090112Lecture04

090112Lecture04 - Lecture 2 Wi 09 Upsampling Upsampling a...

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2 Lecture 2 Wi 09 Upsampling Upsampling a signal x [ m ] by an integer factor L > 0 is an operation that inserts L -1 zeros between successive samples of x [ m ]. [] , . 0, otherwise nn x yn LL = Z L x [ m ] y [ n ] 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 x [ m ] y [ n ] n m L = 2 6 7 8 9 10
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3 Lecture 2 Wi 09 Transform analysis of upsampling : () [] n n n n n L kL k L Yz y nz n xz L xkz Xz =−∞ = = = = Z Then, ( ) jj L Ye Xe ωω =
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4 Lecture 2 Wi 09 Pictorial view of upsampling in frequency domain L = 2 2 () ( ) jj Ye Xe ω =
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5 Lecture 2 Wi 09 Downsampling 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 x [ m ] y [ n ] n m M = 2 Downsampling a signal x [ m ] by an integer factor M > 0 is an operation that keeps every M th sample and discards the rest. [] [ ] . yn xnM = M x [ m ] y [ n ]
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6 Lecture 2 Wi 09 Transform analysis of downsampling / () [] 1, 0, , 2 , where [ ] . 0, otherwise n n n n mM M m M Yz ynz xnM z xmc mz M cm =−∞ = = = ± = " 1 2/ 0 1 . M jk m M M k e M π = = We can show that (see H.W. 1 problem) 0 1 2 3 4 5 6 7 89 10 4 c 2 [ m ] m
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7 Lecture 2 Wi 09 Transform analysis of downsampling -2 / 1 2/ / 0 1 / 0 1 1 / 0 1 1 / 0 ( ) [] 1 1 1 ( ) 1 () mM M m M j kmM mk M km M j kM M m M jk M M k Yz xmc mz xm e z M xme z M xm e z M Xe z M π =−∞ ∞− = −∞ == = = = = = = ∑∑ 1 / 0 2 1 0 1 So, ( ) ( ) 1 M jj k M j M k M j M k Ye e M M ωπ ω = = = =
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8 Lecture 2 Wi 09 Pictorial view of downsampling in frequency domain M = 2 2 1 0 1 () ( ) j k M j j M k Ye Xe M ωπ ω = =
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9 Lecture 2 Wi 09 An analogy with sampling a CT signal 12 () ( ) j c k k Xe X j TT ω π =−∞ = 2 1 0 1 ( ) j k M j j M k Ye M ωπ = = ( ) j c k k =
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10 Lecture 2 Wi 09 Some Properties of up- and down-samplers • Upsamplers and downsamplers are linear systems. • Upsamplers and downsamplers are shift-variant systems. • Upsamplers and downsamplers do not commute in general. – Can show that upsamplers and downsamplers commute if and only if L and M are relatively prime. • Upsamplers and downsamplers do not commute with LSI systems. – Noble identities, coming up shortly, is the closest we can get. M x [ m ] y [ n ] L x [ m ] y [ n ]
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2 Lecture 3 Wi 09 Interpolation [] x n Discrete-time Processing x ( t ) Ideal sampling Ideal sampling Period = T s yn Period = T/L s Revisit: Interpolation is the method used to increase the sampling rate by an integer factor.
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3 Lecture 3 Wi 09 Interpolation = Upsampling + Filtering 0 π 2 π -2 ππ 0 π 2 π -2 0 π 2 π -2 L x [ n ] y 1 [ n ] Ideal LPF ω c = π / L DC Gain = L v [ n ] L =2
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4 Lecture 3 Wi 09 Why does this work? 12 () ( ) 2 ( ) / 1 2 / ( ) ( ) ( ) / j c k j c k jj L cc kk k Xe X j TT Lk Ye L k L V e X e Xj T T L ω ωω ωπ π =−∞ ∞∞ = = −− == = ∑∑ After ideal lowpass filtering with a cutoff freq. of π / L and a gain of L , 1 2/ 2 ( ) ( ) // j k l L L L l L T T L πω Z
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5 Lecture 3 Wi 09 Interpolation in the time domain 1 / integer [] , i n t e g e r 0, otherwise [][ ] [ ] ] k k kL r nn x vn LL yn vkhn k k xh n k L xrhn rL =−∞ = =− = L x [ n ] y 1 [ n ] Ideal LPF ω c = π / L DC Gain = L v [ n ]
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6 Lecture 3 Wi 09 Practical reconstruction using ZOH
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This note was uploaded on 04/18/2009 for the course ECE 700 taught by Professor Un during the Winter '09 term at Ohio State.

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090112Lecture04 - Lecture 2 Wi 09 Upsampling Upsampling a...

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