EE 381 K14 Multidimensional DSP Multidimensional Resampling Lecture

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EE 381 K-14 Multidimensional DSP Multidimensional Resampling Lecture by Prof. Brian L. Evans Scribe:

EE 381 K-14 Multidimensional DSP Multidimensional Resampling Lecture by Prof. Brian L. Evans Scribe: Serene Banerjee Dept. of Electrical and Comp. Eng. The University of Texas at Austin

One-Dimensional Downsampling • Downsample by M – Input M samples with index – Output

One-Dimensional Downsampling • Downsample by M – Input M samples with index – Output first sample (discard M– 1 samples) • Discards data • May cause aliasing M x[n] xd[n] ki is called a coset ki ={0, 1, …, |M|-1} 2

One-Dimensional Downsampling 1 -WN 1/T -2 -3 /2 - /2 Xc(j. W) W WN

One-Dimensional Downsampling 1 -WN 1/T -2 -3 /2 - /2 Xc(j. W) W WN X ( ) /2 2 X d( ) M=3 3 /2 2 =WT Sample the analog bandlimited signal every T time units Downsampling by M generates baseband plus M-1 copies of baseband period of frequency domain =WT’ Aliasing occurs: avoid aliasing by prefiltering with lowpass filter with gain of 1 and cutoff of /M to extract baseband Fig. 3. 19(a)-(c) Oppenheim & Schafer, 1989. 3

One-Dimensional Upsampling • Upsample by L – Input one sample – Output input sample

One-Dimensional Upsampling • Upsample by L – Input one sample – Output input sample followed by L– 1 zeros L x[n] xu[n] • Adds data • May cause imaging 4

One-Dimensional Upsampling 1 -WN 1/T -2 - 1/T -5 /L -3 /L - /L

One-Dimensional Upsampling 1 -WN 1/T -2 - 1/T -5 /L -3 /L - /L 1/T = L/T -2 - /L Xc(j. W) X ( ) Sample the analog bandlimited signal every T time units 2 X u( ) = X(L ) Upsampling by L gives L images of baseband per 2 period of W WN /L X i( ) /L 3 /L =WT’ 2 =WT’ Apply lowpass interpolation filter with gain of L and cutoff of /L to extract baseband Fig. 3. 22 Oppenheim & Schafer, 1989. 5

1 -D Rational Rate Change • Change sampling rate by rational factor L M

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 /L (anti-imaging filter) • Decimate by M – Lowpass filter with a cutoff of /M (anti-aliasing filter) – Downsample by M 6

1 -D Resampling of Speech • Convert 48 k. Hz speech to 8 k.

1 -D Resampling of Speech • Convert 48 k. Hz speech to 8 k. Hz – 48 k. Hz sampling: 24 k. Hz analog bandwidth – 8 k. Hz sampling: 4 k. Hz analog bandwidth – Lowpass filter with anti-aliasing filter with cutoff at /6 and downsample by 6 • Convert 8 k. Hz speech to 48 k. Hz – Interpolate by 6 7

1 -D Resampling of Audio • Convert CD (44. 1 k. Hz) to DAT

1 -D Resampling of Audio • Convert CD (44. 1 k. Hz) to DAT (48 k. Hz) • Direct implementation 160 fs LPF 0= /160 fs LPF 147 0= /147 160 fs /147 Simplify LPF cascade to one LPF with 0= /160 Impractical because 160 fs = 7. 056 MHz 8

1 -D Resampling of Audio • Practical implementation – Perform resampling in three stages

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 9

Multidimensional Downsampling • Downsample by M – Input | det M | samples –

Multidimensional Downsampling • Downsample by M – Input | det M | samples – Output first sample and discard others • Discards data • May cause aliasing M x[n] xd[n] ki is a distinct coset vector 10

Coset Vectors • Indices in one fundamental tile of M – |det M| coset

Coset Vectors • Indices in one fundamental tile of M – |det M| coset vectors (origin always included) – Not unique for a given M (1, 1) (0, 0) (2, 1) (1, 0) Distinct coset vectors for 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 11

Multidimensional Upsampling • Upsample by L – Input one sample – Output the sample

Multidimensional Upsampling • Upsample by L – Input one sample – Output the sample and then | det L | - 1 zeros • Adds data • May cause imaging L x[n] xu[n] Xu(w) = X(LT w) 12

Example Upsampling Downsampling 13

Example Upsampling Downsampling 13

Conclusion • Rational rate change – In one dimension: – In multiple dimensions: •

Conclusion • Rational rate change – In one dimension: – In multiple dimensions: • Interpolation filter in N dimensions – Passband volume is (2 )N / | det L | – Baseband shape related to LT • Decimation filter in N dimensions – Passband volume is (2 )N / | det M | – Baseband shape related to M-T 14