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

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- 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

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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

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– 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

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Pattern for ±1/2 Full Scale

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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:

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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:

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• 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:

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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