3 Apr 19 EIE3001 Sig Sys Spring 2019 19 Recover using Sampled sequence x p n A

3 apr 19 eie3001 sig sys spring 2019 19 recover using

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3-Apr-19 EIE3001 Sig & Sys, Spring 2019 19 Recover using Sampled sequence x p [ n ] A continuous time signal x ( t )
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An Illustration: Reconstruction via Weighted Sum of Sinc(x) Functions 3-Apr-19 EIE3001 Sig & Sys, Spring 2019 20 Recover using Sampled sequence x p [ n ] A continuous time signal x ( t )
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A Brief Summary 3-Apr-19 EIE3001 Sig & Sys, Spring 2019 21 Original continuous time signal Discretized samples Recovered continuous time signal Modulating with impulse trains generates discrete samples Low pass filter with cutoff frequency chosen to retain the original spectra and reject all the high frequency replica
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Can We Really Recover All Signals? ¾ Apparently not. Given a set of samples, there could be multiple versions of interpolations. ¾ What is the sufficient condition for recovery? ¾ Intuitively, we hope the signal is “smooth”. 3-Apr-19 EIE3001 Sig & Sys, Spring 2019 22
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Band-limited Signal 3-Apr-19 EIE3001 Sig & Sys, Spring 2019 23
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Why Required Band-limited? 3-Apr-19 EIE3001 Sig & Sys, Spring 2019 24
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¾ Sampling Theorem (By Harry Nyquist, 1927): Let x ( t ) be a band-limited signal with X ( j Ȧ ) = 0 for | Ȧ | > Ȧ M . Then x ( t ) can be uniquely determined by its samples x ( nT ), n = 0, ±1, ±2, ±3..., if Ȧ s ±ʌ² T !±Ȧ M where Ȧ s is the sampling frequency and T the sampling period . Specifically: where ±Ȧ M < Ȧ s ; Ȧ M < Ȧ c < Ȧ s - Ȧ M ˗ The above equation is equivalent to multiplying x ( t ) by a periodic impulse training with period T and then passing the result through an idea low pass filter with gain T and cutoff frequency Ȧ c Sampling Theorem 3-Apr-19 EIE3001 Sig & Sys, Spring 2019 25 Typically,
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Aliasing 3-Apr-19 EIE3001 Sig & Sys, Spring 2019 26
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Aliasing in Real Life 3-Apr-19 EIE3001 Sig & Sys, Spring 2019 27 Screw propeller spins at a frequency much higher than human eyes can track It will appear in naked eyes a blurred image of propeller blades However, when captured by a camera, which typically samples at 24 – 30 fps, the blades appear as bended The reason is the sampling rate of the camera is smaller than the spinning rate of the blades, and this causes aliasing in the images captured by the camera ¾ Application of aliasing Using sampling (at camera or through flashing fluorescence) to bring down the frequency so that eyes can track
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Example 3-Apr-19 EIE3001 Sig & Sys, Spring 2019 28 [Video] A water experiment to demonstrate aliasing effect
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Aliasing from a Numerical Example ¾ 5HFDOO WKDW IRU D '7 FRPSOH[ VLQXVRLG³ DGGLQJ ±ʌ WR WKH frequency does not change anything e j ȍ n Ł e j ´ȍµ±ʌ¶ n ¾ If we have a CT complex sinusoid, and if we sample it at interval of T (or equivalently at a sampling frequency of Ȧ s ±ʌ ² T ), we cannot tell any difference if we add a frequency of Ȧ s to the frequency of the CT complex sinusoid 3-Apr-19 EIE3001 Sig & Sys, Spring 2019 29
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Aliasing from a Numerical Example We can illustrate this by the diagram below, where two sinusoids, one at low frequency and one at high frequency, give identical samples: 3-Apr-19 EIE3001 Sig & Sys, Spring 2019 30
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Other Interpolation Methods ¾ Using ideal low pass filter or the sinc function to reconstruct from the discrete samples may not be convenience, since the sinc function has infinitely long tails to infinity.
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