Sampling of Continuous-Time Signals Quote of the Day

Sampling of Continuous-Time Signals Quote of the Day Optimist: "The glass is half full." Pessimist: "The glass is half empty." Engineer: "That glass is twice as large as it needs to be." Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck, 1999-2000 Prentice Hall Inc. Signal Types Analog signals: continuous in time and amplitude Example: voltage, current, temperature, Digital signals: discrete both in time and amplitude Example: attendance of this class, digitizes analog signals, Discrete-time signal: discrete in time, continuous in amplitude Example:hourly change of temperature in Austin Theory for digital signals would be too complicated Requires inclusion of nonlinearities into theory Theory is based on discrete-time continuous-amplitude signals Most convenient to develop theory Good enough approximation to practice with some care In practice we mostly process digital signals on processors Need to take into account finite precision effects Our text book is about the theory hence its title Discrete-Time Signal Processing Copyright (C) 2005 Gner Arslan 351M Digital Signal Processing

2 Periodic (Uniform) Sampling Sampling is a continuous to discrete-time conversion -3 -2 -1 0 1 2 3 4 Most common sampling is periodic xn x c nT n T is the sampling period in second fs = 1/T is the sampling frequency in Hz Sampling frequency in radian-per-second s=2fs rad/sec Use [.] for discrete-time and (.) for continuous time signals This is the ideal case not the practical but close enough In practice it is implement with an analog-to-digital converters We get digital signals that are quantized in amplitude and time Copyright (C) 2005 Gner Arslan 351M Digital Signal Processing 3 Periodic Sampling Sampling is, in general, not reversible Given a sampled signal one could fit infinite continuous signals through the samples 1 0.5 0 -0.5 -1 0

20 40 60 80 100 Fundamental issue in digital signal processing If we loose information during sampling we cannot recover it Under certain conditions an analog signal can be sampled without loss so that it can be reconstructed perfectly Copyright (C) 2005 Gner Arslan 351M Digital Signal Processing 4 Sampling Demo In this movie the video camera is sampling at a fixed rate of 30 frames/second. Observe how the rotating phasor aliases to different speeds as it spins faster. p t e j2 fot pn pnT pn / fs e

j2 fo n fs Demo from DSP First: A Multimedia Approach by McClellan, Schafer, Yoder Copyright (C) 2005 Gner Arslan 351M Digital Signal Processing 5 Representation of Sampling Mathematically convenient to represent in two stages Impulse train modulator Conversion of impulse train to a sequence s(t) xc(t) Convert impulse train to discretetime sequence x xc(t) x[n]=xc(nT) x[n] s(t)

-3T-2T-T 0 T 2T3T4T Copyright (C) 2005 Gner Arslan t -3 -2 -1 0 1 2 3 4 351M Digital Signal Processing n 6 Continuous-Time Fourier Transform Continuous-Time Fourier transform pair is defined as X c j x c t e j t dt 1 j t xc t X j e d c

2 We write xc(t) as a weighted sum of complex exponentials Remember some Fourier Transform properties Time Convolution (frequency domain multiplication) x( t) y( t) X( j ) Y( j ) Frequency Convolution (time domain multiplication) x( t)y( t) X( j ) Y( j ) Modulation (Frequency shift) x( t)e j ot X j o Copyright (C) 2005 Gner Arslan 351M Digital Signal Processing 7 Frequency Domain Representation of Sampling Modulate (multiply) continuous-time signal with pulse train: x s t x c t s t x t t nT c s( t)

t nT n n Lets take the Fourier Transform of xs(t) and s(t) 1 Xs j Xc j S j 2 2 S j k s T k Fourier transform of pulse train is again a pulse train Note that multiplication in time is convolution in frequency We represent frequency with = 2f hence s = 2fs 1 X s j X c j k s T k Copyright (C) 2005 Gner Arslan 351M Digital Signal Processing

8 Frequency Domain Representation of Sampling Convolution with pulse creates replicas at pulse location: 1 X s j X c j k s T k This tells us that the impulse train modulator Creates images of the Fourier transform of the input signal Images are periodic with sampling frequency If s< N sampling maybe irreversible due to aliasing of images Xc j -N N s>2N Xs j 3s -2s s -N N s 2s

3s Xs j s<2N 3s Copyright (C) 2005 Gner Arslan -2s s -N N s 2s 351M Digital Signal Processing 3s 9 Nyquist Sampling Theorem Let xc(t) be a bandlimited signal with X c ( j ) 0 for N Then xc(t) is uniquely determined by its samples x[n]= xc(nT) if

2 2fs 2N T N is generally known as the Nyquist Frequency The minimum sampling rate that must be exceeded is known as the Nyquist Rate s Low pass filter s>2N Xs j 3s -2s s -N N s 2s 3s Xs j s<2N 3s Copyright (C) 2005 Gner Arslan

-2s s -N N s 2s 351M Digital Signal Processing 3s 10 Demo Aliasing and Folding Demo Samplemania from John Hopkins University Copyright (C) 2005 Gner Arslan 351M Digital Signal Processing 11