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A 108dB DR, 120dB THD and 0.5Vrms Output Audio DAC with
Inter-Symbol-Interference Shaping Algorithm in 45nm
11
Lars Risbo, Rahmi Hezar, BurakKelleci, Halil Kiper, Mounir Fares
Texas Instruments
1
Motivation• Audio: Low-level performance is crucial
– We care about -140dB spurs
• Multi-level Sigma-delta was a huge improvement over R2R Nyquist DACs used for early CDplayers
22
• However: Rotation selection/DWA cause still audio issues due to ISI/asymmetrical switching
• Need fast design cycles – no time for analog tweaking. Use digital processing
• Time to retire the good ole rotation scheme?
Outline
• Introduction to Dynamic Element Matching– Multi-bit DAC error sources– DWA/rotation– Vectorized sigma-delta modulation– High-order DEM
33
g
• Theoretical foundation for analysing ISI in 1-bit signals• Comparing DEMs
– Analysis of the DWA• FM tones
– 2nd order DEM– Modified Mismatch Shaper– The ISI shaper
• Practical results in 45nm
The Nyquist-sampling multi-bit DAC(from the CD childhood in the 1980ies)
• Pros– No out of band noise– Low jitter sensitivity– Very simple digital
• Cons– DAC control signals are
grossly non-linear• Individual DAC outputs
contain grosss amounts r
44
contain grosss amounts of THD, but cancels out in the final sum (ideal condition)
• For example, the MSB DAC output is the sign of the input signal (always a square wave)
– DAC reference mismatch causes cross-over distortion
• High THD at low levels• Laser trimming needed
to get better than ~12-14 bits THD
Bin
ary
split
ter
1-bit SDM DAC (1990’ies)
P
55
• Pros– Immune to mismatch (only
one 1bit DAC)– No cross-over distortion– Immune to static amplitude
non-linearities on the 1-bit signal
• Cons– Huge amount of out of band
noise– Low-pass filter is demanding
• High precision• Complex• Burns much power
– Very sensitive to dynamic/ISI errors of the DAC
– High jitter sensitivity (unless the filter is done as Switched Cap)
Oversampled multibit DACs
66
• Pros– Reduced out of band noise
compared to 1bit– Relaxed filter requirements– Reduced jitter sensitivity– Individual 1bDAC signals contain
the audio signal plus high-pass shaped noise
– Mismatch error is high-pass shaped (1-st order using simple DWA rotation and 2nd order using 2nd order DEM with higher complexity)
• Tolerant to DAC mismatch
• Cons– Still has many of the draw backs of
the 1-bit DAC– Many segments needed to reduce
the out of band noise and jitter sensitivity
– Still high sensitivity to dynamic errors on the 1bDACs
– The filter is still a challenge– Complex Splitter/DEM needed
DAC using 4:1 weighting on elements Bob Adams (ADI), ISSCC’98
3204:~12 effective levels
(incl. The AFIR)6x4=24 segments used
ADI:64 effective levels
8+16=24 segments used___
ADI has about 12dB less out of band noise
77
4:1 weighting mapswell into a processwith good matching
Still mismatchshaping
More efficient”middle of the road” approach betweenunit- and power of two weighting
Dominant Error SourcesDAC Element MismatchDAC Asymmetrical Switching (ISI)Clock JitterAmplifier Nonlinearity
88
Amplifier Nonlinearity
Impact of DAC Non-linearity
99
We need to reduce out of band noise
Push in-band down
Example: DAC prototype problem
• Issue: bumps on the THD+N vs level graph
• Ruins the DNR datapoint at -60dBFS input signal
• Bump is due to
-80
-90
-85
d
TTTT
1010
• Bump is due to harmonics
• What is the root cause?– Problem identified to be
due to modulator tones-100 +0-90 -80 -70 -60 -50 -40 -30 -20 -10
dBFS
-105
-100
-95
Br B
FM modulation theory• A sinusoidal carrier at frequency f0 is FM modulated by a signal with
frequency f and amplitude A (e.g. audio signal)• The FM modulation produces side bands at offset frequencies being
harmonics of the audio signal frequency f• The relative amplitude of the harmonic side-bands are given by Bessel
functions– Jn(KFMA/f)– Where n is the harmonic offset, A is the modulation amplitude and KFM is the
voltage-to-frequency scaling constant (Hz/Volt), f is the modulation frequency
1111
g q y g ( ) q y(audio signal) and J() is the Bessel function
• AM (amplitude) modulation only produces 1st order sidebands and not the higher order sidebands
Freq.f0
f0+f f0+2f f0+3ff0-ff0-3f f0-2f
Validation of the FM theory using the DAC prototype
• 0.4% DC offset moves the carrier from 0Hz to ~19.5kHz (KFM=4.88MHz/FS)• Side-bands at 19.5kHz+-n*1kHz as expected, side-band amplitudes match
the harmonics for zero DC offset• Red graph with DC offset is a translation of the blue (zero DC offset) graph• This behaviour excludes any amplitude nonlinearity as the mechanism
behind the harmonic distortion-60
-65
1212
0 32k2k 4k 6k 8k 10k 12k 14k 16k 18k 20k 22k 24k 26k 28k 30k19.532k1.984k
Hz
-140
-135
-130
-125
-120
-115
-110
-105
-100
-95
-90
-85
-80
-75
-70
-111.798
-131.178
dBr A
-60dB 1kHz+0.4% DC-60dB, 1kHz, 0% DC
Comparison to measurements
• Plot showing the 2nd harmonic amplitude versus input amplitude for KFM=2.44MHz/FS (FS=Full scale digital input) and input frequency f=1kHz
• KFM was found by applying a ll DC d i th -110
-105
-100
e
MeasuredBessel function
1313
small DC and measuring the tone frequency
• Measurements match FM theory very well
• Such strongly frequency and amplitude dependent THD cannot be explained by the usual on-linearities such a cross-over distortion etc.
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01-130
-125
-120
-115
110
Sine Amplitude ref. FSH
arm
onic
am
plitu
de
- 8 0
- 1 0 0
- 9 5
- 9 0
- 8 5
dBr
B
TTTT
GLA enabled – KFM=4.88MHz/FSGLA disabled – KFM=2*4.88MHz/FS
THD+N vs Amplitude plots• Theoretical plot uses KFM and tone
amplitude matched from DC-input measurements
• THD+N drops off at higher amplitudes, sinces the harmonics spread across a wider and wider bands
• THD+N drops off fast at low amplitudes
• Measurements match theory very
Measured
1414
- 1 0 0 + 0- 9 0 - 8 0 - 7 0 - 6 0 - 5 0 - 4 0 - 3 0 - 2 0 - 1 0d B F S
- 1 0 5Measurements match theory very well
• Blue graph is for disabled GLA where the KFM doubles – which gives a shift on the THD+N ”bump”
– Again mathcing measurements!• ISSUE: The FM harmomics
dominate the THD+N at -60dB, ie. The DNR is degraded by 5-7dB due to the tone problem
• We need to suppres the tone!!– The tone is the root cause
-100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0-105
-100
-95
-90
-85
-80
Sine Amplitude [dBFS]
THD+N
- A-W
eigh
ted
[dB
]
THD cf. FM theory, A-Weighting, fin=1kHz
4.88kHz/mFS (with GLA)2*4.88kHz/mFS (without GLA)FM-theory
The popular DWA a perfect tone generator..
• Simple & popular scheme– Barrel shifter
• A.k.a ”segment rotation”
1515
rotation• Variable rotation
speed:– Proportional to the
signal xn– ”VCO” operation– Frequency Modulated
idle tones
Generic mis-match shaper
1616
DWA:H(z)=1/(z-1)1st order ΣΔ
DWA: 1st order sigma-delta modulator
• The DWA (rotation scheme) is identical to a vectorized mismatch shaper for when– The loop filter is a pure integrator
There is no dither
1717
– There is no dither• Shares the same tone issues as scalar 1st
order sigma-deltas
Root cause of the FM tones
• The popular and simple DWA (GLA) is a rotation scheme:– circular pointer– Pointer index incremented by the segment
1818
y gcount every sample
– The rotation speed is proportional to the signal
• This is almost like a VCO– We simply get FM modulation...
ANALYSIS OF 1-BIT SIGNALS
1919
ANALYSIS OF 1 BIT SIGNALS-INTER-SYMBOL INTERFERENCE-THE TRANSITION DENSITY
1-bit signals: ISI-errors and modeling• 1-bit signal: (0,1) alphabet• Errors depending on the
previous symbol• Can be modelled in z-
domain as a function of the present and past symbol:ISI =f(s s )
Segmentwaveform
sn sn+1 sn+2
2020
ISIn=f(sn,sn-1)– Defined by a 4 entry table
for all transitions (0→0, 0 → 1,1 → 0,1 → 1)
• f() can be linear, e.g. for symmetrical switching:
ISIn=f(sn,sn-1)=a0+ a1sn + a2sn-1
ISI-errorwaveform
Ts
Discrete-time model:ISIn+1=Area/Ts
The transition function
• Linear ISI function:– Fits to 3 of 4 transitions (3
linear equations and 3 unknowns (a0,a1,a2))
– The 4th transition has to fit
• We choose for analysis to assign all the nonlinearityto the rising edge (the 4th transition)
• normalizing the error to
2121
– Any deviation on the 4th transition represents a non-linearity
• Any ISI function can bedecomposed into a linear function and an separate non-linear error on the 4th transition
normalizing the error to unity we get the Transition function:
)1( 1−−=Γ nnn ss
Definition:Transition & Segment densities
• As part of the analysiswe define the segment density:
• A good DEM makesequal use of all segments, so δ– δs tracks the DAC { }ns smean=δ
2222
• Now define the transition density:
signal (assume nearDC signal)
– δs is the same for all segments{ }),( 1−Γ= nnt ssmeanδ
Theoretical maximum transition density
1tδ
s
tδ
δ=
s
t
δ
δ
−=1
0.5
2323
sδ1
0,0Allowed Region
0.5
It takes a ’1’ to get a 0->1 transition so the count of rising edges cannot exceed the count of ’1’s
Equality if no 1->1 transitions
For no 0->0 transitions (linear fit):
This gives:
Overestimates when all 4 transitions occur, hence it is an upper bound
11 1),( −− −=Γ nnn sss
st mean δδ −=Γ= 1)(
Max. Rate pattern: Midscale
2424
Pattern for ±1/3 Full Scale
2525
Pattern for ±1/2 Full Scale
2626
Pattern for ±1/5 Full Scale
2727 2828
ANALYSIS OF THE DWA
Why the DWA produces max. Transition rate
• Assume pn=1, ie pointing at element 1 (like the drawing)
• For the next time step n+1 the pointer is advanced to pn=pn+xn
• Element 1 can only be turned on in this next time step if time step n and n+1 together use mroe segments than what we have:
2929
segments than what we have:– xn+xn+1>M
• Consequently, the DWA never produces 1->1 transitions when the DAC is below midscale
• For symmetry reasons: never 0->0 when above midscale
• Consequently: The DWA always lives on the maximal transition density curve
DWA: Max. Theoretical rate
ation
-100%
3030
+100%
corre
lat
Never
% correlation
Never
DWA: Large signal behavior
• ISI causes an errorproportional to the transition density
• Abs(x) nonlinearity:
3131
• Abs(x) nonlinearity:– Even harmonics– Constant THD ratio
DWA: Frequency Modulated tones
3232Audio-band tones near idle operation!!
FM modulation theory• A sinusoidal carrier at frequency f0 is FM modulated by a signal with
frequency f and amplitude A (e.g. audio signal)• The FM modulation produces side bands at offset frequencies being
harmonics of the audio signal frequency f• The relative amplitude of the harmonic side-bands are given by Bessel
functions– Jn(KFMA/f)– Where n is the harmonic offset, A is the modulation amplitude and KFM is the
voltage-to-frequency scaling constant (Hz/Volt), f is the modulation frequency
3333
g q y g ( ) q y(audio signal) and J() is the Bessel function
• AM (amplitude) modulation only produces 1st order sidebands and not the higher order sidebands
Freq.f0
f0+f f0+2f f0+3ff0-ff0-3f f0-2f
FM idle tones : low level THD issue-60
-90
-85
-80
-75
-70
-65
d
-60dB sine, No DC:FM of idle tone gives harmonics
-60dB sine+0.4% DC:
FM of 19.5kHz idle tone ”carrier”
Earlier Si-results using DWA:
3434
0 32k2k 4k 6k 8k 10k 12k 14k 16k 18k 20k 22k 24k 26k 28k 30k19.532k1.984k
Hz
-140
-135
-130
-125
-120
-115
-110
-105
-100
-95
-111.798
-131.178
dBr A
3rd harm.”carrier”
The FM-math works: Bessel functions
-110
-105
-100
litud
e
MeasuredBessel function
Earlier Si-results using DWA:
3535
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01-130
-125
-120
-115
Sine Amplitude ref. FS
Har
mon
ic a
mpl
DWA: Small signal behaviorTones, ISI & de-correlation
• The (potentially) non-linear part of the ISI canbe modelled by the transition function:
• The ISI is linear (100% correlated with the segment signal) insideeach half-plane (above & below midscale))1(=Γ ss
3636
• 2nd order non-linearity– ISI causes in-band FM
tones as shown– FM sensitivity is fclk
– E.g a 1%FS DC input gives a tone at 1% of fclk
)• However, near idle:
– The noise shaper activitycauses frequent crossingsof the midscale line
– De-correlation between the ISI and segment signal
– ISI-tones not high-passshaped by the DWA
)1( 1−−=Γ nnn ss
DWA+ISI Simulation examples:DAC output including ISI error
T_rise=10psA_DC=0V
T_rise=10psA_DC=1mV
3737
T_rise=20psA_DC=0V
T_rise=10psA_DC=10mV
20ps/Ts=0.006%
DWA Simulation example:Spectrum for a single segment
-60dB FS sine -6dB FS sine
Γn (ISI)Γ (ISI)
3838
sn - segmentsn -segment
Γn (ISI)
FM tone harmonicsDe-correlation (no high-passShaping of ISI error)
ISI: strong even harmonics
Summary: DWA+ISI is a bad cocktail..
• Large signal:– V-shape transition rate / mean ISI error vs. signal xn– Abs(xn) type non-linearity - Even-order harmonics– Constant THD ratio vs. level
• Small-signal:– Signal xn jitters across the midscale point due to the
3939
Signal xn jitters across the midscale point due to the Noise shaper activity
– De-correlation of the ISI-error: no high-pass shaping– FM tone frequency proportional to xn: (100% mod ->
fclk)– THD due to FM tones – both even and odd order
• Harmonics follows FM theory (Bessel function)• nm-scale design:
– Increased clock does not help: ISI goes up and erroris concentrated in tones (not spread spectrum)
2nd order DEM: not much better
• Same FM tone issues
• 2nd order just slightly lower
4040
slightly loweramplitude of the tones
Idle tones: hard to break up
• Tones persist evenafter adding 4 LSB of dither in the noiseshaper
4141
• Dither needs to beadded in the DEM loop to be effective– Rotation cannot be
used– Need a full vector
quantizer
The Modified Mismatch Shaper(MMS)
• Proposed by Shui & Schreier (JSSC March 1999)
• Vectorized DEM, but restrict number of elements changing state to L
4242
elements changing state to L– Keep L constant to linearize the ISI
• The ISI error becomes a harmlessconstant if:– All segments have matching ISI– Rising and falling ISI error must match
MMS Simulation, single segment spectra
-60dB FS -6dB FS
Γ (ISI) Γ (ISI)
4343
Γn (ISI)
sn -segment
Γn (ISI)
sn -segment
Substantial THD improvement over DWA, but more noise
New algorithm: ISI-Shaper(ISSCC 2011)
1/2
TransitionDensity/rate
LinearNo in-band tones
4444
Audio band
+F.S.0-F.S.
ISI-Shaping target
Max. Max.
Linear range
The novel ISI shaping algorithm
HMLF(z)
xnFrom the modulator
M
MismatchShaping
ditherM
VQ
M-segment
Unit Element
DACM
M
DAC Output
S
M
4545
MLoop
z-1
HILF(z) 0.5
M
M
Rtran
ISIShaping
Loop
Sign
M
ISI
Sn-1 Sn ISIn0 0 00 1 11 0 01 1 0M
ISI Shaper: single segment spectra
-60dB FS -6dB FS
4646
2nd order shaping of both the segment and ISI signalHuge reduction of THD – FM tones are gone
Just for comparison: DWA sucks..
-60dB FS sine -6dB FS sine
Γn (ISI)Γ (ISI)
4747
sn - segmentsn -segment
Γn (ISI)
FM tone harmonicsDe-correlation (no high-passShaping of ISI error)
ISI: strong even harmonics
Transition density simulation
• DWA on the theoretical max limit as expected
• 2nd order has lower
4848
rate– Very irregular without
dither• MMS and ISI-shaper;
– Constant rate
Prior-art solutions to ISI• ”DC Dither” to shift tones out of band
– Popular but just moves the problem • Return-to-zero (RTZ)
– Increased current, sensitive to timing accuracy/jitter• Sample and hold de-glitching
4949
p g g– Used to de-glitch R2R DACs in early CD players
• Mismatch shaping with reduced transition rate[4]– Reduce ISI at the cost of mismatch shaping
• Pulse Width Modulation (ISSCC’2010)– Great but requires high frequency clock
• Re-spins and layout tweaking of the analog...
45nm Audio DAC chip implementation
5050
Analog implementation:”plain vanilla”
5151
Output at -6dB using DWA
5252
Output at -6dB using ISI ShaperCS-DAC Direct OutputUsing ext. I2V
5353
Output at -60dB using DWA
5454
Output at -60 dB using ISI Shaper
5555
45nm DAC: Amplitude sweepNear constant THD ratioUsing DWA as expectedDue to ISI errors
5656
45nm DAC summary
5757
Conclusions• DWA+ISI gives high amplitude THD and low level
tones/noise problems– Concentrates ISI energy in in-band tones
• ISI is hard to fight using analog techniques– Goes against the desire for fast cycle design of SoCs
5858
– Goes against the desire to clock fast in nm scale CMOS• The novel ISI shaper:
– Excellent audio performance – even in 45nm– helps fast cycle SoC design– Digitally assisted analog solution to fight ISI– Provides robustness to analog imperfection