2nd & higher-order2

Second and Higher-OrderDelta-Sigma Modulators

MEAD March 2008

Richard [email protected]

ANALOGDEVICES

R. SCHREIER ANALOG DEVICES, INC.

2

Overview1 MOD2: The 2nd-Order Modulator

MOD2 from MOD1 NTF (predicted & actual) SQNR performance Stability Deadbands, Distortion & Tones (audio demo) Topological Variants

2 Higher-Order Modulators MODN from MOD1 NTF Zero Optimization Stability SQNR limits for binary and multi-bit modulators Topology Overview


3

1. MOD2 from MOD1 Replace the quantizer in MOD1 with another copy of

MOD1 in a recursive fashion:

z-1

Q

z-1

Replace Q with MOD1

z-1

z-1

U V

E

X1

V U 1 z 1( )E+=

E1

V X1 E+ X1 1 z 1( )E1+= =E 1 z 1( )E1=V U 1 z 1( )2E1+=


4

Simplified Diagram Combine feedback paths, absorb feedback delay into

second integrator

and the STF is

MOD2s NTF is the square of MOD1s NTF

Q1z1z

z1U V

E

V z( ) z 1 U z( ) 1 z 1( )2E z( )+=

NTF z( ) 1 z 1( )2= STF z( ) z 1=


5

NTF Comparison

103 102 101100

80

60

40

20

0N

TFe

j2

f(

)(d

B)

Normalized Frequency

MOD1

MOD2Twice as much attenuationat all frequencies


6

Predicted Performance In-band quantization noise power

Quantization noise drops as the 5th power of OSR!SQNR increases at 15 dB per octave increase in OSR.

IQNP NTF e j2f( ) 2 See f( ) fd0

1 2 OSR( )

=2f( )4 2e2 fd

0

1 2 OSR( )

4e2

5 OSR( )5----------------------=


7

Predicted SQNR vs. OSR

For and binary quantization, the predictedSQNR of MOD2 is 94.2 dB

23 24 25 26 27 28 29 2100

20

40

60

80

100

120

140

OSR

SQNR

(dB)

MOD1MOD2

OSR 128=


8

MOD2 Simulated PSDHalf-scale sine-wave input with

Observed PSD similar to theory (40 dB/decade slope)But 3rd harmonic is visible and in-band PSD is slightly higher.

f f s 500

103 102 101Normalized Frequency

dBFS

/NBW

140

120

100

80

60

40

20

0SQNR = 85 dB@ OSR = 128

40 dB/decade

Theoretical PSDk = 1

Simulated spectrum(smoothed)

NBW = 5.7106


9

Gain of the Quantizer in MOD2 The effective quantizer gain can be computed from the

simulation data using[S&T Eq. 2.5]

For the preceding simulation, . alters the NTF:

k v y, y y, ---------------E y[ ]E y2[ ]---------------= =

k 0.63=k 1

-1 0 1-1

0

1

NTFk z( ) NTF1 z( )k 1 k( )NTF1 z( )+------------------------------------------------=


10

Revised PSD Prediction

Agreement is now excellent

103 102 101

Theoretical PSDk = 0.63

Normalized Frequency

dBFS

/NBW

NBW = 5.7106140

120

100

80

60

40

20

0

Simulated Spectrum(smoothed)


11

Variable Quantizer Gain When the input is small (below -12 dBFS), the effective

gain of the quantizer is The small-signal NTF is thus

This NTF has 2.5 dB les quantization noise suppressionthan the NTF derived from the assumptionthat

Thus the SQNR should be about 2.5 dB lower than . As the input signal increases, k decreases and the

suppression of quantization noise degradesSQNR increases less quickly than the signal power, and eventually theSQNR saturates and then decreases as the signal power is increased.

k 0.75=

NTF z( ) z 1( )2

z2 0.5z 0.25+--------------------------------------=

1 z 1( )2k 1=


12

Simulated SQNR of MOD2

Well-modeled by ideal formula; less erratic than MOD1Saturation at high signal levels due to decreased quantizer gain andaltered NTF. (Worse with low-frequency inputs.)

100 80 60 40 20 00

20

40

60

80

100

Input Amplitude (dBFS)

SQNR

(dB)

Low-frequency inputHigh-frequency input

15A2 OSR( )524

--------------------------------

SP A2 2=

IQNP 4 1 3( )

5 OSR( )5----------------------=

SQNR SPIQNP---------------=


13

Stability of MOD2 Known to be stable with DC inputs up to full-scale,

but the state bounds blow up asHein [ISCAS 1991]:

(output of 1st integrator)

(output of 2nd integrator)

However, with a hostile input (whose magnitude is lessthan 30% of full-scale) MOD2 can be driven unstable!

As a result of this input-dependent stability, it is wise tokeep the input below 70-90% of full-scale

u 1u 1

x1 u 2+

x25 u( )2

8 1 u( )-----------------------

0 100 200-0.5

0

0.5

u u = 0.3


14

Deadbands, Distortion & TonesAudio Comparison of MOD1 and MOD2

-1

1

10 100 1kHz-100

-80

-60

-40

-20

0

-1

1

102

104

-100

-80

-60

-40

-20

0

NBW = 3.0 Hz

-1

1

10 100 1kHz-100

-80

-60

-40

-20

0

NBW = 0.8 Hz


15

ObservationsTones

Quantization noise of MOD1 is distinctly non-whiteAudible tones when input is near zero, or near other simple rationalfractions of full-scale.

MOD2 is better than MOD1 in terms of its tendencytoward tonal behavior

Dead-bands

MOD1 has dead-bands whose widths are proportionalto 1/A, where A is the gain of the internal op-amp

MOD2 has dead-bands whose widths are proportionalto 1/A2

Dead-band behavior is less problematic in MOD2


16

Topological VariantDelaying Integrators

+ Delaying integrators reduce the settling requirementsCan decouple the integration phase from the driving phase

Q1z1

1z1U V

E

-2-1

NTF z( ) 1 z 1( )2= STF z( ) z 2=


17

Topological VariantFeed-Forward

+ Output of first integrator has no DC componentDynamic range requirements of this integrator are relaxed.

Although near , forInstability is more likely.

Qzz1

1z1U V

E

NTF z( ) 1 z 1( )2= STF z( ) 2z 1 z 2=

STF 1 0= STF 3= =


18

Topological VariantFeed-Forward with Extra Input Feed-In

+ No DC component in either integrators outputReduced dynamic range requirements in both integrators.

+ Perfectly flat STFNo increased risk of instability.

Qzz1

1z1U V

E

NTF z( ) 1 z 1( )2= STF z( ) 1=


19

Topological VariantError Feedback

+ Simple Very sensitive to gain errors

Only suitable for digital implementations.

QU V

E

NTF z( ) 1 z 1( )2= STF z( ) 1=

z-1z-1

2E


20

2. MODN from MOD2

NTF of MODN is the Nth power of MOD1s NTF

Q1z1z

z1U Vz

z1

V E 1z 1----------- V

zz 1----------- V

zz 1----------- V U+( )+ + +=

V z( ) z 1 U z( ) 1 z 1( )N E z( )+=

1 z 1( )NV 1 z 1( )NE 1 z 1( )N 1 1 z 1( )N 2 1+ + +( )z 1 V z 1 U+=1 z 1( )NV 1 z 1( )NE 1 z

1( )N 1

1 z 1 1---------------------------------- z 1 V z 1 U+=

1 z 1( )NV 1 z 1( )NE 1 z1

( )N 1z 1

---------------------------------- z 1 V z 1 U+=


21

NTF Comparison

NTF

ej2

f(

)(d

B)

Normalized Frequency10-3 10-2 10-1

-100

-80

-60

-40

-20

-0

20

40

MOD1

MOD2

MOD3

MOD4

MOD5


22

Predicted Performance In-band quantization noise power

Quantization noise drops as the (2N+1)th power ofOSR!

SQNR increases at (6N+3) dB per octave increase in OSR.

IQNP NTF e j2f( ) 2 See f( ) fd0

1 2 OSR( )

=2f( )2N 2e2 fd

0

1 2 OSR( )

2N

2N 1+( ) OSR( )2N 1+--------------------------------------------------e2

=


23

Improving NTF PerformanceZero Optimization

Minimize the rms in-band value of H by finding the aiwhich minimize the integral of over the passband.

Normalize passband edge to 1 for ease of calculation.H 2

x2 a12

( )2 xd1

1 , n = 2x2 x2 a1

2( )2 xd

1

1 , n = 3x2 a1

2( )2 x2 a22( )2 xd

1

1 , n = 4

-1 1a-a

H f( ) 2 i.e. Find the ai which minimize the integral

f f B


24

Solutions Up to Order = 8Order Optimal Zero Placement Relative to fB SQNR Improvement

1 0 0 dB

2 3.5 dB

3 0, 8 dB

4 13 dB

5 0, [Y. Yang] 18 dB

6 0.23862, 0.66121, 0.93247 23 dB7 0, 0.40585, 0.74153, 0.94911 28 dB8 0.18343, 0.52553, 0.79667, 0.96029 34 dB

13

-------

35---

37---

37---

2 335------

59---

59---

2 521------


25

Topological Implication Apply feedback around pairs integrators:

1z1

zz1

-g

1z1

1z1

-g

2 Delaying Integrators Non-delaying + DelayingIntegrators (LDI Loop)

Poles are the roots of

1 g 1z 1-----------

2+ 0=

i.e. z 1 j g=Not quite on the unit circle,but fairly close if g


27

The 3rd-Order Modulator is Unstablewith a Binary Quantizer

The quantizer input grows without boundThe modulator is unstable, even with an arbitrarily small input.

0 10 20 30 40-1

0

1

0 10 20 30 40-200-100

0100200

Sample Number

v

yHUGE!

Longstringsof +1/-1


28

Solutions to the Stability ProblemHistorical Order

1 Use multi-bit quantizationOriginally considered undesirable because the inherent linearity of a1-bit DAC is lost when a multi-bit quantizer is used.Less of an issue now that mismatch-shaping is available.

2 Use a more general NTF (not pure differentiation)Lower the NTF gain so that quantization error is amplified less.A common rule of thumb is to limit the maximum NTF gain to ~1.5.Unfortunately, limiting the NTF gain reduces the amount by whichquantization noise is attenuated.

3 Use a multi-stage (MASH) architectureMore on this later in the course.

Combinations of the above are possible


29

Multi-bit Quantization Can show that a modulator with NTF H and unity STF

is guaranteed to be stable if at all times,where and

In MODN,, so

, whereand thus .

Thus implies .MODN is guaranteed to be stable with an n-bit quantizer if the inputmagnitude is less than /2.This result is extremely conservative.

Similarly, guarantees the modulator isstable for inputs up to 50% of full-scale.

u umax

h 1 H 1( ) 2N= =nlev 2N= umax nlev 1 h 1+ 1= =

nlev 2N 1+=


30

Proof of CriterionBy Induction

Assume STF = 1 and . Assume for . [Induction Hypothesis]Then

Thus

And by induction for all i > 0. QED

h 1

n( ) u n( ) umax( )e i( ) 1 i n

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