio station frequency (e.g 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) Channel Transmitter Receiver Transmitter Home Service Provider upstream downstream

Demands for Broadband Access Courtesy of Milos Milosevic (Schlumberger)

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

channel frequency response a subchannel frequency magnitude a carrier 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) Harmonically related carriers

Periodic Signals f(t) is periodic if, for some positive constant T 0 For all values of t, f(t) = f(t + T 0 ) Smallest value of T 0 is the period of f(t). sin(2  f o t) = sin(2  f 0 t + 2  ) = sin(2  f 0 t + 4  ): period 2 . 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 T 0 would give the same value as integrating from –T 0 /2 to T 0 /2

Sinusoids f 0 (t) = C 0 cos(2  f 0 t +   ) f n (t) = C n cos(2  n f 0 t +  n ) The frequency, n f 0, is the nth harmonic of f 0 Fundamental frequency in Hertz is f 0 Fundamental frequency in rad/s is  = 2  f 0 C n cos(n  0 t +  n ) = C n cos(  n ) cos(n  0 t) - C n sin(  n ) sin(n  0 t) = a n cos(n  0 t) + b n sin(n  0 t)

Fourier Series General representation of a periodic signal Fourier series coefficients Compact Fourier series

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

Example #1 Fundamental period T 0 =  Fundamental frequency f 0 = 1/T 0 = 1/  Hz  0 = 2  /T 0 = 2 rad/s 0  1 e -t/2 f(t)f(t)

Example #2 Fundamental period T 0 =  Fundamental frequency f 0 = 1/T 0 = 1/  Hz  0 = 2  /T 0 =  rad/s 0  A f(t)f(t) -A

Example #3 Fundamental period T 0 =  Fundamental frequency f 0 = 1/T 0 = 1/  Hz  0 = 2  /T 0 = 1 rad/s  1 f(t)f(t) 

Fourier Analysis

Periodic Signals For all t, x(t + T) = x(t) x(t) is a period signal Periodic signals have a Fourier series representation C n computes 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 =  + j  Laplace transform consists of an algebraic expression and a region of convergence (ROC) For the substitution s = j  or s = j 2  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 0 1 1/2-1/2 x rect(x) 0 1 1/2-1/2 x (x)(x)

Useful Functions Sinc function –Even function –Zero crossings at –Amplitude decreases proportionally to 1/x 0 1 x sinc(x) 

Fourier Transform Pairs 0 1  /2-  /2 t f(t)f(t) 0   F()F()             F

Fourier Transform Pairs 0 1 t f(t) = 1  F(  ) = 2  (  ) (2  ) F (2  ) means that the area under the spike is (2  ) 0

Fourier Transform Pairs 0  F()F() 00  0 0 t f(t) F ()() ()()

Fourier Transform Pairs 1 t sgn(t)

Fourier Transform Properties

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

0 1  /2-  /2 t f(t)f(t) 0   F()F()             Duality Forward/inverse transforms are similar Example: rect(t/  )   sinc(   / 2) –Apply duality  sinc(t  /2)  2  rect(-  /  ) –rect(·) is even  sinc(t  /2)  2  rect(  /  )

Scaling Same as Laplace transform 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 -  t 0 (so t 0 is the slope of the linear phase)

Frequency-shifting Property

Modulation

Example: y(t) = f(t) cos(  0 t) f(t) is an ideal lowpass signal Assume  1 <<  0 Demodulation is modulation followed by lowpass filtering Similar derivation for modulation with sin(  0 t) 0 1  --  F()F()  0 Y()Y()  -   -   -   +      F      -     +     F   

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

Time Integration Property

Summary Definition of Fourier Transform Two ways to find Fourier Transform –Use definitions –Use properties