Multidimensional Resampling Lecture by: Dr. Vishal Monga One-Dimensional Downsampling Downsample by M M Input M samples with index = K − x[n] xd[n] ki ,0 , M 1 Output first sample = (discard M–1 samples) xd [n] x[Mn ] Discards data ki is called a coset May cause aliasing ki ={0, 1, …, |M|-1} M −1 = 1 −1 − π X d ( ) ∑ X (M ( 2 ki )) M i=0 2 One-Dimensional DownsamplingΩ Xc(j ) 1 Sample the analog −Ω Ω Ω N N bandlimited signal X (ω) every T time units 1/ T Downsampling by M −2 π −π/2 π/2 2 π ω=Ω T generates baseband ω plus M-1 copies of 1 X d( ) M=3 MT baseband per period of frequency domain −3π/2 3π/2 2 π ω=Ω T’ Aliasing occurs: avoid aliasing by pre- filtering with lowpass filter with gain of Fig. 3.19(a)-(c) Oppenheim & 1 and cutoff of π/M to extract baseband Schafer, 1989. 3 One-Dimensional Upsampling Upsample by L L Input one sample x[n] xu[n] Output input sample followed by L–1 zeros = X u ( ) X (L ) Adds data May cause imaging −1 −1 ∈ ℑ = x[L n] if L n xu [n] 0 otherwise 4 One-Dimensional Upsampling Ω Xc(j ) 1 Sample the analog −Ω Ω Ω N N bandlimited signal X (ω) every T time units 1/ T Upsampling by L gives −2 π −π π 2 π ω=Ω T L images of baseband ω ω π ω X u( ) = X(L ) per 2 period of 1/ T Apply lowpass interpolation filter −5π/ −3π/ −π/ π/ 3π/ ω=Ω T’ L L L L L with gain of L and ω X i( ) π 1/ T = L/T cutoff of /L to extract baseband ω=Ω Fig. 3.22 Oppenheim & −2π −π/ L π/ L 2π T’ 5 Schafer, 1989. 1-D Rational Rate Change Change sampling rate Interpolate by L -1 by rational factor L M Upsample by L Upsample by L Lowpass filter with a π Downsample by M cutoff of /L (anti-imaging filter) Aliasing and imaging Decimate by M Change sampling rate -1 Lowpass filter with a by rational factor L M cutoff of π/M Interpolate by L (anti-aliasing filter) Decimate by M Downsample by M 6 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 π/6 and downsample by 6 Convert 8 kHz speech to 48 kHz Interpolate by 6 7 1-D Resampling of Audio Convert CD (44.1 kHz) to DAT (48 kHz) L = 48 kHz = 160 M 44 .1 kHz 147 Direct implementation LPF LPF 147 160 ω πππ ω πππ 0= /160 0= /147 fs 160 fs 160 fs 160 fs 160 fs /147 ω π Simplify LPF cascade to one LPF with 0= /160 Impractical because 160 fs = 7.056 MHz 8 1-D Resampling of Audio Practical implementation × × × × × L = 160 = 5 2 2 2 2 2 = 8 4 5 M 147 7× 7×3 7 3 7 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 M Input | det M | samples x[n] xd[n] Output first sample and discard others = xd [n] x[Mn ] Discards data ki is a distinct May cause aliasing coset vector det M −1 = 1 −T − π X d ( ) ∑ X (M ( 2 k i )) det M i=0 10 Coset Vectors Indices in one fundamental tile of M |det M| coset vectors (origin always included) Not unique for a given M (1,1) (2,1) 2 2 M = 2 0 (0,0) (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 L Input one sample x[n] xu[n] Output the sample and then | det L | - 1 zeros ωωω T ωωω Xu( ) = X(L ) Adds data May cause imaging −1 −1 ∈ = x[L n] if L n Rℑ xu [n] 0 otherwise 12 −1 −1 ∈ = x[L n] if L n Rℑ xu [n] 0 otherwise Example Upsampling 1 1 R = 1 −1 = xd [n] x[Mn ] Downsampling 13 Conclusion Rational rate change −1 In one dimension: H = L M −1 In multiple dimensions: H = L M Interpolation filter in N dimensions Passband volume is (2π) N / | det L | Baseband shape related to LT Decimation filter in N dimensions Passband volume is (2π) N / | det M | Baseband shape related to M-T 14.