11: Multirate Systems

11: Multirate Systems • Multirate Systems • Building blocks • Resampling Cascades • Noble Identities • Noble Identities Proof • Upsampled z-transform • Downsampled z-transform • Downsampled Spectrum • Power Spectral Density + • Perfect Reconstruction • Commutators • Summary 11: Multirate Systems • MATLAB routines

DSP and Digital Filters (2017-9045) Multirate: 11 – 1 / 14 Multirate Systems

11: Multirate Systems Multirate systems include more than one sample rate • Multirate Systems • Building blocks • Resampling Cascades • Noble Identities • Noble Identities Proof • Upsampled z-transform • Downsampled z-transform • Downsampled Spectrum • Power Spectral Density + • Perfect Reconstruction • Commutators • Summary • MATLAB routines

DSP and Digital Filters (2017-9045) Multirate: 11 – 2 / 14 Multirate Systems

11: Multirate Systems Multirate systems include more than one sample rate • Multirate Systems • Building blocks • Resampling Cascades Why bother?: • Noble Identities • Noble Identities Proof • May need to change the sample rate • Upsampled z-transform • Downsampled z-transform e.g. Audio sample rates include 32, 44.1, 48, 96 kHz • Downsampled Spectrum • Power Spectral Density + • Can relax analog or digital ﬁlter requirements • Perfect Reconstruction • Commutators e.g. Audio DAC increases sample rate so that the reconstruction ﬁlter • Summary • MATLAB routines can have a more gradual cutoff

DSP and Digital Filters (2017-9045) Multirate: 11 – 2 / 14 Multirate Systems

DSP and Digital Filters (2017-9045) Multirate: 11 – 2 / 14 Building blocks

11: Multirate Systems • Multirate Systems Downsample • Building blocks y[m] = x[Km] • Resampling Cascades • Noble Identities • Noble Identities Proof • Upsampled z-transform • Downsampled z-transform • Downsampled Spectrum • Power Spectral Density + • Perfect Reconstruction • Commutators • Summary • MATLAB routines

DSP and Digital Filters (2017-9045) Multirate: 11 – 3 / 14 Building blocks

11: Multirate Systems • Multirate Systems Downsample • Building blocks y[m] = x[Km] • Resampling Cascades • Noble Identities u n K | n • Noble Identities Proof Upsample v[n] = K • Upsampled z-transform 0 else • Downsampled z-transform ( • Downsampled Spectrum • Power Spectral Density + • Perfect Reconstruction • Commutators • Summary • MATLAB routines

DSP and Digital Filters (2017-9045) Multirate: 11 – 3 / 14 Building blocks

11: Multirate Systems • Multirate Systems Downsample • Building blocks y[m] = x[Km] • Resampling Cascades • Noble Identities u n K | n • Noble Identities Proof Upsample v[n] = K • Upsampled z-transform 0 else • Downsampled z-transform ( • Downsampled Spectrum • Power Spectral Density + Example: • Perfect Reconstruction Downsample by 3 then upsample by 4 • Commutators • Summary w[n] • MATLAB routines

DSP and Digital Filters (2017-9045) Multirate: 11 – 3 / 14 Building blocks

11: Multirate Systems • Multirate Systems Downsample • Building blocks y[m] = x[Km] • Resampling Cascades • Noble Identities u n K | n • Noble Identities Proof Upsample v[n] = K • Upsampled z-transform 0 else • Downsampled z-transform ( • Downsampled Spectrum • Power Spectral Density + Example: • Perfect Reconstruction Downsample by 3 then upsample by 4 • Commutators • Summary w[n] x[m] • MATLAB routines

0 0

DSP and Digital Filters (2017-9045) Multirate: 11 – 3 / 14 Building blocks

11: Multirate Systems • Multirate Systems Downsample • Building blocks y[m] = x[Km] • Resampling Cascades • Noble Identities u n K | n • Noble Identities Proof Upsample v[n] = K • Upsampled z-transform 0 else • Downsampled z-transform ( • Downsampled Spectrum • Power Spectral Density + Example: • Perfect Reconstruction Downsample by 3 then upsample by 4 • Commutators • Summary w[n] x[m] y[r] • MATLAB routines

0 0 0

DSP and Digital Filters (2017-9045) Multirate: 11 – 3 / 14 Building blocks

11: Multirate Systems • Multirate Systems Downsample • Building blocks y[m] = x[Km] • Resampling Cascades • Noble Identities u n K | n • Noble Identities Proof Upsample v[n] = K • Upsampled z-transform 0 else • Downsampled z-transform ( • Downsampled Spectrum • Power Spectral Density + Example: • Perfect Reconstruction Downsample by 3 then upsample by 4 • Commutators • Summary w[n] x[m] y[r] • MATLAB routines

0 0 0

• We use different index variables (n, m, r) for different sample rates

DSP and Digital Filters (2017-9045) Multirate: 11 – 3 / 14 Building blocks

11: Multirate Systems • Multirate Systems Downsample • Building blocks y[m] = x[Km] • Resampling Cascades • Noble Identities u n K | n • Noble Identities Proof Upsample v[n] = K • Upsampled z-transform 0 else • Downsampled z-transform ( • Downsampled Spectrum • Power Spectral Density + Example: • Perfect Reconstruction Downsample by 3 then upsample by 4 • Commutators • Summary w[n] x[m] y[r] • MATLAB routines

0 0 0

• We use different index variables (n, m, r) for different sample rates • Use different colours for signals at different rates (sometimes)

DSP and Digital Filters (2017-9045) Multirate: 11 – 3 / 14 Building blocks

11: Multirate Systems • Multirate Systems Downsample • Building blocks y[m] = x[Km] • Resampling Cascades • Noble Identities u n K | n • Noble Identities Proof Upsample v[n] = K • Upsampled z-transform 0 else • Downsampled z-transform ( • Downsampled Spectrum • Power Spectral Density + Example: • Perfect Reconstruction Downsample by 3 then upsample by 4 • Commutators • Summary w[n] x[m] y[r] • MATLAB routines

0 0 0

• We use different index variables (n, m, r) for different sample rates • Use different colours for signals at different rates (sometimes) • Synchronization: all signals have a sample at n = 0.

DSP and Digital Filters (2017-9045) Multirate: 11 – 3 / 14 Resampling Cascades

11: Multirate Systems • Multirate Systems • Building blocks Successive downsamplers or upsamplers • Resampling Cascades can be combined • Noble Identities • Noble Identities Proof • Upsampled z-transform • Downsampled z-transform • Downsampled Spectrum • Power Spectral Density + • Perfect Reconstruction • Commutators • Summary • MATLAB routines

DSP and Digital Filters (2017-9045) Multirate: 11 – 4 / 14 Resampling Cascades

11: Multirate Systems • Multirate Systems • Building blocks Successive downsamplers or upsamplers • Resampling Cascades can be combined • Noble Identities • Noble Identities Proof • Upsampled z-transform • Downsampled z-transform Upsampling can be exactly inverted • Downsampled Spectrum • Power Spectral Density + • Perfect Reconstruction • Commutators • Summary • MATLAB routines

DSP and Digital Filters (2017-9045) Multirate: 11 – 4 / 14 Resampling Cascades

11: Multirate Systems • Multirate Systems • Building blocks Successive downsamplers or upsamplers • Resampling Cascades can be combined • Noble Identities • Noble Identities Proof • Upsampled z-transform • Downsampled z-transform Upsampling can be exactly inverted • Downsampled Spectrum • Power Spectral Density + • Perfect Reconstruction • Commutators Downsampling destroys information • Summary • MATLAB routines permanently ⇒ uninvertible

DSP and Digital Filters (2017-9045) Multirate: 11 – 4 / 14 Resampling Cascades

11: Multirate Systems • Multirate Systems • Building blocks Successive downsamplers or upsamplers • Resampling Cascades can be combined • Noble Identities • Noble Identities Proof • Upsampled z-transform • Downsampled z-transform Upsampling can be exactly inverted • Downsampled Spectrum • Power Spectral Density + • Perfect Reconstruction • Commutators Downsampling destroys information • Summary • MATLAB routines permanently ⇒ uninvertible

Resampling can be interchanged iff P and Q are coprime (surprising!)

DSP and Digital Filters (2017-9045) Multirate: 11 – 4 / 14 Resampling Cascades

11: Multirate Systems • Multirate Systems • Building blocks Successive downsamplers or upsamplers • Resampling Cascades can be combined • Noble Identities • Noble Identities Proof • Upsampled z-transform • Downsampled z-transform Upsampling can be exactly inverted • Downsampled Spectrum • Power Spectral Density + • Perfect Reconstruction • Commutators Downsampling destroys information • Summary • MATLAB routines permanently ⇒ uninvertible

Resampling can be interchanged iff P and Q are coprime (surprising!)

1 P Proof: Left side: y[n] = w Q n = x Q n if Q | n else y[n] = 0. h i h i

[Note: a | b means “a divides into b exactly”]

DSP and Digital Filters (2017-9045) Multirate: 11 – 4 / 14 Resampling Cascades

11: Multirate Systems • Multirate Systems • Building blocks Successive downsamplers or upsamplers • Resampling Cascades can be combined • Noble Identities • Noble Identities Proof • Upsampled z-transform • Downsampled z-transform Upsampling can be exactly inverted • Downsampled Spectrum • Power Spectral Density + • Perfect Reconstruction • Commutators Downsampling destroys information • Summary • MATLAB routines permanently ⇒ uninvertible

Resampling can be interchanged iff P and Q are coprime (surprising!)

1 P Proof: Left side: y[n] = w Q n = x Q n if Q | n else y[n] = 0. h i hP i Right side: v[n] = u [Pn] = x Q n if Q | Pn. h i [Note: a | b means “a divides into b exactly”]

DSP and Digital Filters (2017-9045) Multirate: 11 – 4 / 14 Resampling Cascades

11: Multirate Systems • Multirate Systems • Building blocks Successive downsamplers or upsamplers • Resampling Cascades can be combined • Noble Identities • Noble Identities Proof • Upsampled z-transform • Downsampled z-transform Upsampling can be exactly inverted • Downsampled Spectrum • Power Spectral Density + • Perfect Reconstruction • Commutators Downsampling destroys information • Summary • MATLAB routines permanently ⇒ uninvertible

Resampling can be interchanged iff P and Q are coprime (surprising!)

1 P Proof: Left side: y[n] = w Q n = x Q n if Q | n else y[n] = 0. h i hP i Right side: v[n] = u [Pn] = x Q n if Q | Pn. But {Q | Pn ⇒ Q | n} iff P andh Qi are coprime. [Note: a | b means “a divides into b exactly”]

DSP and Digital Filters (2017-9045) Multirate: 11 – 4 / 14 Noble Identities

11: Multirate Systems • Multirate Systems • Building blocks Resamplers commute with addition • Resampling Cascades • Noble Identities and multiplication • Noble Identities Proof • Upsampled z-transform • Downsampled z-transform • Downsampled Spectrum • Power Spectral Density + • Perfect Reconstruction • Commutators • Summary • MATLAB routines

DSP and Digital Filters (2017-9045) Multirate: 11 – 5 / 14 Noble Identities

11: Multirate Systems • Multirate Systems • Building blocks Resamplers commute with addition • Resampling Cascades • Noble Identities and multiplication • Noble Identities Proof • Upsampled z-transform • Downsampled z-transform • Downsampled Spectrum • Power Spectral Density + Delays must be multiplied by the • Perfect Reconstruction resampling ratio • Commutators • Summary • MATLAB routines

DSP and Digital Filters (2017-9045) Multirate: 11 – 5 / 14 Noble Identities

11: Multirate Systems • Multirate Systems • Building blocks Resamplers commute with addition • Resampling Cascades • Noble Identities and multiplication • Noble Identities Proof • Upsampled z-transform • Downsampled z-transform • Downsampled Spectrum • Power Spectral Density + Delays must be multiplied by the • Perfect Reconstruction resampling ratio • Commutators • Summary • MATLAB routines Noble identities: Exchange resamplers and ﬁlters

DSP and Digital Filters (2017-9045) Multirate: 11 – 5 / 14 Noble Identities

11: Multirate Systems • Multirate Systems • Building blocks Resamplers commute with addition • Resampling Cascades • Noble Identities and multiplication • Noble Identities Proof • Upsampled z-transform • Downsampled z-transform • Downsampled Spectrum • Power Spectral Density + Delays must be multiplied by the • Perfect Reconstruction resampling ratio • Commutators • Summary • MATLAB routines Noble identities: Exchange resamplers and ﬁlters

1 2 Example: H(z) = h[0] + h[1]z− + h[2]z− + ··· H(z3) = h[0] + h[1]z−3 + h[2]z−6 + ···

DSP and Digital Filters (2017-9045) Multirate: 11 – 5 / 14 Noble Identities

11: Multirate Systems • Multirate Systems • Building blocks Resamplers commute with addition • Resampling Cascades • Noble Identities and multiplication • Noble Identities Proof • Upsampled z-transform • Downsampled z-transform • Downsampled Spectrum • Power Spectral Density + Delays must be multiplied by the • Perfect Reconstruction resampling ratio • Commutators • Summary • MATLAB routines Noble identities: Exchange resamplers and ﬁlters

Corrollary

1 2 Example: H(z) = h[0] + h[1]z− + h[2]z− + ··· H(z3) = h[0] + h[1]z−3 + h[2]z−6 + ···

DSP and Digital Filters (2017-9045) Multirate: 11 – 5 / 14 Noble Identities Proof

11: Multirate Systems • Multirate Systems Deﬁne hQ[n] to be the Q • Building blocks impulse response of H(z ). • Resampling Cascades • Noble Identities • Noble Identities Proof • Upsampled z-transform • Downsampled z-transform • Downsampled Spectrum • Power Spectral Density + • Perfect Reconstruction • Commutators • Summary • MATLAB routines

DSP and Digital Filters (2017-9045) Multirate: 11 – 6 / 14 Noble Identities Proof

11: Multirate Systems • Multirate Systems Deﬁne hQ[n] to be the Q • Building blocks impulse response of H(z ). • Resampling Cascades • Noble Identities Assume that h[r] is of length M + 1 so that hQ[n] is of length QM + 1. • Noble Identities Proof • Upsampled z-transform We know that hQ[n] = 0 except when Q | n and that h[r] = hQ[Qr]. • Downsampled z-transform • Downsampled Spectrum • Power Spectral Density + • Perfect Reconstruction • Commutators • Summary • MATLAB routines

DSP and Digital Filters (2017-9045) Multirate: 11 – 6 / 14 Noble Identities Proof

11: Multirate Systems • Multirate Systems Deﬁne hQ[n] to be the Q • Building blocks impulse response of H(z ). • Resampling Cascades • Noble Identities Assume that h[r] is of length M + 1 so that hQ[n] is of length QM + 1. • Noble Identities Proof • Upsampled z-transform We know that hQ[n] = 0 except when Q | n and that h[r] = hQ[Qr]. • Downsampled z-transform • Downsampled Spectrum w r v Qr • Power Spectral Density + [ ] = [ ] • Perfect Reconstruction • Commutators • Summary • MATLAB routines

DSP and Digital Filters (2017-9045) Multirate: 11 – 6 / 14 Noble Identities Proof

11: Multirate Systems • Multirate Systems Deﬁne hQ[n] to be the Q • Building blocks impulse response of H(z ). • Resampling Cascades • Noble Identities Assume that h[r] is of length M + 1 so that hQ[n] is of length QM + 1. • Noble Identities Proof • Upsampled z-transform We know that hQ[n] = 0 except when Q | n and that h[r] = hQ[Qr]. • Downsampled z-transform • Downsampled Spectrum w r v Qr QM h s x Qr s • Power Spectral Density + [ ] = [ ] = s=0 Q[ ] [ − ] • Perfect Reconstruction • Commutators P • Summary • MATLAB routines

DSP and Digital Filters (2017-9045) Multirate: 11 – 6 / 14 Noble Identities Proof

11: Multirate Systems • Multirate Systems Deﬁne hQ[n] to be the Q • Building blocks impulse response of H(z ). • Resampling Cascades • Noble Identities Assume that h[r] is of length M + 1 so that hQ[n] is of length QM + 1. • Noble Identities Proof • Upsampled z-transform We know that hQ[n] = 0 except when Q | n and that h[r] = hQ[Qr]. • Downsampled z-transform • Downsampled Spectrum w r v Qr QM h s x Qr s • Power Spectral Density + [ ] = [ ] = s=0 Q[ ] [ − ] • Perfect Reconstruction M = m=0 hQ[Qm]x[Qr − Qm] • Commutators P • Summary • MATLAB routines P

DSP and Digital Filters (2017-9045) Multirate: 11 – 6 / 14 Noble Identities Proof

11: Multirate Systems • Multirate Systems Deﬁne hQ[n] to be the Q • Building blocks impulse response of H(z ). • Resampling Cascades • Noble Identities Assume that h[r] is of length M + 1 so that hQ[n] is of length QM + 1. • Noble Identities Proof • Upsampled z-transform We know that hQ[n] = 0 except when Q | n and that h[r] = hQ[Qr]. • Downsampled z-transform • Downsampled Spectrum w r v Qr QM h s x Qr s • Power Spectral Density + [ ] = [ ] = s=0 Q[ ] [ − ] • Perfect Reconstruction M M = m=0 hQ[Qm]x[Qr − Qm] = m=0 h[m]x[Q(r − m)] • Commutators P • Summary • MATLAB routines P P

DSP and Digital Filters (2017-9045) Multirate: 11 – 6 / 14 Noble Identities Proof

11: Multirate Systems • Multirate Systems Deﬁne hQ[n] to be the Q • Building blocks impulse response of H(z ). • Resampling Cascades • Noble Identities Assume that h[r] is of length M + 1 so that hQ[n] is of length QM + 1. • Noble Identities Proof • Upsampled z-transform We know that hQ[n] = 0 except when Q | n and that h[r] = hQ[Qr]. • Downsampled z-transform • Downsampled Spectrum w r v Qr QM h s x Qr s • Power Spectral Density + [ ] = [ ] = s=0 Q[ ] [ − ] • Perfect Reconstruction M M • = m=0 hQ[Qm]x[Qr − Qm] = m=0 h[m]x[Q(r − m)] Commutators M P • Summary = m=0 h[m]u[r − m] • MATLAB routines P P P

DSP and Digital Filters (2017-9045) Multirate: 11 – 6 / 14 Noble Identities Proof

11: Multirate Systems • Multirate Systems Deﬁne hQ[n] to be the Q • Building blocks impulse response of H(z ). • Resampling Cascades • Noble Identities Assume that h[r] is of length M + 1 so that hQ[n] is of length QM + 1. • Noble Identities Proof • Upsampled z-transform We know that hQ[n] = 0 except when Q | n and that h[r] = hQ[Qr]. • Downsampled z-transform • Downsampled Spectrum w r v Qr QM h s x Qr s • Power Spectral Density + [ ] = [ ] = s=0 Q[ ] [ − ] • Perfect Reconstruction M M • = m=0 hQ[Qm]x[Qr − Qm] = m=0 h[m]x[Q(r − m)] Commutators M P • Summary , = m=0 h[m]u[r − m] = y[r] • MATLAB routines P P P

DSP and Digital Filters (2017-9045) Multirate: 11 – 6 / 14 Noble Identities Proof

11: Multirate Systems • Multirate Systems Deﬁne hQ[n] to be the Q • Building blocks impulse response of H(z ). • Resampling Cascades • Noble Identities Assume that h[r] is of length M + 1 so that hQ[n] is of length QM + 1. • Noble Identities Proof • Upsampled z-transform We know that hQ[n] = 0 except when Q | n and that h[r] = hQ[Qr]. • Downsampled z-transform • Downsampled Spectrum w r v Qr QM h s x Qr s • Power Spectral Density + [ ] = [ ] = s=0 Q[ ] [ − ] • Perfect Reconstruction M M • = m=0 hQ[Qm]x[Qr − Qm] = m=0 h[m]x[Q(r − m)] Commutators M P • Summary , = m=0 h[m]u[r − m] = y[r] • MATLAB routines P P UpsampledP Noble Identity:

DSP and Digital Filters (2017-9045) Multirate: 11 – 6 / 14 Noble Identities Proof

11: Multirate Systems • Multirate Systems Deﬁne hQ[n] to be the Q • Building blocks impulse response of H(z ). • Resampling Cascades • Noble Identities Assume that h[r] is of length M + 1 so that hQ[n] is of length QM + 1. • Noble Identities Proof • Upsampled z-transform We know that hQ[n] = 0 except when Q | n and that h[r] = hQ[Qr]. • Downsampled z-transform • Downsampled Spectrum w r v Qr QM h s x Qr s • Power Spectral Density + [ ] = [ ] = s=0 Q[ ] [ − ] • Perfect Reconstruction M M • = m=0 hQ[Qm]x[Qr − Qm] = m=0 h[m]x[Q(r − m)] Commutators M P • Summary , = m=0 h[m]u[r − m] = y[r] • MATLAB routines P P UpsampledP Noble Identity:

We know that v[n] = 0 except when Q | n and that v[Qr] = x[r].

DSP and Digital Filters (2017-9045) Multirate: 11 – 6 / 14 Noble Identities Proof

11: Multirate Systems • Multirate Systems Deﬁne hQ[n] to be the Q • Building blocks impulse response of H(z ). • Resampling Cascades • Noble Identities Assume that h[r] is of length M + 1 so that hQ[n] is of length QM + 1. • Noble Identities Proof • Upsampled z-transform We know that hQ[n] = 0 except when Q | n and that h[r] = hQ[Qr]. • Downsampled z-transform • Downsampled Spectrum w r v Qr QM h s x Qr s • Power Spectral Density + [ ] = [ ] = s=0 Q[ ] [ − ] • Perfect Reconstruction M M • = m=0 hQ[Qm]x[Qr − Qm] = m=0 h[m]x[Q(r − m)] Commutators M P • Summary , = m=0 h[m]u[r − m] = y[r] • MATLAB routines P P UpsampledP Noble Identity:

We know that v[n] = 0 except when Q | n and that v[Qr] = x[r]. QM w[n] = s=0 hQ[s]v[n − s] P

DSP and Digital Filters (2017-9045) Multirate: 11 – 6 / 14 Noble Identities Proof

11: Multirate Systems • Multirate Systems Deﬁne hQ[n] to be the Q • Building blocks impulse response of H(z ). • Resampling Cascades • Noble Identities Assume that h[r] is of length M + 1 so that hQ[n] is of length QM + 1. • Noble Identities Proof • Upsampled z-transform We know that hQ[n] = 0 except when Q | n and that h[r] = hQ[Qr]. • Downsampled z-transform • Downsampled Spectrum w r v Qr QM h s x Qr s • Power Spectral Density + [ ] = [ ] = s=0 Q[ ] [ − ] • Perfect Reconstruction M M • = m=0 hQ[Qm]x[Qr − Qm] = m=0 h[m]x[Q(r − m)] Commutators M P • Summary , = m=0 h[m]u[r − m] = y[r] • MATLAB routines P P UpsampledP Noble Identity:

We know that v[n] = 0 except when Q | n and that v[Qr] = x[r]. QM M w[n] = s=0 hQ[s]v[n − s] = m=0 hQ[Qm]v[n − Qm] P P

DSP and Digital Filters (2017-9045) Multirate: 11 – 6 / 14 Noble Identities Proof

11: Multirate Systems • Multirate Systems Deﬁne hQ[n] to be the Q • Building blocks impulse response of H(z ). • Resampling Cascades • Noble Identities Assume that h[r] is of length M + 1 so that hQ[n] is of length QM + 1. • Noble Identities Proof • Upsampled z-transform We know that hQ[n] = 0 except when Q | n and that h[r] = hQ[Qr]. • Downsampled z-transform • Downsampled Spectrum w r v Qr QM h s x Qr s • Power Spectral Density + [ ] = [ ] = s=0 Q[ ] [ − ] • Perfect Reconstruction M M • = m=0 hQ[Qm]x[Qr − Qm] = m=0 h[m]x[Q(r − m)] Commutators M P • Summary , = m=0 h[m]u[r − m] = y[r] • MATLAB routines P P UpsampledP Noble Identity:

We know that v[n] = 0 except when Q | n and that v[Qr] = x[r]. QM M w[n] = s=0 hQ[s]v[n − s] = m=0 hQ[Qm]v[n − Qm] = M h[m]v[n − Qm] Pm=0 P P

DSP and Digital Filters (2017-9045) Multirate: 11 – 6 / 14 Noble Identities Proof

11: Multirate Systems • Multirate Systems Deﬁne hQ[n] to be the Q • Building blocks impulse response of H(z ). • Resampling Cascades • Noble Identities Assume that h[r] is of length M + 1 so that hQ[n] is of length QM + 1. • Noble Identities Proof • Upsampled z-transform We know that hQ[n] = 0 except when Q | n and that h[r] = hQ[Qr]. • Downsampled z-transform • Downsampled Spectrum w r v Qr QM h s x Qr s • Power Spectral Density + [ ] = [ ] = s=0 Q[ ] [ − ] • Perfect Reconstruction M M • = m=0 hQ[Qm]x[Qr − Qm] = m=0 h[m]x[Q(r − m)] Commutators M P • Summary , = m=0 h[m]u[r − m] = y[r] • MATLAB routines P P UpsampledP Noble Identity:

We know that v[n] = 0 except when Q | n and that v[Qr] = x[r]. QM M w[n] = s=0 hQ[s]v[n − s] = m=0 hQ[Qm]v[n − Qm] = M h[m]v[n − Qm] Pm=0 P If Q ∤ n, thenP v[n − Qm] = 0 ∀m so w[n]=0= y[n]

DSP and Digital Filters (2017-9045) Multirate: 11 – 6 / 14 Noble Identities Proof

11: Multirate Systems • Multirate Systems Deﬁne hQ[n] to be the Q • Building blocks impulse response of H(z ). • Resampling Cascades • Noble Identities Assume that h[r] is of length M + 1 so that hQ[n] is of length QM + 1. • Noble Identities Proof • Upsampled z-transform We know that hQ[n] = 0 except when Q | n and that h[r] = hQ[Qr]. • Downsampled z-transform • Downsampled Spectrum w r v Qr QM h s x Qr s • Power Spectral Density + [ ] = [ ] = s=0 Q[ ] [ − ] • Perfect Reconstruction M M • = m=0 hQ[Qm]x[Qr − Qm] = m=0 h[m]x[Q(r − m)] Commutators M P • Summary , = m=0 h[m]u[r − m] = y[r] • MATLAB routines P P UpsampledP Noble Identity:

We know that v[n] = 0 except when Q | n and that v[Qr] = x[r]. QM M w[n] = s=0 hQ[s]v[n − s] = m=0 hQ[Qm]v[n − Qm] = M h[m]v[n − Qm] Pm=0 P If Q ∤ n, thenP v[n − Qm] = 0 ∀m so w[n]=0= y[n] M If Q | n = Qr, then w[Qr] = m=0 h[m]v[Qr − Qm] P DSP and Digital Filters (2017-9045) Multirate: 11 – 6 / 14 Noble Identities Proof

11: Multirate Systems • Multirate Systems Deﬁne hQ[n] to be the Q • Building blocks impulse response of H(z ). • Resampling Cascades • Noble Identities Assume that h[r] is of length M + 1 so that hQ[n] is of length QM + 1. • Noble Identities Proof • Upsampled z-transform We know that hQ[n] = 0 except when Q | n and that h[r] = hQ[Qr]. • Downsampled z-transform • Downsampled Spectrum w r v Qr QM h s x Qr s • Power Spectral Density + [ ] = [ ] = s=0 Q[ ] [ − ] • Perfect Reconstruction M M • = m=0 hQ[Qm]x[Qr − Qm] = m=0 h[m]x[Q(r − m)] Commutators M P • Summary , = m=0 h[m]u[r − m] = y[r] • MATLAB routines P P UpsampledP Noble Identity:

We know that v[n] = 0 except when Q | n and that v[Qr] = x[r]. QM M w[n] = s=0 hQ[s]v[n − s] = m=0 hQ[Qm]v[n − Qm] = M h[m]v[n − Qm] Pm=0 P If Q ∤ n, thenP v[n − Qm] = 0 ∀m so w[n]=0= y[n] M If Q | n = Qr, then w[Qr] = m=0 h[m]v[Qr − Qm] = M h[m]x[r − m] = u[r] mP=0 DSP and Digital Filters (2017-9045) P Multirate: 11 – 6 / 14 Noble Identities Proof

11: Multirate Systems • Multirate Systems Deﬁne hQ[n] to be the Q • Building blocks impulse response of H(z ). • Resampling Cascades • Noble Identities Assume that h[r] is of length M + 1 so that hQ[n] is of length QM + 1. • Noble Identities Proof • Upsampled z-transform We know that hQ[n] = 0 except when Q | n and that h[r] = hQ[Qr]. • Downsampled z-transform • Downsampled Spectrum w r v Qr QM h s x Qr s • Power Spectral Density + [ ] = [ ] = s=0 Q[ ] [ − ] • Perfect Reconstruction M M • = m=0 hQ[Qm]x[Qr − Qm] = m=0 h[m]x[Q(r − m)] Commutators M P • Summary , = m=0 h[m]u[r − m] = y[r] • MATLAB routines P P UpsampledP Noble Identity:

We know that v[n] = 0 except when Q | n and that v[Qr] = x[r]. QM M w[n] = s=0 hQ[s]v[n − s] = m=0 hQ[Qm]v[n − Qm] = M h[m]v[n − Qm] Pm=0 P If Q ∤ n, thenP v[n − Qm] = 0 ∀m so w[n]=0= y[n] M If Q | n = Qr, then w[Qr] = m=0 h[m]v[Qr − Qm] = M h[m]x[r − m] = u[r] = y[Qr] , mP=0 DSP and Digital Filters (2017-9045) P Multirate: 11 – 6 / 14 Upsampled z-transform

11: Multirate Systems −n • Multirate Systems V (z) = n v[n]z • Building blocks • Resampling Cascades P • Noble Identities • Noble Identities Proof • Upsampled z-transform • Downsampled z-transform • Downsampled Spectrum • Power Spectral Density + • Perfect Reconstruction • Commutators • Summary • MATLAB routines

DSP and Digital Filters (2017-9045) Multirate: 11 – 7 / 14 Upsampled z-transform

11: Multirate Systems V z v n z−n u n z−n • Multirate Systems ( ) = n [ ] = n s.t.K|n [ K ] • Building blocks • Resampling Cascades P P • Noble Identities • Noble Identities Proof • Upsampled z-transform • Downsampled z-transform • Downsampled Spectrum • Power Spectral Density + • Perfect Reconstruction • Commutators • Summary • MATLAB routines

DSP and Digital Filters (2017-9045) Multirate: 11 – 7 / 14 Upsampled z-transform

11: Multirate Systems V z v n z−n u n z−n • Multirate Systems ( ) = n [ ] = n s.t.K|n [ K ] • Building blocks • Resampling Cascades = P u[m]z−KmP • Noble Identities m • Noble Identities Proof • Upsampled z-transform P • Downsampled z-transform • Downsampled Spectrum • Power Spectral Density + • Perfect Reconstruction • Commutators • Summary • MATLAB routines

DSP and Digital Filters (2017-9045) Multirate: 11 – 7 / 14 Upsampled z-transform

11: Multirate Systems V z v n z−n u n z−n • Multirate Systems ( ) = n [ ] = n s.t.K|n [ K ] • Building blocks • Resampling Cascades = P u[m]z−KmP= U(zK ) • Noble Identities m • Noble Identities Proof • Upsampled z-transform P • Downsampled z-transform jω jKω • Downsampled Spectrum Spectrum: V (e ) = U(e ) • Power Spectral Density + • Perfect Reconstruction • Commutators • Summary • MATLAB routines

DSP and Digital Filters (2017-9045) Multirate: 11 – 7 / 14 Upsampled z-transform

11: Multirate Systems V z v n z−n u n z−n • Multirate Systems ( ) = n [ ] = n s.t.K|n [ K ] • Building blocks • Resampling Cascades = P u[m]z−KmP= U(zK ) • Noble Identities m • Noble Identities Proof • Upsampled z-transform P • Downsampled z-transform jω jKω • Downsampled Spectrum Spectrum: V (e ) = U(e ) • Power Spectral Density + Spectrum is horizontally shrunk and replicated K times. • Perfect Reconstruction • Commutators • Summary • MATLAB routines

DSP and Digital Filters (2017-9045) Multirate: 11 – 7 / 14 Upsampled z-transform

11: Multirate Systems V z v n z−n u n z−n • Multirate Systems ( ) = n [ ] = n s.t.K|n [ K ] • Building blocks • Resampling Cascades = P u[m]z−KmP= U(zK ) • Noble Identities m • Noble Identities Proof • Upsampled z-transform P • Downsampled z-transform jω jKω • Downsampled Spectrum Spectrum: V (e ) = U(e ) • Power Spectral Density + Spectrum is horizontally shrunk and replicated K times. • Perfect Reconstruction • Commutators • Summary • MATLAB routines

Example:

0.5

0 -2 0 2 ω

DSP and Digital Filters (2017-9045) Multirate: 11 – 7 / 14 Upsampled z-transform

11: Multirate Systems V z v n z−n u n z−n • Multirate Systems ( ) = n [ ] = n s.t.K|n [ K ] • Building blocks • Resampling Cascades = P u[m]z−KmP= U(zK ) • Noble Identities m • Noble Identities Proof • Upsampled z-transform P • Downsampled z-transform jω jKω • Downsampled Spectrum Spectrum: V (e ) = U(e ) • Power Spectral Density + Spectrum is horizontally shrunk and replicated K times. • Perfect Reconstruction • Commutators • Summary • MATLAB routines

Example: K = 3: three images of the original spectrum in all.

1 1

0.5 0.5

0 0 -2 0 2 -2 0 2 ω ω

DSP and Digital Filters (2017-9045) Multirate: 11 – 7 / 14 Upsampled z-transform

11: Multirate Systems V z v n z−n u n z−n • Multirate Systems ( ) = n [ ] = n s.t.K|n [ K ] • Building blocks • Resampling Cascades = P u[m]z−KmP= U(zK ) • Noble Identities m • Noble Identities Proof • Upsampled z-transform P • Downsampled z-transform jω jKω • Downsampled Spectrum Spectrum: V (e ) = U(e ) • Power Spectral Density + • Spectrum is horizontally shrunk and replicated K times. Perfect Reconstruction 1 • Commutators Total energy unchanged; power (= energy/sample) multiplied by K • Summary • MATLAB routines

Example: K = 3: three images of the original spectrum in all.

1 1

0.5 0.5

0 0 -2 0 2 -2 0 2 ω ω

DSP and Digital Filters (2017-9045) Multirate: 11 – 7 / 14 Upsampled z-transform

11: Multirate Systems V z v n z−n u n z−n • Multirate Systems ( ) = n [ ] = n s.t.K|n [ K ] • Building blocks • Resampling Cascades = P u[m]z−KmP= U(zK ) • Noble Identities m • Noble Identities Proof • Upsampled z-transform P • Downsampled z-transform jω jKω • Downsampled Spectrum Spectrum: V (e ) = U(e ) • Power Spectral Density + • Spectrum is horizontally shrunk and replicated K times. Perfect Reconstruction 1 • Commutators Total energy unchanged; power (= energy/sample) multiplied by K • Summary • MATLAB routines

Example: K = 3: three images of the original spectrum in all. 2 2 1 jω 1 jω Energy unchanged: 2π U(e ) dω = 2π V (e ) dω

1 R 1 R

0.5 0.5

0 0 -2 0 2 -2 0 2 ω ω

DSP and Digital Filters (2017-9045) Multirate: 11 – 7 / 14 Upsampled z-transform

11: Multirate Systems V z v n z−n u n z−n • Multirate Systems ( ) = n [ ] = n s.t.K|n [ K ] • Building blocks • Resampling Cascades = P u[m]z−KmP= U(zK ) • Noble Identities m • Noble Identities Proof • Upsampled z-transform P • Downsampled z-transform jω jKω • Downsampled Spectrum Spectrum: V (e ) = U(e ) • Power Spectral Density + • Spectrum is horizontally shrunk and replicated K times. Perfect Reconstruction 1 • Commutators Total energy unchanged; power (= energy/sample) multiplied by K • Summary • MATLAB routines Upsampling normally followed by a LP ﬁlter to remove images.

Example: K = 3: three images of the original spectrum in all. 2 2 1 jω 1 jω Energy unchanged: 2π U(e ) dω = 2π V (e ) dω

1 R 1 R

0.5 0.5

0 0 -2 0 2 -2 0 2 ω ω

DSP and Digital Filters (2017-9045) Multirate: 11 – 7 / 14 Downsampled z-transform

11: Multirate Systems • Multirate Systems Deﬁne cK [n] = δK|n[n] • Building blocks • Resampling Cascades • Noble Identities • Noble Identities Proof • Upsampled z-transform • Downsampled z-transform • Downsampled Spectrum • Power Spectral Density + • Perfect Reconstruction • Commutators • Summary • MATLAB routines

DSP and Digital Filters (2017-9045) Multirate: 11 – 8 / 14 Downsampled z-transform

11: Multirate Systems 1 K−1 j2πkn • K Multirate Systems Deﬁne cK [n] = δK|n[n] = K k=0 e • Building blocks • Resampling Cascades • Noble Identities P • Noble Identities Proof • Upsampled z-transform • Downsampled z-transform • Downsampled Spectrum • Power Spectral Density + • Perfect Reconstruction • Commutators • Summary • MATLAB routines

DSP and Digital Filters (2017-9045) Multirate: 11 – 8 / 14 Downsampled z-transform

11: Multirate Systems 1 K−1 j2πkn • K Multirate Systems Deﬁne cK [n] = δK|n[n] = K k=0 e • Building blocks • Resampling Cascades • Noble Identities x[n] KP| n • Noble Identities Proof Now deﬁne xK [n] = • Upsampled z-transform (0 K ∤ n • Downsampled z-transform • Downsampled Spectrum • Power Spectral Density + • Perfect Reconstruction • Commutators • Summary • MATLAB routines

DSP and Digital Filters (2017-9045) Multirate: 11 – 8 / 14 Downsampled z-transform

11: Multirate Systems 1 K−1 j2πkn • K Multirate Systems Deﬁne cK [n] = δK|n[n] = K k=0 e • Building blocks • Resampling Cascades • Noble Identities x[n] KP| n • Noble Identities Proof Now deﬁne xK [n] = = cK [n]x[n] • Upsampled z-transform (0 K ∤ n • Downsampled z-transform • Downsampled Spectrum • Power Spectral Density + • Perfect Reconstruction • Commutators • Summary • MATLAB routines

DSP and Digital Filters (2017-9045) Multirate: 11 – 8 / 14 Downsampled z-transform

11: Multirate Systems 1 K−1 j2πkn • K Multirate Systems Deﬁne cK [n] = δK|n[n] = K k=0 e • Building blocks • Resampling Cascades • Noble Identities x[n] KP| n • Noble Identities Proof Now deﬁne xK [n] = = cK [n]x[n] • Upsampled z-transform (0 K ∤ n • Downsampled z-transform −n • Downsampled Spectrum XK (z) = n xK [n]z • Power Spectral Density + • Perfect Reconstruction • Commutators P • Summary • MATLAB routines

DSP and Digital Filters (2017-9045) Multirate: 11 – 8 / 14 Downsampled z-transform

11: Multirate Systems 1 K−1 j2πkn • K Multirate Systems Deﬁne cK [n] = δK|n[n] = K k=0 e • Building blocks • Resampling Cascades • Noble Identities x[n] KP| n • Noble Identities Proof Now deﬁne xK [n] = = cK [n]x[n] • Upsampled z-transform (0 K ∤ n • Downsampled z-transform 1 K−1 j2πkn −n K −n • Downsampled Spectrum XK (z) = n xK [n]z = K n k=0 e x[n]z • Power Spectral Density + • Perfect Reconstruction • Commutators P P P • Summary • MATLAB routines

DSP and Digital Filters (2017-9045) Multirate: 11 – 8 / 14 Downsampled z-transform

11: Multirate Systems 1 K−1 j2πkn • K Multirate Systems Deﬁne cK [n] = δK|n[n] = K k=0 e • Building blocks • Resampling Cascades • Noble Identities x[n] KP| n • Noble Identities Proof Now deﬁne xK [n] = = cK [n]x[n] • Upsampled z-transform (0 K ∤ n • Downsampled z-transform 1 K−1 j2πkn −n K −n • Downsampled Spectrum XK (z) = n xK [n]z = K n k=0 e x[n]z • Power Spectral Density + −j2πk −n • 1 K−1 Perfect Reconstruction = =0 x[n] e K z • Commutators PK k n P P • Summary • MATLAB routines P P

DSP and Digital Filters (2017-9045) Multirate: 11 – 8 / 14 Downsampled z-transform

11: Multirate Systems 1 K−1 j2πkn • K Multirate Systems Deﬁne cK [n] = δK|n[n] = K k=0 e • Building blocks • Resampling Cascades • Noble Identities x[n] KP| n • Noble Identities Proof Now deﬁne xK [n] = = cK [n]x[n] • Upsampled z-transform (0 K ∤ n • Downsampled z-transform 1 K−1 j2πkn −n K −n • Downsampled Spectrum XK (z) = n xK [n]z = K n k=0 e x[n]z • Power Spectral Density + −j2πk −n −j2πk • 1 K−1 1 K−1 Perfect Reconstruction = =0 x[n] e K z = =0 X(e K z) • Commutators PK k n P P K k • Summary • MATLAB routines P P P

DSP and Digital Filters (2017-9045) Multirate: 11 – 8 / 14 Downsampled z-transform

11: Multirate Systems 1 K−1 j2πkn • K Multirate Systems Deﬁne cK [n] = δK|n[n] = K k=0 e • Building blocks • Resampling Cascades • Noble Identities x[n] KP| n • Noble Identities Proof Now deﬁne xK [n] = = cK [n]x[n] • Upsampled z-transform (0 K ∤ n • Downsampled z-transform 1 K−1 j2πkn −n K −n • Downsampled Spectrum XK (z) = n xK [n]z = K n k=0 e x[n]z • Power Spectral Density + −j2πk −n −j2πk • 1 K−1 1 K−1 Perfect Reconstruction = =0 x[n] e K z = =0 X(e K z) • Commutators PK k n P P K k • Summary • MATLAB routines P P P From previous slide:

K XK(z) = Y (z )

DSP and Digital Filters (2017-9045) Multirate: 11 – 8 / 14 Downsampled z-transform

11: Multirate Systems 1 K−1 j2πkn • K Multirate Systems Deﬁne cK [n] = δK|n[n] = K k=0 e • Building blocks • Resampling Cascades • Noble Identities x[n] KP| n • Noble Identities Proof Now deﬁne xK [n] = = cK [n]x[n] • Upsampled z-transform (0 K ∤ n • Downsampled z-transform 1 K−1 j2πkn −n K −n • Downsampled Spectrum XK (z) = n xK [n]z = K n k=0 e x[n]z • Power Spectral Density + −j2πk −n −j2πk • 1 K−1 1 K−1 Perfect Reconstruction = =0 x[n] e K z = =0 X(e K z) • Commutators PK k n P P K k • Summary • MATLAB routines P P P From previous slide:

K XK(z) = Y (z ) 1 ⇒ Y (z) = XK (z K )

DSP and Digital Filters (2017-9045) Multirate: 11 – 8 / 14 Downsampled z-transform

11: Multirate Systems 1 K−1 j2πkn • K Multirate Systems Deﬁne cK [n] = δK|n[n] = K k=0 e • Building blocks • Resampling Cascades • Noble Identities x[n] KP| n • Noble Identities Proof Now deﬁne xK [n] = = cK [n]x[n] • Upsampled z-transform (0 K ∤ n • Downsampled z-transform 1 K−1 j2πkn −n K −n • Downsampled Spectrum XK (z) = n xK [n]z = K n k=0 e x[n]z • Power Spectral Density + −j2πk −n −j2πk • 1 K−1 1 K−1 Perfect Reconstruction = =0 x[n] e K z = =0 X(e K z) • Commutators PK k n P P K k • Summary • MATLAB routines P P P From previous slide:

K XK(z) = Y (z ) 1 1 K−1 −j2πk 1 K K K ⇒ Y (z) = XK (z ) = K k=0 X(e z ) P

DSP and Digital Filters (2017-9045) Multirate: 11 – 8 / 14 Downsampled z-transform

11: Multirate Systems 1 K−1 j2πkn • K Multirate Systems Deﬁne cK [n] = δK|n[n] = K k=0 e • Building blocks • Resampling Cascades • Noble Identities x[n] KP| n • Noble Identities Proof Now deﬁne xK [n] = = cK [n]x[n] • Upsampled z-transform (0 K ∤ n • Downsampled z-transform 1 K−1 j2πkn −n K −n • Downsampled Spectrum XK (z) = n xK [n]z = K n k=0 e x[n]z • Power Spectral Density + −j2πk −n −j2πk • 1 K−1 1 K−1 Perfect Reconstruction = =0 x[n] e K z = =0 X(e K z) • Commutators PK k n P P K k • Summary • MATLAB routines P P P From previous slide:

K XK(z) = Y (z ) 1 1 K−1 −j2πk 1 K K K ⇒ Y (z) = XK (z ) = K k=0 X(e z ) Frequency Spectrum: P 1 K−1 j(ω−2πk) jω K Y (e ) = K k=0 X(e ) P

DSP and Digital Filters (2017-9045) Multirate: 11 – 8 / 14 Downsampled z-transform

11: Multirate Systems 1 K−1 j2πkn • K Multirate Systems Deﬁne cK [n] = δK|n[n] = K k=0 e • Building blocks • Resampling Cascades • Noble Identities x[n] KP| n • Noble Identities Proof Now deﬁne xK [n] = = cK [n]x[n] • Upsampled z-transform (0 K ∤ n • Downsampled z-transform 1 K−1 j2πkn −n K −n • Downsampled Spectrum XK (z) = n xK [n]z = K n k=0 e x[n]z • Power Spectral Density + −j2πk −n −j2πk • 1 K−1 1 K−1 Perfect Reconstruction = =0 x[n] e K z = =0 X(e K z) • Commutators PK k n P P K k • Summary • MATLAB routines P P P From previous slide:

K XK(z) = Y (z ) 1 1 K−1 −j2πk 1 K K K ⇒ Y (z) = XK (z ) = K k=0 X(e z ) Frequency Spectrum: P 1 K−1 j(ω−2πk) jω K Y (e ) = K k=0 X(e ) 1 jω jω 2π jω 4π K K − K K − K = K PX(e ) + X(e ) + X(e ) + ···

DSP and Digital Filters (2017-9045) Multirate: 11 – 8 / 14 Downsampled z-transform

11: Multirate Systems 1 K−1 j2πkn • K Multirate Systems Deﬁne cK [n] = δK|n[n] = K k=0 e • Building blocks • Resampling Cascades • Noble Identities x[n] KP| n • Noble Identities Proof Now deﬁne xK [n] = = cK [n]x[n] • Upsampled z-transform (0 K ∤ n • Downsampled z-transform 1 K−1 j2πkn −n K −n • Downsampled Spectrum XK (z) = n xK [n]z = K n k=0 e x[n]z • Power Spectral Density + −j2πk −n −j2πk • 1 K−1 1 K−1 Perfect Reconstruction = =0 x[n] e K z = =0 X(e K z) • Commutators PK k n P P K k • Summary • MATLAB routines P P P From previous slide:

K XK(z) = Y (z ) 1 1 K−1 −j2πk 1 K K K ⇒ Y (z) = XK (z ) = K k=0 X(e z ) Frequency Spectrum: P 1 K−1 j(ω−2πk) jω K Y (e ) = K k=0 X(e ) 1 jω jω 2π jω 4π K K − K K − K = K PX(e ) + X(e ) + X(e ) + ··· Average of K aliased versions, each expanded in ω by a factor of K.

DSP and Digital Filters (2017-9045) Multirate: 11 – 8 / 14 Downsampled z-transform

11: Multirate Systems 1 K−1 j2πkn • K Multirate Systems Deﬁne cK [n] = δK|n[n] = K k=0 e • Building blocks • Resampling Cascades • Noble Identities x[n] KP| n • Noble Identities Proof Now deﬁne xK [n] = = cK [n]x[n] • Upsampled z-transform (0 K ∤ n • Downsampled z-transform 1 K−1 j2πkn −n K −n • Downsampled Spectrum XK (z) = n xK [n]z = K n k=0 e x[n]z • Power Spectral Density + −j2πk −n −j2πk • 1 K−1 1 K−1 Perfect Reconstruction = =0 x[n] e K z = =0 X(e K z) • Commutators PK k n P P K k • Summary • MATLAB routines P P P From previous slide:

DSP and Digital Filters (2017-9045) Multirate: 11 – 8 / 14 Downsampled Spectrum

11: Multirate Systems 1 K−1 j(ω−2πk) • jω K Multirate Systems Y (e ) = K k=0 X(e ) • Building blocks • Resampling Cascades • Noble Identities P • Noble Identities Proof • Upsampled z-transform • Downsampled z-transform • Downsampled Spectrum • Power Spectral Density + • Perfect Reconstruction • Commutators • Summary • MATLAB routines

DSP and Digital Filters (2017-9045) Multirate: 11 – 9 / 14 Downsampled Spectrum

11: Multirate Systems 1 K−1 j(ω−2πk) • jω K Multirate Systems Y (e ) = K k=0 X(e ) • Building blocks • Resampling Cascades • Noble Identities P • Noble Identities Proof Example 1:

• Upsampled z-transform 1 • Downsampled z-transform K = 3 • Downsampled Spectrum Not quite limited to π 0.5 • Power Spectral Density + ± K • Perfect Reconstruction 0 -2 0 2 • Commutators ω • Summary • MATLAB routines

DSP and Digital Filters (2017-9045) Multirate: 11 – 9 / 14 Downsampled Spectrum

11: Multirate Systems 1 K−1 j(ω−2πk) • jω K Multirate Systems Y (e ) = K k=0 X(e ) • Building blocks • Resampling Cascades • Noble Identities P • Noble Identities Proof Example 1:

DSP and Digital Filters (2017-9045) Multirate: 11 – 9 / 14 Downsampled Spectrum

11: Multirate Systems 1 K−1 j(ω−2πk) • jω K Multirate Systems Y (e ) = K k=0 X(e ) • Building blocks • Resampling Cascades • Noble Identities P • Noble Identities Proof Example 1:

• Upsampled z-transform 1 1 • Downsampled z-transform K = 3 • Downsampled Spectrum Not quite limited to π 0.5 0.5 • Power Spectral Density + ± K • Perfect Reconstruction 0 0 Shaded region shows aliasing -2 0 2 -2 0 2 • Commutators ω ω • Summary 1 2 1 1 2 Energy decreases: jω jω • MATLAB routines 2π Y (e ) dω ≈ K × 2π X(e ) dω R R

DSP and Digital Filters (2017-9045) Multirate: 11 – 9 / 14 Downsampled Spectrum

11: Multirate Systems 1 K−1 j(ω−2πk) • jω K Multirate Systems Y (e ) = K k=0 X(e ) • Building blocks • Resampling Cascades • Noble Identities P • Noble Identities Proof Example 1:

0 -2 0 2 ω

DSP and Digital Filters (2017-9045) Multirate: 11 – 9 / 14 Downsampled Spectrum

11: Multirate Systems 1 K−1 j(ω−2πk) • jω K Multirate Systems Y (e ) = K k=0 X(e ) • Building blocks • Resampling Cascades • Noble Identities P • Noble Identities Proof Example 1:

, 0 0 No aliasing: -2 0 2 -2 0 2 ω ω

DSP and Digital Filters (2017-9045) Multirate: 11 – 9 / 14 Downsampled Spectrum

11: Multirate Systems 1 K−1 j(ω−2πk) • jω K Multirate Systems Y (e ) = K k=0 X(e ) • Building blocks • Resampling Cascades • Noble Identities P • Noble Identities Proof Example 1:

, 0 0 No aliasing: -2 0 2 -2 0 2 ω ω

π π No aliasing: If all energy is in r K ≤|ω| < (r + 1) K for some integer r

DSP and Digital Filters (2017-9045) Multirate: 11 – 9 / 14 Downsampled Spectrum

11: Multirate Systems 1 K−1 j(ω−2πk) • jω K Multirate Systems Y (e ) = K k=0 X(e ) • Building blocks • Resampling Cascades • Noble Identities P • Noble Identities Proof Example 1:

, 0 0 No aliasing: -2 0 2 -2 0 2 ω ω

π π No aliasing: If all energy is in r K ≤|ω| < (r + 1) K for some integer r π Normal case (r = 0): If all energy in 0 ≤|ω| ≤ K

DSP and Digital Filters (2017-9045) Multirate: 11 – 9 / 14 Downsampled Spectrum

11: Multirate Systems 1 K−1 j(ω−2πk) • jω K Multirate Systems Y (e ) = K k=0 X(e ) • Building blocks • Resampling Cascades • Noble Identities P • Noble Identities Proof Example 1:

, 0 0 No aliasing: -2 0 2 -2 0 2 ω ω

π π No aliasing: If all energy is in r K ≤|ω| < (r + 1) K for some integer r π Normal case (r = 0): If all energy in 0 ≤|ω| ≤ K 1 1 Downsampling: Total energy multiplied by ≈ K (= K if no aliasing) Average power ≈ unchanged (= energy/sample)

DSP and Digital Filters (2017-9045) Multirate: 11 – 9 / 14 ∫ ∫ ∫ ∫

Power Spectral Density +

11: Multirate Systems Example: Signal in ω ∈ ±0.4π + Tone @ ω = ±0.1π + White noise • Multirate Systems • Building blocks • Resampling Cascades original rate 1 • Noble Identities • Noble Identities Proof • Upsampled z-transform 0.5

• =0.6 +0.1 =0.5 Downsampled z-transform ∫ • Downsampled Spectrum 0 • Power Spectral Density + PSD , -3 -2 -1 0 1 2 3 Frequency (rad/samp) • Perfect Reconstruction • Commutators • Summary • MATLAB routines

DSP and Digital Filters (2017-9045) Multirate: 11 – 10 / 14 ∫ ∫ ∫ ∫

Power Spectral Density +

11: Multirate Systems Example: Signal in ω ∈ ±0.4π + Tone @ ω = ±0.1π + White noise • Multirate Systems • Building blocks • Resampling Cascades Power Energy/sample Average PSD original rate = = 1 • Noble Identities • Noble Identities Proof • Upsampled z-transform 0.5

DSP and Digital Filters (2017-9045) Multirate: 11 – 10 / 14 ∫ ∫ ∫ ∫

Power Spectral Density +

11: Multirate Systems Example: Signal in ω ∈ ±0.4π + Tone @ ω = ±0.1π + White noise • Multirate Systems • Building blocks • Resampling Cascades Power = Energy/sample = Average PSD original rate • 1 Noble Identities 1 π • Noble Identities Proof = 2 PSD(ω)dω • Upsampled z-transform π −π 0.5

• =0.6 +0.1 =0.5 Downsampled z-transform ∫ • Downsampled Spectrum R 0 • Power Spectral Density + PSD , -3 -2 -1 0 1 2 3 Frequency (rad/samp) • Perfect Reconstruction • Commutators • Summary • MATLAB routines

DSP and Digital Filters (2017-9045) Multirate: 11 – 10 / 14 ∫ ∫ ∫ ∫

Power Spectral Density +

DSP and Digital Filters (2017-9045) Multirate: 11 – 10 / 14 ∫ ∫ ∫ ∫

Power Spectral Density +

DSP and Digital Filters (2017-9045) Multirate: 11 – 10 / 14 ∫ ∫ ∫

Power Spectral Density +

• =0.6 +0.1 =0.5 Downsampled z-transform Component powers: ∫ • Downsampled Spectrum R 0 • Power Spectral Density + Signal = , Tone = , Noise = PSD , -3 -2 -1 0 1 2 3 0.3 0.2 0.1 Frequency (rad/samp) • Perfect Reconstruction • Commutators • Summary Upsampling: • MATLAB routines upsample × 2 Same energy per second 0.4 0.2 = 0.13 + 0.18 =0.3 +0.18 =0.13 ⇒ Power is ÷K ∫ 0

PSD , -3 -2 -1 0 1 2 3 Frequency (rad/samp)

DSP and Digital Filters (2017-9045) Multirate: 11 – 10 / 14 ∫ ∫

Power Spectral Density +

• =0.6 +0.1 =0.5 Downsampled z-transform Component powers: ∫ • Downsampled Spectrum R 0 • Power Spectral Density + Signal = , Tone = , Noise = PSD , -3 -2 -1 0 1 2 3 0.3 0.2 0.1 Frequency (rad/samp) • Perfect Reconstruction • Commutators • Summary Upsampling: • MATLAB routines upsample × 2 upsample × 3 Same energy 0.3 0.4 per second 0.2 0.2 0.1 = 0.13 + 0.18 =0.3 +0.18 =0.13 ∫ = 0.056 + 0.14 =0.2 +0.14 =0.056 ⇒ Power is ÷K ∫ 0 0

PSD , -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 Frequency (rad/samp) PSD , Frequency (rad/samp)

DSP and Digital Filters (2017-9045) Multirate: 11 – 10 / 14 ∫

Power Spectral Density +

• =0.6 +0.1 =0.5 Downsampled z-transform Component powers: ∫ • Downsampled Spectrum R 0 • Power Spectral Density + Signal = , Tone = , Noise = PSD , -3 -2 -1 0 1 2 3 0.3 0.2 0.1 Frequency (rad/samp) • Perfect Reconstruction • Commutators • Summary Upsampling: • MATLAB routines upsample × 2 upsample × 3 Same energy 0.3 0.4 per second 0.2 0.2 0.1 = 0.13 + 0.18 =0.3 +0.18 =0.13 ∫ = 0.056 + 0.14 =0.2 +0.14 =0.056 ⇒ Power is ÷K ∫ 0 0

PSD , -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 Frequency (rad/samp) PSD , Frequency (rad/samp) Downsampling:

downsample ÷ 2 Average power 0.6 is unchanged. 0.4 0.2 = 0.5 + 0.1 =0.6 +0.1 =0.5 ∫

0 PSD , -3 -2 -1 0 1 2 3 Frequency (rad/samp)

DSP and Digital Filters (2017-9045) Multirate: 11 – 10 / 14 Power Spectral Density +

• =0.6 +0.1 =0.5 Downsampled z-transform Component powers: ∫ • Downsampled Spectrum R 0 • Power Spectral Density + Signal = , Tone = , Noise = PSD , -3 -2 -1 0 1 2 3 0.3 0.2 0.1 Frequency (rad/samp) • Perfect Reconstruction • Commutators • Summary Upsampling: • MATLAB routines upsample × 2 upsample × 3 Same energy 0.3 0.4 per second 0.2 0.2 0.1 = 0.13 + 0.18 =0.3 +0.18 =0.13 ∫ = 0.056 + 0.14 =0.2 +0.14 =0.056 ⇒ Power is ÷K ∫ 0 0

PSD , -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 Frequency (rad/samp) PSD , Frequency (rad/samp) Downsampling:

downsample ÷ 2 downsample ÷ 3 Average power 0.6 0.6 is unchanged. 0.4 0.4 0.2 0.2 = 0.5 + 0.1 =0.6 +0.1 =0.5 ∫ = 0.49 + 0.11 =0.6 +0.11 =0.49 ∃ aliasing in ∫ 0 0 PSD , the case. -3 -2 -1 0 1 2 3 PSD , -3 -2 -1 0 1 2 3 ÷3 Frequency (rad/samp) Frequency (rad/samp)

DSP and Digital Filters (2017-9045) Multirate: 11 – 10 / 14

Perfect Reconstruction

11: Multirate Systems

• Multirate Systems x[n] • Building blocks • Resampling Cascades • Noble Identities • Noble Identities Proof • Upsampled z-transform • Downsampled z-transform • Downsampled Spectrum • Power Spectral Density + • Perfect Reconstruction • Commutators • Summary • MATLAB routines

DSP and Digital Filters (2017-9045) Multirate: 11 – 11 / 14

Perfect Reconstruction

11: Multirate Systems

• Multirate Systems x[n] • Building blocks u[m] • Resampling Cascades • Noble Identities • Noble Identities Proof • Upsampled z-transform • Downsampled z-transform • Downsampled Spectrum • Power Spectral Density + • Perfect Reconstruction • Commutators • Summary • MATLAB routines

DSP and Digital Filters (2017-9045) Multirate: 11 – 11 / 14

Perfect Reconstruction

11: Multirate Systems

• Multirate Systems x[n] • Building blocks u[m] • Resampling Cascades

• Noble Identities p[n] • Noble Identities Proof • Upsampled z-transform • Downsampled z-transform • Downsampled Spectrum • Power Spectral Density + • Perfect Reconstruction • Commutators • Summary • MATLAB routines

DSP and Digital Filters (2017-9045) Multirate: 11 – 11 / 14

Perfect Reconstruction

11: Multirate Systems

• Multirate Systems x[n] • Building blocks u[m] • Resampling Cascades

• Noble Identities p[n] • Noble Identities Proof • Upsampled z-transform v[m] • Downsampled z-transform • Downsampled Spectrum • Power Spectral Density + • Perfect Reconstruction • Commutators • Summary • MATLAB routines

DSP and Digital Filters (2017-9045) Multirate: 11 – 11 / 14

Perfect Reconstruction

11: Multirate Systems

• Multirate Systems x[n] • Building blocks u[m] • Resampling Cascades

• Noble Identities p[n] • Noble Identities Proof • Upsampled z-transform v[m] • Downsampled z-transform q[n] • Downsampled Spectrum • Power Spectral Density + • Perfect Reconstruction • Commutators • Summary • MATLAB routines

DSP and Digital Filters (2017-9045) Multirate: 11 – 11 / 14

Perfect Reconstruction

11: Multirate Systems

• Multirate Systems x[n] • Building blocks u[m] • Resampling Cascades

• Noble Identities p[n] • Noble Identities Proof • Upsampled z-transform v[m] • Downsampled z-transform q[n] • Downsampled Spectrum

• Power Spectral Density + w[m] • Perfect Reconstruction • Commutators • Summary • MATLAB routines

DSP and Digital Filters (2017-9045) Multirate: 11 – 11 / 14 Perfect Reconstruction

11: Multirate Systems

• Multirate Systems x[n] • Building blocks u[m] • Resampling Cascades

• Noble Identities p[n] • Noble Identities Proof • Upsampled z-transform v[m] • Downsampled z-transform q[n] • Downsampled Spectrum

• Power Spectral Density + w[m] • Perfect Reconstruction

y n • Commutators [ ] • Summary • MATLAB routines

DSP and Digital Filters (2017-9045) Multirate: 11 – 11 / 14 Perfect Reconstruction

11: Multirate Systems

• Multirate Systems x[n] • Building blocks u[m] • Resampling Cascades

• Noble Identities p[n] • Noble Identities Proof • Upsampled z-transform v[m] • Downsampled z-transform q[n] • Downsampled Spectrum

• Power Spectral Density + w[m] • Perfect Reconstruction

y n • Commutators [ ] • Summary • MATLAB routines 1 Input sequence x[n] is split into three streams at 3 the sample rate: u[m] = x[3m], v[m] = x[3m − 1], w[m] = x[3m − 2]

DSP and Digital Filters (2017-9045) Multirate: 11 – 11 / 14 Perfect Reconstruction

11: Multirate Systems

• Multirate Systems x[n] • Building blocks u[m] • Resampling Cascades

• Noble Identities p[n] • Noble Identities Proof • Upsampled z-transform v[m] • Downsampled z-transform q[n] • Downsampled Spectrum

• Power Spectral Density + w[m] • Perfect Reconstruction

y n • Commutators [ ] • Summary • MATLAB routines 1 Input sequence x[n] is split into three streams at 3 the sample rate: u[m] = x[3m], v[m] = x[3m − 1], w[m] = x[3m − 2]

Following upsampling, the streams are aligned by the delays and then added to give: y[n] = x[n − 2]

DSP and Digital Filters (2017-9045) Multirate: 11 – 11 / 14 Perfect Reconstruction

11: Multirate Systems

• Multirate Systems x[n] • Building blocks u[m] • Resampling Cascades

• Noble Identities p[n] • Noble Identities Proof • Upsampled z-transform v[m] • Downsampled z-transform q[n] • Downsampled Spectrum

• Power Spectral Density + w[m] • Perfect Reconstruction

y n • Commutators [ ] • Summary • MATLAB routines 1 Input sequence x[n] is split into three streams at 3 the sample rate: u[m] = x[3m], v[m] = x[3m − 1], w[m] = x[3m − 2]

Following upsampling, the streams are aligned by the delays and then added to give: y[n] = x[n − 2]

Perfect Reconstruction: output is a delayed scaled replica of the input

DSP and Digital Filters (2017-9045) Multirate: 11 – 11 / 14

Commutators

11: Multirate Systems

• Multirate Systems x[n] • Building blocks u[m] • Resampling Cascades

• Noble Identities v[m] • Noble Identities Proof • Upsampled z-transform w[m] • Downsampled z-transform • Downsampled Spectrum • Power Spectral Density + • Perfect Reconstruction

y n • Commutators [ ] • Summary • MATLAB routines

DSP and Digital Filters (2017-9045) Multirate: 11 – 12 / 14

Commutators

11: Multirate Systems

• Multirate Systems x[n] • Building blocks u[m] • Resampling Cascades

• Noble Identities v[m] • Noble Identities Proof • Upsampled z-transform w[m] • Downsampled z-transform • Downsampled Spectrum • Power Spectral Density + • Perfect Reconstruction

y n • Commutators [ ] • Summary • MATLAB routines The combination of delays and downsamplers can be regarded as a commutator that distributes values in sequence to u, w and v.

DSP and Digital Filters (2017-9045) Multirate: 11 – 12 / 14

Commutators

11: Multirate Systems

• Multirate Systems x[n] • Building blocks u[m] • Resampling Cascades

• Noble Identities v[m] • Noble Identities Proof • Upsampled z-transform w[m] • Downsampled z-transform • Downsampled Spectrum • Power Spectral Density + • Perfect Reconstruction

DSP and Digital Filters (2017-9045) Multirate: 11 – 12 / 14

Commutators

11: Multirate Systems

• Multirate Systems x[n] • Building blocks u[m] • Resampling Cascades

• Noble Identities v[m] • Noble Identities Proof • Upsampled z-transform w[m] • Downsampled z-transform • Downsampled Spectrum • Power Spectral Density + • Perfect Reconstruction

DSP and Digital Filters (2017-9045) Multirate: 11 – 12 / 14 Commutators

11: Multirate Systems

• Multirate Systems x[n] • Building blocks u[m] • Resampling Cascades

• Noble Identities v[m] • Noble Identities Proof

• w[m] Upsampled z-transform 1 • Downsampled z-transform v[m + 3 ] • Downsampled Spectrum 2

• Power Spectral Density + w[m + 3 ] • Perfect Reconstruction

y n • Commutators [ ] • Summary • MATLAB routines The combination of delays and downsamplers can be regarded as a commutator that distributes values in sequence to u, w and v. − 1 − 2 Fractional delays, z 3 and z 3 are needed to synchronize the streams. The output commutator takes values from the streams in sequence. For clarity, we omit the fractional delays and regard each terminal, ◦, as holding its value until needed.

DSP and Digital Filters (2017-9045) Multirate: 11 – 12 / 14 Commutators

11: Multirate Systems

• Multirate Systems x[n] • Building blocks u[m] • Resampling Cascades

• Noble Identities v[m] • Noble Identities Proof

• w[m] Upsampled z-transform 1 • Downsampled z-transform v[m + 3 ] • Downsampled Spectrum 2

• Power Spectral Density + w[m + 3 ] • Perfect Reconstruction

DSP and Digital Filters (2017-9045) Multirate: 11 – 12 / 14 Commutators

11: Multirate Systems

• Multirate Systems x[n] • Building blocks u[m] • Resampling Cascades

• Noble Identities v[m] • Noble Identities Proof

• w[m] Upsampled z-transform 1 • Downsampled z-transform v[m + 3 ] • Downsampled Spectrum 2

• Power Spectral Density + w[m + 3 ] • Perfect Reconstruction

1 The commutator direction is against the direction of the z− delays.

DSP and Digital Filters (2017-9045) Multirate: 11 – 12 / 14 Summary

DSP and Digital Filters (2017-9045) Multirate: 11 – 13 / 14 Summary

11: Multirate Systems Multirate Building Blocks • • Multirate Systems 1:K • Building blocks ◦ Upsample: X(z) → X(zK) • Resampling Cascades • Noble Identities Invertible, Inserts K − 1 zeros between samples • Noble Identities Proof • Upsampled z-transform Shrinks and replicates spectrum • Downsampled z-transform Follow by LP ﬁlter to remove images • Downsampled Spectrum • Power Spectral Density + K:1 1 K−1 −j2πk 1 • K K Perfect Reconstruction ◦ Downsample: X(z) → K k=0 X(e z ) • Commutators th • Summary Destroys information and energy, keeps every K sample • P MATLAB routines Expands and aliasses the spectrum Spectrum is the average of K aliased expanded versions Precede by LP ﬁlter to prevent aliases

DSP and Digital Filters (2017-9045) Multirate: 11 – 13 / 14 Summary

• Equivalences ◦ Noble Identities: H(z) ←→ H(zK ) ◦ Interchange P : 1 and 1: Q iff P and Q coprime

DSP and Digital Filters (2017-9045) Multirate: 11 – 13 / 14 Summary

• Equivalences ◦ Noble Identities: H(z) ←→ H(zK ) ◦ Interchange P : 1 and 1: Q iff P and Q coprime • Commutators ◦ Combine delays and down/up sampling

DSP and Digital Filters (2017-9045) Multirate: 11 – 13 / 14 Summary

• Equivalences ◦ Noble Identities: H(z) ←→ H(zK ) ◦ Interchange P : 1 and 1: Q iff P and Q coprime • Commutators ◦ Combine delays and down/up sampling

For further details see Mitra: 13.

DSP and Digital Filters (2017-9045) Multirate: 11 – 13 / 14 MATLAB routines

11: Multirate Systems • Multirate Systems resample change sampling rate • Building blocks • Resampling Cascades • Noble Identities • Noble Identities Proof • Upsampled z-transform • Downsampled z-transform • Downsampled Spectrum • Power Spectral Density + • Perfect Reconstruction • Commutators • Summary • MATLAB routines

DSP and Digital Filters (2017-9045) Multirate: 11 – 14 / 14