EECS 247 Lecture 11: Digital Filters © 2002 B. Boser 1A/DDSPDigital Filters• Advantages of digital filters– Dynamic range– No coefficient errors, aging– Programmable– Always work on first silicon if …• FIR filters– Linear phase– Synthesis• FIR / IIR comparison• Implementation issues– Coefficient rounding– Intermediate result dynamic range– Limit cyclesEECS 247 Lecture 11: Digital Filters © 2002 B. Boser 2A/DDSPAnalog versus Digital DR• It’s much less expensive to add dynamic range to digital circuits than analog circuits• To double the dynamic range of a digital datapath, we need to add only a bit to an already-wide datapath:15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 015 14 13 12 11 10 9 8 7 6 5 4 3 2 1 016+6dB DREECS 247 Lecture 11: Digital Filters © 2002 B. Boser 3A/DDSPAnalog versus Digital DRFor comparison, consider summing the outputs of 4 identical analog circuits with identical inputs:A1A2A3A4ΣvOUTvINEECS 247 Lecture 11: Digital Filters © 2002 B. Boser 4A/DDSPAnalog versus Digital DRAnalog noise is typically uncorrelated in each of the blocks A1-A4:A1A2A3A4ΣvOUTvINSignal grows 4XNoise grows 2X+6dB Dynamic RangeEECS 247 Lecture 11: Digital Filters © 2002 B. Boser 5A/DDSPAnalog versus Digital DR• Doubling analog DR is very expensive:– 4X the power– 4X the area• Doubling digital DR is relatively cheap,– And cost/function decreases by 29%/year (3dB/year)!• Practical circuits tolerate very little loss of DR due to finite datapath precision in their DSP sections– Analog dynamic range is too precious to lose– Digital DR loss of 5% (~ 0.4dB) of total noise power is typical• Why use analog filters at all?EECS 247 Lecture 11: Digital Filters © 2002 B. Boser 6A/DDSPADC Dynamic Range• The figure shows the DR of the best standalone ADCs in 2000• Dynamic range decreases as converter bandwidth increases• From 1975-1995, ADC performance at any sampling frequency improved by 2dB/yearADC Sampling Frequency (Hz)10410610840806012014010020Dynamic Range (dB | Bits)613102023163EECS 247 Lecture 11: Digital Filters © 2002 B. Boser 7A/DDSPADC Dynamic Range• ADCs embedded in IC “Systems on a Chip” (SoCs) have less DR than the best standalone ADCs• The embedded ADC performance level is shown in red• Analog-digital crosstalk and design risk issues limit embedded ADC DR to about 100dB• 1 GHz, 30dB DR levels are much more forgiving and the performance gap narrowsADC Sampling Frequency (Hz)10410610840806012014010020Dynamic Range (dB)embedded ADCsEECS 247 Lecture 11: Digital Filters © 2002 B. Boser 8A/DDSPADC Dynamic Range• Minimization of analog signal processing is a key goal of mixed-signal IC architecture• However, analog signal processing is almost unavoidable “above the red line”ADC Sampling Frequency (Hz)10410610840806012014010020Dynamic Range (dB)embedded ADCsEECS 247 Lecture 11: Digital Filters © 2002 B. Boser 9A/DDSPPractical Constraints• Only few ADC design teams in the world can produce “green line” dynamic range• If your SoC architecture requires one of those teams to succeed, think again! • Mixed-signal SoC architectures fail when their architects choose to ignore long-established, empirically-proven performance scaling lawsEECS 247 Lecture 11: Digital Filters © 2002 B. Boser 10A/DDSPFIR Filters• Only finite zeros• Linear phase if coefficients are symmetric• Implement with delays, multipliers, adders• Lack of good analog delays prevents widespread use of analog FIR filters• Good synthesis tools (e.g. Remez-Exchange algorithm)EECS 247 Lecture 11: Digital Filters © 2002 B. Boser 11A/DDSPFIR Filter Phase Response• Consider the Nth-order FIR filter with transfer function:H(z) = a0+ a1z-1 + a2z-2 +…+ aN-2z2-N + aN-1z1-N + aNz-N• Suppose the filter coefficients are symmetric about the middle term, i.e.:H(z) = a0+ a1z-1 + a2z-2 +…+ a2z2-N + a1z1-N + a0z-NEECS 247 Lecture 11: Digital Filters © 2002 B. Boser 12A/DDSPFIR Filter Phase ResponseH(z) = a0+ a1z-1 + a2z-2 +…+ a2z2-N + a1z1-N + a0z-N= a0(1+z-N) + a1(z-1+z1-N) + a2(z-2 + z2-N) +…= a0z-N/2(zN/2+z-N/2) + a1z-N/2 (z-1+N/2+z1-N/2) ++ a2z-N/2 (z-2+N/2 + z2-N/2) +…= z-N/2[ a0(zN/2+z-N/2) + a1(z-1+N/2+z1-N/2) + a2(z-2+N/2 + z2-N/2) +…]EECS 247 Lecture 11: Digital Filters © 2002 B. Boser 13A/DDSPFIR Filter Phase Response• The term in brackets [] is a sum of cosine terms with no phase shift:H(ejωT) = e-jωNT/2[ 2a0cos(ωNT/2) + more real cos terms]θ(ω) = - ωNT/2 τGR = NT/2• The constant group delay of the symmetric coefficient FIR filter is obvious:half the filter impulse response durationEECS 247 Lecture 11: Digital Filters © 2002 B. Boser 14A/DDSPCoefficient Symmetry• Three classes of zero groupings produce symmetric coefficients and linear phase• The first is real axis zeroes at r and 1/r:H(z) = z-2-(r+1/r)z-1+1EECS 247 Lecture 11: Digital Filters © 2002 B. Boser 15A/DDSPCoefficient Symmetry• Conjugate pairs of unit circle zeroes provide linear phase:H(z) = z-2- 2z-1cos θ +1θEECS 247 Lecture 11: Digital Filters © 2002 B. Boser 16A/DDSPCoefficient Symmetry• Finally, groups of four zeroes at re±jθand (1/r)e±jθprovide linear phase• The filter coefficients for these 4 zeroes are: θ1-2(r+1/r)cosθ4+r2+1/r2-2(r+1/r)cosθ1EECS 247 Lecture 11: Digital Filters © 2002 B. Boser 17A/DDSPFIR Filter Phase Response• Another interesting case involves antisymmetric filter coefficients:• It’s easy to show thatH(z) = a0+ a1z-1 + a2z-2 +…- a2z2-N- a1z1-N- a0z-NH(ejωT) = e-jωNT/2ejπ/2[ 2a0sin(ωNT/2) + more sin terms]EECS 247 Lecture 11: Digital Filters © 2002 B. Boser 18A/DDSPFIR Filter Phase Response• For the antisymmetric coefficient caseθ(ω) = - ωNT/2 τGR = NT/2π2• It’s still linear phase, but with the frequency independent 90° phase shift characteristic of differentiatorsEECS 247 Lecture 11: Digital Filters © 2002 B. Boser 19A/DDSPLinear Phase FIR Examplefs = 1e6;Fp = 0.10*fs; Fs = 0.13*fs;Rp = 0.1; Rs = 60;x = (10^(Rp/20)1)/(10^(Rp/20)+1); y = 10^(-Rs/20); [N,fo,ao,W]=remezord( …[Fp Fs],[1 0],[x y],fs);b = remez(N, fo, ao, W);Hr = tf(b, 1, 1/fs);Hr = Hr / 10^(rpass/40);0 1 2 3 4 5x 105-70-60-50-40-30-20-100Frequency 0...fs/2 [Hz]Magnitude [dB]0 2 4 6 8 10x 104-0.12-0.1-0.08-0.06-0.04-0.020Frequency 0...fs/2 [Hz]Magnitude [dB]EECS 247 Lecture 11: Digital Filters © 2002 B. Boser 20A/DDSPz-Plane-2 -1 0 1
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