Digital 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 cycles
A/D DSP
EECS 247 Lecture 11: Digital Filters
© 2002 B. Boser 1
Analog 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
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3
2
1
0
4
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+6dB DR 16 15 14 13 12 11 10 9
A/D DSP
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Analog versus Digital DR For comparison, consider summing the outputs of 4 identical analog circuits with identical inputs: vIN A1
A2
A3
A4
Σ vOUT A/D DSP
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Analog versus Digital DR Analog noise is typically uncorrelated in each of the blocks A1-A4: vIN A1
A2
Signal grows 4X Noise grows 2X +6dB Dynamic Range A/D DSP
EECS 247 Lecture 11: Digital Filters
A3
A4
Σ vOUT © 2002 B. Boser 4
Analog 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? A/D DSP
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• 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/year
A/D DSP
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Dynamic Range (dB | Bits)
ADC Dynamic Range 140
23
120
20
100
16
80
13
60
10
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6
20
3
104 108 106 ADC Sampling Frequency (Hz)
© 2002 B. Boser 6
ADC 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
•
Dynamic Range (dB)
•
140
1 GHz, 30dB DR levels are much more forgiving and the performance gap narrows
A/D DSP
120 100 80 60 40
embedded ADCs
20
104 108 106 ADC Sampling Frequency (Hz)
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ADC Dynamic Range 140 • 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”
Dynamic Range (dB)
•
120 100 80 60 40
embedded ADCs
20
104 108 106 ADC Sampling Frequency (Hz)
A/D DSP
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Practical 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 laws
A/D DSP
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FIR 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) A/D DSP
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FIR 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-N
A/D DSP
EECS 247 Lecture 11: Digital Filters
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FIR Filter Phase Response H(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) +…] A/D DSP
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FIR 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]
• The constant group delay of the symmetric coefficient FIR filter is obvious: θ(ω) = - ωNT/2
τGR = NT/2 half the filter impulse response duration
A/D DSP
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Coefficient 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+1
A/D DSP
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Coefficient Symmetry • Conjugate pairs of unit circle zeroes provide linear phase:
θ
H(z) = z-2- 2z-1cos θ +1
A/D DSP
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Coefficient Symmetry • Finally, groups of four zeroes at re±jθ and (1/r)e±jθ provide linear phase • The filter coefficients for these 4 zeroes are:
θ
1 -2(r+1/r)cosθ 4+r2+1/r2 -2(r+1/r)cosθ 1 A/D DSP
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FIR Filter Phase Response • Another interesting case involves antisymmetric filter coefficients: H(z) = a0 + a1z-1 + a2z-2 +…- a2z2-N - a1z1-N - a0z-N
• It’s easy to show that H(ejωT) = e-jωNT/2ejπ/2 [ 2a0sin(ωNT/2) + more sin terms]
A/D DSP
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FIR Filter Phase Response • For the antisymmetric coefficient case θ(ω) =
π - ωNT/2 2
τGR = NT/2
• It’s still linear phase, but with the frequency independent 90° phase shift characteristic of differentiators
A/D DSP
EECS 247 Lecture 11: Digital Filters
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Linear Phase FIR Example fs = 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); 0 [N,fo,ao,W]=remezord( … -10 [Fp Fs],[1 0],[x y],fs); -20 b = remez(N, fo, ao, W); -30 Hr = tf(b, 1, 1/fs); -40 Hr = Hr / 10^(rpass/40);
0
Magnitude [dB]
-0.02 -0.04 -0.06 -0.08 -0.1 -0.12 2
4 6 Frequency 0...f /2 [Hz] s
8
10 4
x 10
Magnitude [dB]
0
-50 -60 -70
A/D DSP
0
1
2 3 Frequency 0...f /2 [Hz] s
EECS 247 Lecture 11: Digital Filters
4
5 5
x 10
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z-Plane Pole-Zero Map 1 0.8 0.6
Imaginary Axis
0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -2
-1
0
1
2
3
4
5
Real Axis
A/D DSP
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Phase Response 2500
Linear?
Phase [degrees]
2000
1500
1000
500
0
A/D DSP
0
1
2 3 Frequency [Hz]
4
5 5
x 10
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FIR / IIR Comparison 0.04
91st
FIR IIR
0.02
order linear phase FIR
0
Magnitude [dB]
-0.02
0
or
-0.04 -0.06
FIR IIR
-0.08 -0.1
-10
-0.12
order elliptic IIR
Magnitude [dB]
-0.14
7th
-20
-0.16
1
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4 5 6 7 Frequency 0...f /2 [Hz] s
8
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10 4
x 10
-30 -40 -50 -60 -70
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0
1
2 3 Frequency 0...f /2 [Hz] s
4
5 5
x 10
© 2002 B. Boser 22
FIR Coefficient Rounding 0
8 Bits 16 Bits 24 Bits Floating Point
-20
Magnitude [dB]
-40
-60
-80
-100
-120 0
A/D DSP
1
2 3 Frequency 0...f /2 [Hz] s
EECS 247 Lecture 11: Digital Filters
4
5 5
x 10
© 2002 B. Boser 23
FIR Coefficient Precision • Finite precision FIR filters add transfer functions of two filters – The infinite precision FIR filter – A rounding error FIR filter
• The infinite precision FIR dominates the passband response • The rounding error FIR filter sets stopband attenuation when the infinite precision FIR response is much smaller
A/D DSP
EECS 247 Lecture 11: Digital Filters
© 2002 B. Boser 24
FIR Coefficient Precision • Random rounding errors transform to white “stopband noise” – Stopband attenuation increases by about 6dB for each bit of coefficient precision
• If you don’t like the highest bump in the stopband response, generate a new pattern of rounding error – Use slightly different dc gain – Or slightly different Parks-McClellan (remez) bands
• Trial and error can improve filter stopbands by several dB at a given coefficient precision
A/D DSP
EECS 247 Lecture 11: Digital Filters
© 2002 B. Boser 25
Filter Dynamic Range • Digital filters need more numeric dynamic range than the signals they process – They must not overload – They must not surprise you with quantization noise
• Digital multiplier/accumulators are multiplexed – Difference equations are added up term-by-term, giving us “intermediate transfer functions” to worry about – Intermediate overload is as bad as overload
• Let’s look at an IIR bandstop filter example… A/D DSP
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© 2002 B. Boser 26
2nd-Order Bandstop Filter • Bandstop filters have transfer functions: z −2 − z −1 ( 2 cos Θ) + 1 r z − z −1 ( 2r cos Θ) + 1 2πf P Θ≈ fs πf r ≈1− P Qf s
H (z) =
2 −2
θ
• Their gains are close to unity at both dc and fs/2 A/D DSP
EECS 247 Lecture 11: Digital Filters
© 2002 B. Boser 27
2nd-Order Bandstop Filter • Bandstop design specifications: – fS=1MHz – fP=20kHz – QP=100
A/D DSP
EECS 247 Lecture 11: Digital Filters
θ
© 2002 B. Boser 28
Direct Form Realization Y (z) z −2 − z −1 (2 cos Θ) + 1 = 2 −2 X ( z ) r z − z −1 (2 r cos Θ) + 1
[
]
[
]
Y ( z ) r 2 z −2 − z −1 ( 2 r cos Θ) + 1 = X ( z ) z −2 − z −1 ( 2 cos Θ) + 1 yk −2 r 2 z −2 − yk −1 ( 2 r cos Θ) + yk = xk −2 − xk −1 ( 2 cos Θ) + xk
Note: Direct form realizations are not ideal for higher order IIR filters. Lattice filters (and variants), which simulate LC ladders, are less susceptible to finite coefficient precision and dynamic range. A/D DSP
EECS 247 Lecture 11: Digital Filters
© 2002 B. Boser 29
2nd-Order Bandstop Filter • We can build a direct form bandstop this way (method #1):
yk = xk − xk −1 ( 2 cos Θ) + xk −2 + yk −1 ( 2 r cos Θ) − yk −2 r 2 z −2 1442443 intermedia te result 1 444 14 44 4 24 3 intermedia te result 2 14444444244444443 intermedia te result 3
• Or this way (method #2):
yk = yk −1 ( 2 r cos Θ) − yk −2 r 2 z −2 + xk − xk −1 ( 2 cos Θ) + xk −2 • The order of addition matters if we overload in the middle! A/D DSP
EECS 247 Lecture 11: Digital Filters
© 2002 B. Boser 30
2nd-Order Bandstop Filter • Proceeding from left to right, the difference equation generates 3 intermediate transfer functions plus the complete bandstop transfer function • All 4 of these transfer function magnitude responses for method 1 are shown on the next slide A/D DSP
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© 2002 B. Boser 31
Method #1 Bandstop Responses 20 intermediate responses
Gain (dB)
0 complete bandstop - 20
- 40
- 60 0
100
200
300
400
500
Frequency (kHz) A/D DSP
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Method #1 Bandstop Responses 20 intermediate responses
Gain (dB)
0
- 20
12dB max. gain - 40
- 60 0
100
200
300
400
500
Frequency (kHz) A/D DSP
EECS 247 Lecture 11: Digital Filters
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Method #1 Bandstop Responses • The complete bandstop filter never exceeds unity gain for sinusoidal inputs • Intermediate gains exceed 12dB – That’s 2-bits above the input MSB
• Let’s examine the area of the notch in more detail … A/D DSP
EECS 247 Lecture 11: Digital Filters
© 2002 B. Boser 34
Method #1 Bandstop Responses 20
Gain (dB)
0
- 20
no “high Q surprises” - 40
- 60 19.5
20.5 Frequency (kHz)
A/D DSP
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Method #2 Bandstop Responses • Next, we’ll examine all the intermediate transfer functions for the method #2 difference equation • The following figures show significantly different intermediate frequency responses and somewhat lower maximum intermediate gains A/D DSP
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© 2002 B. Boser 36
Method #2 Bandstop Responses 0
Gain (dB)
- 50
9.5dB max. gain
- 100
- 150
- 200 0
100
200
300
400
500
Frequency (kHz) A/D DSP
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Method #2 Bandstop Responses 20
Gain (dB)
0
- 20
- 40
- 60 19.5
20.5 Frequency (kHz)
A/D DSP
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Avoiding Overload • Sinusoidal steady-state responses for intermediate results are easy to compute and provide useful insight, but sinusoidal inputs are “never” worst cases for overload • Absolute values of filter impulse response coefficients can give worst-case conditions for overload and intermediate overload A/D DSP
EECS 247 Lecture 11: Digital Filters
© 2002 B. Boser 39
Digital Filter Models • The order of arithmetic operations in digital signal processing matters • Digital filter models must be “cycle true” • Unanticipated filter overloads are inexcusable design errors – Real-world chip developments must never be lateto-market because of such easy-to-avoid errors! A/D DSP
EECS 247 Lecture 11: Digital Filters
© 2002 B. Boser 40
Biquad Quantization Noise • Suppose we build our bandstop difference equation with B-bit registers and a BxB = 2B hardware multiplier:
y k = Gx k − Gx k −1 ( 2 cos Θ) + Gx k −2 + y k −1 ( 2 r cos Θ) − yk −2 r 2 z −2 • Build up the difference equation leaving partial results in a 2B-bit accumulator – Accumulate y(k)’s with a minimum number (i.e. 1) of rounding operations – If each of the 5 products above is rounded to B-bits, you’ll have 5X more quantization noise power
• Output noise from rounding operations can be large for high Q digital biquads A/D DSP
EECS 247 Lecture 11: Digital Filters
© 2002 B. Boser 41
Digital Filter Models • Datapath rounding operations can degrade digital filter dynamic ranges by surprisingly large amounts • Digital filter models must be “bit true” • Bit true and cycle true models require that filter models (and modelers) provide exact test vectors for integrated digital filters A/D DSP
EECS 247 Lecture 11: Digital Filters
© 2002 B. Boser 42
Limit Cycles • A disadvantage of digital IIR filters relative to digital FIR filters is that their responses get strange as they settle in response to transients • As settling error approaches rounding error, offsets and oscillations can occur – Non-zero offsets lead to “dead zones” – Oscillations are called “limit cycles”
• A combination of rounding (or truncation) and feedback is required for limit cycles A/D DSP
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© 2002 B. Boser 43
Limit Cycles • We’ll look for limit cycles in the bandpass filter: H(z) =
0.125 (z-2-1) z-2 + 0.75
• Note that this filter passes frequencies near fS/4 A/D DSP
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© 2002 B. Boser 44
Bandpass Magnitude Response 20
Gain (dB)
0
- 20
- 40
- 60 0
A/D DSP
100
200 300 Frequency (kHz)
EECS 247 Lecture 11: Digital Filters
400
500
© 2002 B. Boser 45
Bandpass Transient Response • Let’s examine the bandpass filter’s response to the initial condition y(1)=y(2)=10 • The bandpass filter output should decay to 0 – The floating point filter output does – The fixed point filter output doesn’t – Let’s take a look…
A/D DSP
EECS 247 Lecture 11: Digital Filters
© 2002 B. Boser 46
Bandpass Transient Response 10
floating point filter
Filter Output
5
0
-5
rounded filter - 10 0
10
20
30
40
50
Sample Number (time) A/D DSP
EECS 247 Lecture 11: Digital Filters
© 2002 B. Boser 47
Limit Cycles • This bandpass filter limit cycle oscillation occurs right at fs/4 – Right in the middle of the filter passband – Could this be a low-level input to the filter at fs/4?
• IIR filter designers must evaluate and be wary of limit cycle oscillations
A/D DSP
EECS 247 Lecture 11: Digital Filters
© 2002 B. Boser 48
Digital Filter Models • Bit true and cycle true digital filter models allow simulation and evaluation of: – – – –
Overload and intermediate overload Quantization noise Limit cycles and dead zones Finite precision coefficient effects
• Spending time and money on silicon without such models is crazy! A/D DSP
EECS 247 Lecture 11: Digital Filters
© 2002 B. Boser 49
