EE381K-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

7 - 2 M x[n]x[n]xd[n]xd[n] One-Dimensional Downsampling Downsample by M –Input M samples with index –Output first sample (discard M–1 samples) Discards data May cause aliasing k i is called a coset k i ={0, 1, …, |M|-1}

7 - 3 One-Dimensional Downsampling NN  N 1 Xc(j)Xc(j)   1/T X ()X ()  T  X d (  )  T’ Downsampling by M generates baseband plus M-1 copies of baseband per period of frequency domain Sample the analog bandlimited signal every T time units Aliasing occurs: avoid aliasing by pre- filtering with lowpass filter with gain of 1 and cutoff of  /M to extract baseband Fig. 3.19(a)-(c) Oppenheim & Schafer,    M=3

7 - 4 One-Dimensional Upsampling Upsample by L –Input one sample –Output input sample followed by L–1 zeros Adds data May cause imaging L x[n]x[n]xu[n]xu[n]

7 - 5 One-Dimensional Upsampling NN  N 1 Xc(j)Xc(j)   1/T X ()X ()  T   L 1/T X u (  ) = X(L  )  T’  L  L  L  L Upsampling by L gives L images of baseband per 2  period of  Sample the analog bandlimited signal every T time units Apply lowpass interpolation filter with gain of L and cutoff of  /L to extract baseband  L 1/T = L/T X i (  )  L   T’ Fig Oppenheim & Schafer, 1989.

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

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

LPF  0 =  / f s 160 f s /147fsfs LPF  0 =  /147 1-D Resampling of Audio Convert CD (44.1 kHz) to DAT (48 kHz) Direct implementation Simplify LPF cascade to one LPF with  0 =  /160 Impractical because 160 f s = MHz

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

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

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

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

Example Upsampling Downsampling

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 L T Decimation filter in N dimensions –Passband volume is  2  N / | det M | –Baseband shape related to M -T