Signals and Systems: Chapter 4 Fourier transform

Department of Electrical Engineering University of Arkansas EE 2000 SIGNALS AND SYSTEMS ELEG 3124 SYSTEMS AND SIGNALS Ch Fourier Transform Dr Jingxian Wu wuj@uark.edu (These slides are taken from Dr Jingxian Wu, University of Arkansas, 2020.) OUTLINE • Introduction • Fourier Transform • Properties of Fourier Transform • Applications of Fourier Transform INTRODUCTION: MOTIVATION • Motivation: – Fourier series: periodic signals can be decomposed as the summation of orthogonal complex exponential signals + T x (t ) =  cn exp  jn0t  cn =  x (t ) exp  jn0t dt n = − T • each harmonic contains a unique frequency: n/T x(t) t Time domain • time domain ➔ frequency domain Frequency domain (T =  ) How about aperiodic signals ? INTRODUCTION: TRANSFER FUNCTION • System transfer function e e jt H () jt h(t ) H ( ) =  h(t ) exp jt dt + − • System with periodic inputs e + jn0t c e n = − h(t ) + jn0t n e jn0t H (n0 ) h(t ) jn0t c e  n H ( n ) n = − + x(t ) h(t ) jn0t c e  n H ( n ) n = − OUTLINE • Introduction • Fourier Transform • Properties of Fourier Transform • Applications of Fourier Transform FOURIER TRANSFORM • Fourier Transform + X ( ) =  x(t )e − jt dt − – given x(t), we can find its Fourier transform X ( ) • Inverse Fourier Transform x (t ) = 2  + − X ( )e jt d – given X ( ) , we can find the time domain signal x(t) – signal is decomposed into the “weighted summation” of complex exponential functions (integration is the extreme case of summation) x(t ) ➔ X ( ) FOURIER TRANSFORM • Example – Find the Fourier transform of x(t ) = rect (t /  ) x(t) x(t) t t FOURIER TRANSFORM • Example – Find the Fourier transform of x(t ) = exp( −a | t |) a0 FOURIER TRANSFORM • Example – Find the Fourier transform of x(t ) = exp( −at )u(t ) a0 10 FOURIER TRANSFORM • Example – Find the Fourier transform of x(t ) =  (t − a) 34 PROPERTY: PARSAVAL’S THEOREM • Review: signal energy + E =  | x(t ) |2 dt − • Parsaval’s theorem + | x ( t ) | dt = − 2  + − | X ( ) |2 d 35 PROPERTY: PARSAVAL’S THEOREM • Example: – Find the energy of the signal x(t ) = exp( −2t )u(t ) 36 PROPERTY: PERIODIC SIGNAL • Fourier transform of periodic signal – Periodic signal can be written as Fourier series x(t ) = + c n = − n exp  jn0t  – Perform Fourier transform on both sides X ( ) = 2 +  c  ( − n ) n = − n 37 OUTLINE • Introduction • Fourier Transform • Properties of Fourier Transform • Applications of Fourier Transform 38 APPLICATIONS: FILTERING • Filtering – Filtering is the process by which the essential and useful part of a signal is separated from undesirable components • Passing a signal through a filter (system) • At the output of the filter, some undesired part of the signal (e.g noise) is removed – Based on the convolution property, we can design filter that only allow signal within a certain frequency range to pass through x(t )  h(t ) x(t ) X ( ) h(t ) H ( ) filter filter time domain X ( ) H ( ) frequency domain 39 APPLICATIONS: FILTERING • Classifications of filters Passband Stop band Low pass filter Stop Passband Stop band band Band pass filter Stop band Passband High pass filter Passband Stop Passband band Band stop (Notch) filter 40 APPLICATION: FILTERING • A filtering example – A demo of a notch filter X ( ) H ( ) Corrupted sound Filter X ( ) H ( ) Filtered sound 41 APPLICATIONS: FILTERING • Example – Find out the frequency response of the RC circuit – What kind of filters it is? RC circuit 42 APPLICATION: SAMPLING THEOREM • Sampling theorem: time domain – Sampling: convert the continuous-time signal to discrete-time signal x(t ) p (t ) = +   (t − nT ) n = − sampling period xs (t ) = x(t ) p(t ) Sampled signal 43 APPLICATION: SAMPLING THEOREM • Sampling theorem: frequency domain – Fourier transform of the impulse train • impulse train is periodic Fourier series + p(t ) =   (t − nTs ) = Ts n = − + jns t  e  n = − 2 s = Ts • Find Fourier transform on both sides 2 P( ) = Ts +  ( − n ) n = − s • Time domain multiplication ➔ Frequency domain convolution x (t ) p (t )  X ( )  P( ) 2 x(t ) p(t )  Ts +  X ( − n ) n = − s 44 APPLICATION: SAMPLING THEOREM • Sampling theorem: frequency domain – Sampling in time domain ➔ Repetition in frequency domain Time domain Frequency domain 45 APPLICATION: SAMPLING THEOREM • Sampling theorem – If the sampling rate is twice of the bandwidth, then the original signal can be perfectly reconstructed from the samples s  2B s  2B s = 2B s  2B Frequency domain 46 APPLICATION: AMPLITUDE MODULATION • What is modulation? – The process by which some characteristic of a carrier wave is varied in accordance with an information-bearing signal Information bearing signal • Three signals: modulation Modulated signal Carrier wave – Information bearing signal (modulating signal) • Usually at low frequency (baseband) • E.g speech signal: 20Hz – 20KHz – Carrier wave • Usually a high frequency sinusoidal (passband) • E.g AM radio station (1050KHz) FM radio station (100.1MHz), 2.4GHz, etc – Modulated signal: passband signal 47 APPLICATION: AMPLITUDE MODULATION • Amplitude Modulation (AM) s(t ) = Ac m(t ) cos(2f ct ) – A direct product between message signal and carrier signal m(t ) s(t ) Mixer Ac cos(2f ct ) Local Oscillator Amplitude modulation 48 APPLICATION: AMPLITUDE MODULATION • Amplitude Modulation (AM) S( f ) = Ac M ( f − f c ) + M ( f + f c ) Amplitude modulation ... filters Passband Stop band Low pass filter Stop Passband Stop band band Band pass filter Stop band Passband High pass filter Passband Stop Passband band Band stop (Notch) filter 40 APPLICATION:... • Fourier Transform • Properties of Fourier Transform • Applications of Fourier Transform FOURIER TRANSFORM • Fourier Transform + X ( ) =  x(t )e − jt dt − – given x(t), we can find its Fourier. .. a0 FOURIER TRANSFORM • Example – Find the Fourier transform of x(t ) = exp( −at )u(t ) a0 10 FOURIER TRANSFORM • Example – Find the Fourier transform of x(t ) =  (t − a) 11 FOURIER TRANSFORM: