Downsampling Section 6, Nick Antipa, 3/9/2018 Based on slides by Jon Tamir
Some notes from Giulia Fanti, Frank Ong, Michael Lustig, Al Midterm 1
()*+ */0%1/) • ! " = ∑%&' , - . )*+ 5)*+ ()*+ = 3 , - .*/0%1/) + 3 , - .*/0%1/) + 3 , - .*/0%1/) %&' %&) %&5) • What is the problem with this? My favorite example of aliasing And
• http://i.imgur.com/8X8Fcoy.gifv Downsampling
x[n] N y[n]=x[nN]
• Compresses in the time domain • Expands in the frequency domain Alternative derivation of downsampling DTFT • Recall: A N-periodic sequence has a discrete Fourier series (DFS):
N 1 1 s˜(m)= S˜ ej2⇡mk/N DFS Representation N k kX=0 N 1 ˜ j2⇡km/N Sk = s˜(m)e DFS Coefficients m=0 X Alternative derivation of downsampling DTFT • Impulse train: s˜(m)= (m kN) k X2Z • DFS Coefficients (check):
S˜k =1
• DFS Representation: N 1 1 s˜(m)= ej2⇡mk/N N kX=0 Alternative derivation of downsampling DTFT
x[n] N y[n]=x[nN] Alternative derivation of downsampling DTFT
x[n] N y[n]=x[nN] Downsampling x[n] y[n] N
&' 1. Stretch ! "#$ to !(" ( ) 2. Create (N-1) copies of the stretched versions 3. Shift each copy by successive multiples of 2+ and add 4. Divide by N Decimation (filtering and downsampling) Chirp
Q: for a chirp of length L seconds, ranging from !" to !#, what is the expression for $% & , the continuous-time signal? ! − ! A: $ & = sin 2. ! + # " & & % " 1
Q: How fast do we need to sample this signal?
A: !2 > 2!#
Q: What is 45 6Ω ? (ignore effects of windowing)
Ω A: 9:;& 2Ω" Chirp
* − * ! # = sin 2) * + - + # # " + / Q: What is x[n] if we sample exactly at Nyquist? * − * ! 0 = sin 2) * + - + 10 10 + / Ω 0 0 = sin 0) + 1 − + Ω- 4 4 omega Q: Plot the frequency vs n ) Ω ) + Ω-
n 4 Downsample by 2 (without AA filter)
( - - Recall: ! " = sin "' ) 1 − + (* . .
( 1- 1- A: ! 2" = sin 2"' ) 1 − + (* . .
Q: What is the frequency vs n now? 2' Ω 2" 2" 2' 3 1 − + Ω4 5 5 Ω 2' 3 ' Ω4 Ω ' 3 Ω4 n 5 5 2 Chirp demo Imaging example
image of an important plot 144
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164 90 92 94 96 98 100 Decimated by 2
decimated by 2 decimated by 2 72 . . .
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82 45 46 47 48 49 50 Decimated by 2 again (4 total)
decimated by 4 decimated by 4 36
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blurred image of an important plot 144
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164 90 92 94 96 98 100 Resampling looks fine!
properly resampled image properly resampled image 36
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x[n] N y[n]
• Expands in the time domain • Compresses in the frequency domain
Y (ej!)=X(ej!N ) Interpolation (Upsampling and filtering) Interpolation
1. Smooth discrete-time signal • Low frequency content
2. Upsample by 3 • No longer smooth! Contains high frequencies
3. LPF • Removes high frequencies