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 + • 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 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 • Reduce computational complexity fs FIR filter length ∝ ∆f where ∆f is width of transition band 2 Lower fs ⇒ shorter filter + fewer samples ⇒computation ∝ fs 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 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] • 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]