Downsampling, Upsampling, and Reconstruction • A-to-D and its relation to sampling • Downsampling and its relation to sampling • Upsampling and interpolation • D-to-A and reconstruction filtering • Filters and their relation to convolution Copyright © 2007 by M.H. Perrott All rights reserved. M.H. Perrott©2007 Downsampling, Upsampling, and Reconstruction, Slide 1 Digital Processing of Analog Signals xc(t) A-to-D x[n] Downsample Converter by N 1/T Sample/s 1/(NT) Sample/s Digital Signal r[n] Processing u[n] Operations 1/(NT) Sample/s Upsample D-to-A yc(t) by M Converter M/(NT) Sample/s M/(NT) Sample/s • Digital circuits can perform very complex processing of analog signals, but require – Conversion of analog signals to the digital domain – Conversion of digital signals to the analog domain – Downsampling and upsampling to match sample rates of A-to-D, digital processor, and D-to-A M.H. Perrott©2007 Downsampling, Upsampling, and Reconstruction, Slide 2 Inclusion of Filtering Operations xc(t) A-to-D x[n] Anti-Alias Downsample Converter Filter by 10 Analog 1/T Sample/s 1/T Sample/s 1/(NT) Sample/s Anti-Alias Filter Digital Signal r[n] Processing u[n] Operations 1/(NT) Sample/s Upsample Interpolate y[n] D-to-A yc(t) by 10 with Filter Converter M/(NT) Sample/s M/(NT) Sample/s M/(NT) Sample/s Analog Reconstruction Filter • A-to-D and downsampler require anti-alias filtering – Prevents aliasing • D-to-A and upsampler require interpolation (i.e., reconstruction) filtering – Provides `smoothly’ changing waveforms M.H. Perrott©2007 Downsampling, Upsampling, and Reconstruction, Slide 3 Summary of Sampling Process (Review) p(t) 1 t p(t) T xc(t) xp(t) Impulse Train x[n] to Sequence xc(t) xp(t) x[n] t t n T 1 j2πλ Xc(f) Xp(f) X(e ) A A A T T f f λ 0 -2 -1 0 1 2 -1-2 012 T T T T • Sampling leads to periodicity in frequency domain We need to avoid overlap of replicated signals in frequency domain (i.e., aliasing) M.H. Perrott©2007 Downsampling, Upsampling, and Reconstruction, Slide 4 The Sampling Theorem (Review) xc(t) p(t) xp(t) 1 t t t T T Xc(f) P(f) Xp(f) 1 A A T T f f f -fbw fbw -2 -1 0 1 2 -2 -1 0 1 2 T T T T T T T T -fbw fbw - 1 + f 1 - f T bw T bw • Overlap in frequency domain (i.e., aliasing) is avoided if: • We refer to the minimum 1/T that avoids aliasing as the Nyquist sampling frequency M.H. Perrott©2007 Downsampling, Upsampling, and Reconstruction, Slide 5 A-to-D Converter xc(t) xc(t) A-to-D x[n] Converter Analog 1/T Sample/s Anti-Alias Filter 1 t p(t) T xc(t) xp(t) Quantize xp(t) Impulse Train x[n] Value to Sequence t tn T 1 • Operates using both a sampler and quantizer – Sampler converts continuous-time input signal into a discrete-time sequence – Quantizer converts continuous-valued signal/sequence into a discrete-valued signal/sequence • Introduces quantization noise as discussed in Lab 4 M.H. Perrott©2007 Downsampling, Upsampling, and Reconstruction, Slide 6 Frequency Domain View of A-to-D xc(t) xc(t) A-to-D x[n] Converter Analog 1/T Sample/s Anti-Alias Filter P(f) 1/T f -1/T 0 1/T p(t) Xc(f) x (t) x (t) x (t) x[n] 1 c p Quantize p Impulse Train Value to Sequence f j2πλ 0 Xp(f) X(e ) 1/T 1/T f λ -2/T -1/T 0 1/T 2/T -2 -1 0 1 2 • Analysis of A-to-D same as for sampler – For simplicity, we will ignore the influence of quantization noise in our picture analysis • In lab 4, we will explore the influence of quantization noise using Matlab M.H. Perrott©2007 Downsampling, Upsampling, and Reconstruction, Slide 7 Downsampling Anti-Alias x[n] Downsample r[n] Filter by N 1 N p[n] x[n] xs[n] Remove r[n] Zero Samples n 1 n n N 1 • Similar to sampling, but operates on sequences • Analysis is simplified by breaking into two steps – Multiply input by impulse sequence of period N samples – Remove all samples of xs[n] associated with the zero- valued samples of the impulse sequence, p[n] • Amounts to scaling of time axis by factor 1/N M.H. Perrott©2007 Downsampling, Upsampling, and Reconstruction, Slide 8 Frequency Domain View of Downsampling Anti-Alias x[n] Downsample r[n] Filter by N πλ P(ej2 ) 1/N λ -1 -2/N -1/N0 1/N 2/N 1 p[n] j2πλ X(e ) x[n] x (t) r[n] 1 s Remove Zero Samples λ -1 0 1 j2πλ j2πλ Xs(e ) R(e ) 1/N 1/N λ λ -1 -2/N -1/N 0 1/N 2/N 1 -1 0 1 • Multiplication by impulse sequence leads to replicas of input transform every 1/N Hz in frequency • Removal of zero samples (i.e., scaling of time axis) leads to scaling of frequency axis by factor N M.H. Perrott©2007 Downsampling, Upsampling, and Reconstruction, Slide 9 The Need for Anti-Alias Filtering Anti-Alias x[n] Downsample r[n] Filter by N πλ P(ej2 ) 1/N Undesired λ Signal or -1 -2/N -1/N0 1/N 2/N 1 Noise p[n] j2πλ X(e ) x[n] x (t) r[n] 1 s Remove Zero Samples λ -1 0 1 j2πλ j2πλ Xs(e ) R(e ) 1/N 1/N λ λ -1 -2/N -1/N 0 1/N 2/N 1 -1 0 1 • Removal of anti-alias filter would allow undesired signals or noise to alias into desired signal band What is the appropriate bandwidth of the anti-alias lowpass filter? M.H. Perrott©2007 Downsampling, Upsampling, and Reconstruction, Slide 10 Upsampler u[n] u [n] y[n] Upsample p Interpolate by N with Filter u[n] u [n] y[n] Add p Interpolate n Zero Samples with Filter 1 n n 1 N 1 N • Consists of two operations – Add N-1 zero samples between every sample of the input • Effectively scales time axis by factor N – Filter the resulting sequence, up[n], in order to create a smoothly varying set of sequence samples • Proper choice of the filter leads to interpolation between the non-zero samples of sequence up[n] (discussed in Lab 5) M.H. Perrott©2007 Downsampling, Upsampling, and Reconstruction, Slide 11 Frequency Domain View of Upsampling u[n] u [n] y[n] Upsample p Interpolate by N with Filter u[n] u [n] y[n] Add p Interpolate πλ Zero Samples with Filter πλ U(ej2 ) Y(ej2 ) 1/N 1 λ λ -1 0 1 -1 0 1 j2πλ Up(e ) 1/N λ -1 -2/N -1/N 0 1/N 2/N 1 • Addition of zero samples (scaling of time axis) leads to scaling of frequency axis by factor 1/N • Interpolation filter removes all replicas of the signal transform except for the baseband copy M.H. Perrott©2007 Downsampling, Upsampling, and Reconstruction, Slide 12 D-to-A Converter y[n] D-to-A yc(t) yc(t) Converter n 1/T Sample/s Analog 1 Reconstruction Filter n t 1 T y[n] y (t) Sequence to c Impulse Train • Simple analytical model includes two operations – Convert input sequence samples into corresponding impulse train – Filter impulse train to create a smoothly varying signal • Proper choice of the reconstruction filter leads to interpolation between impulse train values M.H. Perrott©2007 Downsampling, Upsampling, and Reconstruction, Slide 13 Frequency Domain View of D-to-A j2πλ Y(e ) y[n] yc(t) yc(t) Yc(f) 1/T D-to-A 1 Converter λ f -2 -1 0 1 2 1/T Sample/s Analog 0 Reconstruction Filter πλ Y(ej2 ) Y(f) 1/T 1/T λ f -2 -1 0 1 2 y[n] y (t) -2/T -1/T 0 1/T 2/T Sequence to c Impulse Train • Conversion from sequence to impulse train amounts to scaling the frequency axis by sample rate of D- to-A (1/T) • Reconstruction filter removes all replicas of the signal transform except for the baseband copy M.H. Perrott©2007 Downsampling, Upsampling, and Reconstruction, Slide 14 A Common Reconstruction Filter y[n] D-to-A yc(t) yc(t) H (f) Converter zoh n t 1/T Sample/s Zero-Order 1 Hold T n t 1 T y[n] y (t) Sequence to c Impulse Train • Zero-order hold circuit operates by maintaining the impulse value across the D-to-A sample period – Easy to implement in hardware How do we analyze this? M.H. Perrott©2007 Downsampling, Upsampling, and Reconstruction, Slide 15 Filtering is Convolution in Time yc(t) yc(t) Hzoh(f) t t T Zero-Order Hold T hzoh(t) t T yc(t) hzoh(t) yc(t) t t t T T T • Recall that multiplication in frequency corresponds to convolution in time • Filtering corresponds to convolution in time between the input and the filter impulse response M.H. Perrott©2007 Downsampling, Upsampling, and Reconstruction, Slide 16 Frequency Domain View of Filtering yc(t) yc(t) Hzoh(f) t t T Zero-Order Hold T hzoh(t) t T yc(t) hzoh(t) yc(t) t t t T T T Yc(f) Hzoh(f) Yc(f) -2 2 T T f f f -2 -1 0 1 2 0 -2 -1 0 1 2 T T T T -1 1 T T T T T T • Zero-order hold is not a great filter, but it’s simple… M.H.