Circuits Circuits Integres Mixtes Analogique-NumeriqueIntegres Mixtes Analogique-Numerique

Hassan.Aboushady@lip6.frHassan.Aboushady@lip6.fr

http://www-asim.lip6.fr/~hassan/ciman.phphttp://www-asim.lip6.fr/~hassan/ciman.php

Conversion Conversion Analogique-Numerique Analogique-Numerique ΣΔΣΔ

entrée analogique

Bruit de Quantification

où

Rapport Signal sur Bruit

!

OSR"2 # SNRdB + 3dB

# +0.5 bit of resolution

3

+ H(z) ADC

DAC

Noise ShapingNoise Shaping

X(s)X(s) X(z)X(z) Y(z)Y(z)

•• The continuous-time analog signal, X(s), is sampled at the input to obtain a The continuous-time analog signal, X(s), is sampled at the input to obtain adiscrete-time signal, X(z).discrete-time signal, X(z).•• H(z) is a discrete-time analog filter, usually implemented in the switched H(z) is a discrete-time analog filter, usually implemented in the switchedcapacitors technique.capacitors technique.•• Y(z) is the discrete-time digital output signal. Y(z) is the discrete-time digital output signal.

Under certain conditions:Under certain conditions:•• The ADC can be modeled as a source of quantization noise. The ADC can be modeled as a source of quantization noise.•• Ideally, in a discrete-time system the DAC does not add error. Ideally, in a discrete-time system the DAC does not add error.

The ADC is put in a feedback loop with a filter H(z):The ADC is put in a feedback loop with a filter H(z):

--++

Linear Mathematical ModelLinear Mathematical Model

+ H(z)X(s)X(s) X(z)X(z) Y(z)Y(z)

+

E(z)E(z)

--++

!

Y (z) = E(z)+H (z) X(z)"Y (z)[ ]

!

1+H (z)[ ]Y (z) = E(z)+H (z)X(z)

!

Y (z) =1

1+H (z)E(z)+

H (z)

1+H (z)X(z)

!

NTF(z) =Y (z)

E(z)=

1

1+H (z)Noise Transfer Function:Noise Transfer Function:

!

STF(z) =Y (z)

X(z)=

H (z)

1+H (z)Signal Transfer Function:Signal Transfer Function:

ffss=1/T=1/T

1st order 1st order ΣΔΣΔ Modulator Modulator

+X(s)X(s) X(z)X(z) Y(z)Y(z)

+

E(z)E(z)

--++

!

Y (z) =1

1+H (z)E(z)+

H (z)

1+H (z)X(z)

!

y[n] = e[n]" e[n "1]+ x[n "1]Inverse z TransformInverse z Transform

!

z"1

1" z"1

ffss=1/T=1/T

!

H (z) =z"1

1" z"1

!

Y (z) = (1" z"1)E(z)+ z

"1X(z)

!

Y (z) = E(z) " z"1E(z) + z

"1X(z)

Quantization errorQuantization error

!

q[n] = e[n]" e[n "1]

!

if fin = 0 " e[n] = e[n #1] " q[n] = 0

!

as fin " # e[n] $ e[n %1] # q[n]"

1st order 1st order ΣΔΣΔ Modulator Modulator

+X(s)X(s) X(z)X(z) Y(z)Y(z)

+

E(z)E(z)

--++

!

Y (z) =1

1+H (z)E(z)+

H (z)

1+H (z)X(z)

!

NTF(z) =Y (z)

E(z)=1" z

"1Noise Transfer Function:Noise Transfer Function:

!

STF(z) =Y (z)

X(z)= z

"1Signal Transfer Function:Signal Transfer Function:

!

z"1

1" z"1

ffss=1/T=1/T

!

H (z) =z"1

1" z"1

!

Y (z) = (1" z"1)E(z)+ z

"1X(z)

1st order 1st order ΣΔΣΔ Noise Transfer Function Noise Transfer Function

!

NTF(") =1# e# j"T

= 2 j e# j"

T

2ej"T

2 # e# j"

T

2

2 j

letlet

!

z" ej#T

!

NTF(z) =1" z"1

!

Noise Power = E( f )2" NTF( f )

2

# fm

fm

$ df!

NTF(") = 2 j e# j"

T

2 sin "T

2

$

% &

'

( )

!

="2

12

# 2

3

2 fm

fs

$

% &

'

( )

3

!

="2

12 fs#

$ fm

fm

% 4sin2 &f

fs

'

( )

*

+ , df

!

="2

12

# 2

3

1

OSR3

!

In -Band

!

NTF( f )2

= 4sin2 "

f

fs

#

$ %

&

' (

44

ffmm

!

fs /2

!

fs

!

OSR"2 # SNRdB + 9dB

# +1.5 bit of resolution

!

fs

!

fs /2

!

NTF( f ) = 2sin "f

fs

#

$ %

&

' (

Integrator Circuit Integrator Circuit

!

H (s) =Vout(s)

Vin(s)=

"1

s RC

Continuous-Time IntegratorContinuous-Time Integrator

Continuous-Time and Discrete-Time ResistorContinuous-Time and Discrete-Time Resistor

1st order 1st order ΣΔΣΔ using Switched Capacitors using Switched Capacitors

2nd order 2nd order ΣΔΣΔ Modulator Modulator

+X(s)X(s) X(z)X(z) Y(z)Y(z)

+

E(z)E(z)

--++

!

NTF(z) =Y (z)

E(z)= 1" z"1( )

2

!

STF(z) =Y (z)

X(z)= z

"1

Noise Transfer Function:Noise Transfer Function:

Signal Transfer Function:Signal Transfer Function:

!

z"1

1" z"1

!

Y (z) = (1" z"1)2E(z)+ z

"1X(z)

ffss=1/T=1/T

!

1

1" z"1 +--

++

2nd order 2nd order ΣΔΣΔ Noise Transfer Function Noise Transfer Function

!

Noise Power = E( f )2" NTF( f )

2

# fm

fm

$ df

!

="2

12

# 4

5

1

OSR5

!

In -Band

!

f / fs

!

NTF( f ) = 4sin2 "

f

fs

#

$ %

&

' (

!

OSR"2 # SNRdB +15dB

# +2.5 bits of resolution

SNR of Ideal SNR of Ideal ΣΔ ΣΔ ModulatorsModulators

!

"2

12

# 2

3

1

OSR3

!

"2

12

# 4

5

1

OSR5

!

"2

12

1

OSR

!

"2

12

# 2n

2n+1

1

OSR2n+1

Conventional ADC:Conventional ADC:

1st Order 1st Order ΣΔΣΔ::

2nd Order 2nd Order ΣΔΣΔ::

nth Order nth Order ΣΔ ??ΣΔ ??

In-Band In-Band Noise Power:Noise Power:

Not Really !!Not Really !!

1-bit 1-bit ΣΔΣΔ

SNR of Ideal SNR of Ideal ΣΔ ΣΔ ModulatorsModulators

!

"2

12

# 2

3

1

OSR3

!

"2

12

# 4

5

1

OSR5

!

"2

12

1

OSR

!

"2

12

# 2n

2n+1

1

OSR2n+1

Conventional ADC:Conventional ADC: 1st Order 1st Order ΣΔΣΔ:: 2nd Order 2nd Order ΣΔΣΔ::

nth Order nth Order ΣΔ ??ΣΔ ??

In-Band In-Band Noise Power:Noise Power:

Not Really !!Not Really !!

For a For a High High SNR:SNR:• Increase oversampling ratio, OSR ↑• Increase order of noise-shaping function, n ↑ • Increase number of bits in the quantizer, Δ ↓

Non-IdealitiesNon-Idealities::•• the sampling frequency is limited• n>2 => stability problems • # bits in the quantizer > 1 => Feedback DAC linearity problems

DAC LinearityDAC Linearity

DigitalDigitalInputInput

AnalogAnalogOutputOutput

DigitalDigitalInputInput

AnalogAnalogOutputOutput

1-bit DAC1-bit DAC Multi-bit DACMulti-bit DAC

Data Weighted AveragingData Weighted Averaging

Switched-Capacitors DAC:Switched-Capacitors DAC:

Current SourcesCurrent Sources DAC:DAC:ConventionalConventional DWADWA

H. Aboushady University of Paris VI

H. Aboushady University of Paris VI

H. Aboushady University of Paris VI

H. Aboushady University of Paris VI

H. Aboushady University of Paris VIOSROSR

n=1

n=2

n=3n=4n=6n=8

H. Aboushady University of Paris VI

H. Aboushady University of Paris VI

H. Aboushady University of Paris VI

n=1

n=2

n=3n=4n=6n=8

OSROSR