EE381K-14 Multidimensional DSP Multidimensional Resampling

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EE381K-14 Multidimensional DSPMultidimensional
Resampling

• Lecture by Prof. Brian L. Evans
• Scribe Serene Banerjee
• Dept. of Electrical and Comp. Eng.
• The University of Texas at Austin

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One-Dimensional Downsampling

• Downsample by M
• Input M samples with index
• Output first sample(discard M1 samples)
• Discards data
• May cause aliasing

ki is called a cosetki 0, 1, , M-1
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One-Dimensional Downsampling
Xc(jW)
1
Sample the analog bandlimited signal every T time
units
W
WN
-WN
X (w)
1/T
Downsampling by M generates baseband plus M-1
copies of baseband per period of frequency domain
wWT
p/2
-p/2
2 p
-2 p
X d(w)
M3
wWT
-3p/2
3p/2
2 p
Aliasing occurs avoid aliasing by pre-filtering
with lowpass filter with gain of 1 and cutoff of
p/M to extract baseband
Fig. 3.19(a)-(c) Oppenheim Schafer, 1989.
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One-Dimensional Upsampling

• Upsample by L
• Input one sample
• Output input samplefollowed by L1 zeros
• Adds data
• May cause imaging

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One-Dimensional Upsampling
Xc(jW)
1
Sample the analog bandlimited signal every T time
units
W
WN
-WN
X (w)
1/T
Upsampling by L givesL images of baseband per 2
p period of w
wWT
p
-p
2 p
-2 p
X u(w) X(L w)
1/T
Apply lowpass interpolation filter with gain of L
and cutoff of p/L to extract baseband
wWT
p/L
3p/L
-p/L
-3p/L
-5p/L
X i(w)
1/T L/T
wWT
Fig. 3.22 Oppenheim Schafer, 1989.
p/L
2p
-p/L
-2p
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1-D Rational Rate Change

• Change sampling rate by rational factor L M -1
• Upsample by L
• Downsample by M
• Aliasing and imaging
• Change sampling rate by rational factor L M -1
• Interpolate by L
• Decimate by M

• Interpolate by L
• Upsample by L
• Lowpass filter with a cutoff of p/L(anti-imaging
  filter)
• Decimate by M
• Lowpass filter with a cutoff of
  p/M(anti-aliasing filter)
• Downsample by M

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1-D Resampling of Speech

• Convert 48 kHz speech to 8 kHz
• 48 kHz sampling 24 kHz analog bandwidth
• 8 kHz sampling 4 kHz analog bandwidth
• Lowpass filter with anti-aliasing filter with
  cutoff at ?/6 and downsample by 6
• Convert 8 kHz speech to 48 kHz
• Interpolate by 6

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1-D Resampling of Audio

• Convert CD (44.1 kHz) to DAT (48 kHz)
• Direct implementation
• Simplify LPF cascade to one LPF with w0p/160
• Impractical because 160 fs 7.056 MHz

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1-D Resampling of Audio

• Practical implementation
• Perform resampling in three stages
• First two stages increase sampling rate
• Alternative Linearly interpolate CD audio
• Interpolation pulse is a triangle (frequency
  response is sinc squared)
• Introduces high frequencies which will alias

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Multidimensional Downsampling

• Downsample by M
• Input det M samples
• Output first sampleand discard others
• Discards data
• May cause aliasing

ki is a distinct coset vector
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Coset Vectors

• Indices in one fundamental tile of M
• det M coset vectors (origin always included)
• Not unique for a given M
• Another choice of coset vectors for this M
  (0, 0) , (0, 1) , (1, 0) , (1, 1)
• Set of distinct coset vectors for M is unique

(1,1)
(2,1)
(0,0)
(1,0)
Distinct coset vectors for M
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Multidimensional Upsampling

• Upsample by L
• Input one sample
• Output the sample and then det L - 1 zeros
• Adds data
• May cause imaging

L
xn
xun
Xu(w) X(LT w)
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Example
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Conclusion

• Rational rate change
• In one dimension
• In multiple dimensions
• Interpolation filter in N dimensions
• Passband volume is (2p)N / det L
• Baseband shape related to LT
• Decimation filter in N dimensions
• Passband volume is (2p)N / det M
• Baseband shape related to M-T