Signal Processing Design of Integrated Analog and...

Signal Processing Design of Integrated Analog and Digital

FiltersProf. Paul Hasler

Types of Integrated Filters

Integrated Filters

Digital Filters(Binary valued)

Analog Filters(Continuous or multivalued)

AnalogContinuous-Time

Filters

AnalogSampled-Data

Filters

Where to divide Analog and Digital?

A/DConverter

Realworld(analog)

DSPProcessor

Computer(digital)

Realworld(analog)

DSPProcessor

Computer(digital)ASP

ICA/D

Specialized A/D

Analog-Digital ComparisonLow SNR: Analog / High SNR: Digital

[Sarpeskar 1997]

Analog-Digital Comparison

16bitA/D F

FT

DSP

10 bit A/D10

10 bit A/D10

10 bit A/D10

10 bit A/D10

filter

filter

filter

filter

Input

Input

DSPApplication

Program

DSPApplication

Program

DSP

SNR < 10bits

• What if exponentially spaced FFT?

Practical Interpretation of CostLow SNR: Analog / High SNR: Digital

[Sarpeskar 1997]

Circuit Issues for FiltersProgrammability / Tunability: flexibility and complexity

Available for digital (clocks/ crystals) as well as some analog (e.g. Floating-Gate) filters

Circuit Issues for Filters

High Signal-to-Noise Ratio (resolution):Ratio of the largest signal and the smallest signal

Largest signal: Harmonic Distortion (continuous-time filters), Range limitationsSmallest Signal: Noise

Programmability / Tunability: flexibility and complexityAvailable for digital (clocks/ crystals) as well as

some analog (e.g. Floating-Gate) filters

Circuit Issues for Filters

High Signal-to-Noise Ratio (resolution):Ratio of the largest signal and the smallest signal

Largest signal: Harmonic Distortion (continuous-time filters), Range limitationsSmallest Signal: Noise

Insensitivity to environmental fluctuations: Power-supply:Power Supply Rejection Ratio (PSRR)Temperature, etc.

Programmability / Tunability: flexibility and complexityAvailable for digital (clocks/ crystals) as well as

some analog (e.g. Floating-Gate) filters

Not a problem for digital filters, but can be the cause of severalproblems to other analog circuits

Circuit Issues for Filters

High Signal-to-Noise Ratio (resolution):Ratio of the largest signal and the smallest signal

Largest signal: Harmonic Distortion (continuous-time filters), Range limitationsSmallest Signal: Noise

Insensitivity to environmental fluctuations: Power-supply:Power Supply Rejection Ratio (PSRR)Temperature, etc.

Programmability / Tunability: flexibility and complexityAvailable for digital (clocks/ crystals) as well as

some analog (e.g. Floating-Gate) filters

Not a problem for digital filters, but can be the cause of severalproblems to other analog circuits

Typically, sampling in amplitude / time results in, • the more complexity is needed ( S/H blocks, anti-aliasing filters), • more power / lower-frequency / area

Design of Analog Filters

• Find the transfer function for a given filter

• “Partition” the transfer function, or an approximation, into simple parts that can be implemented.

• Implement the transfer function in a particular circuit technology

Design of Analog Filters

• Find the transfer function for a given filter

• “Partition” the transfer function, or an approximation, into simple parts that can be implemented.

• Implement the transfer function in a particular circuit technology

H(s) or H(z)

H1(s) H2(s) H3(s)

GND

V1[n]Vout[n]

GND

GND

Design of Analog Filters

H(s) or H(z)

H1(s) H2(s) H3(s)

GND

V1[n]Vout[n]

GND

GND

As basic building blocks we have • integrators, delay elements• first-order (low-pass / bandpass)• second order functions

(low-pass / bandpass / highpass)

The “circuit” design question is how to make these functionswhat inputs / outputs / internal variables

should be voltages / currents, etc.

Design of Analog Filters

H(s) or H(z)

H1(s) H2(s) H3(s)

GND

V1[n]Vout[n]

GND

GND

Design style constrainspartition

Design styleconstrainschoice of transfer function

Choosing H(s) or H(z) for a filter

Ideal lowpass filter• we can get other filters from lowpass

|H(s)|

frequency

Gain in db = 20 log10( amplitude )= 10 log10(signal power)

fpb

Passband

H(s) or H(z) for a lowpass filterLowpasslog |H(s)|

log(frequency)fpb fsb

Tpb

Tsb

1

Passband

Tra

nsiti

on R

egio

n

Stopband

H(s) or H(z) for a lowpass filterLowpasslog |H(s)|

log(frequency)fpb fsb

Tpb

Tsb

1Normalized gain

StopBand FrequencyPassBand Frequency

StopBandMinimum

Gain

PassBandMinimum

Gain

Passband

Tra

nsiti

on R

egio

n

Stopband

H(s) or H(z) for a Highpass filterHighpasslog |H(s)|

log(frequency)fpbfsb

Tpb

Tsb

1Normalized gain

StopBand Frequency PassBand Frequency

StopBandMinimum

Gain

PassBandMinimum

Gain

Stopband

Tra

nsiti

on R

egio

n

Passband

H(s) or H(z) for a Highpass filterBandpasslog |H(s)|

log(frequency)fpb1fsb1

Tpb

Tsb

1Normalized gain

Low StopBandFrequency

Low PassBandFrequency

StopBandMinimum

Gain

PassBandMinimum

Gain

fsb1

High StopBandFrequency

fpb2

High PassBandFrequency

PassbandT

rans

ition

Reg

ion

Tra

nsiti

on R

egio

n

Four Canonical Cont-Time Filters

Butterworth: Maximally flat in passband…moderate rolloff

Chebyshev : Faster rolloff by allowing ripples in passband or stopband

Elliptic: Fastest rollff by appowing ripples in both passband and stopband

Bessel: Near linear phase, slow rolloff

Classic Analog Filters (IIR digital filters):

Four Canonical Cont-Time Filters

Butterworth: Maximally flat in passband…moderate rolloff

Chebyshev : Faster rolloff by allowing ripples in passband or stopband

Elliptic: Fastest rollff by appowing ripples in both passband and stopband

Bessel: Near linear phase, slow rolloff

Other FIR (digital) filters….Other filter design (H(s) or H(z)) techniques: Optimization approaches

Should we choose H(s) or H(z) for our representation?Partially due to particular circuit tradeoffs

(tunability? tools? continuous tunability? Accuracy? power consumption?)

Classic Analog Filters (IIR digital filters):

What happens if we just cascade first order stages?

T(s) = 1

(1 + j sτ)N

What happens if we just cascade first order stages?

T(s) = 1

(1 + j sτ)N

102 103 10410-3

10-2

10-1

100

Frequency (Hz)

Filte

r gai

n

N=1N=9

N=33

17

25

Corner shifts with N

Rolloff is not initiallysharp…..

Butterworth Filter Design

Tpb = √

1

1 + ε2

Definition of ε:Transfer Function: (low-pass)

T(s) = 11 + (-1)N+1 ε sNτN

Tpb = 1/ √2 for ε = 1

(fsb τ = 2π )

√1

1 + ε2 ω2Nτ2N|T(jω)| =

Butterworth Filter Design

Tpb = √

1

1 + ε2

Definition of ε:Transfer Function: (low-pass)

T(s) = 11 + j(-1)N+1 ε2 sNτN

Tpb = 1/ √2 for ε = 1

Need to solve to meet the specification of Tsb at fsb : Filter Order (N)

(fpb τ = 2π )

Tsb2 ( 1 + ε2(fsb / fpb )2N ) = 1

√1

1 + ε2 ω2Nτ2N|T(jω)| =

To meet specifications, one choosesthe next largest integer

Butterworth Filter Design

Tpb = √

1

1 + ε2

Definition of ε:Transfer Function: (low-pass)

T(s) = 11 + j(-1)N+1 ε2 sNτN

Tpb = 1/ √2 for ε = 1

Need to solve to meet the specification of Tsb at fsb : Filter Order (N)

(fsb τ = 2π )

Tsb2 ( 1 + ε2(fsb / fpb )2N ) = 1

√1

1 + ε2 ω2Nτ2N|T(jω)| =

To meet specifications, one choosesthe next largest integer

Pole locations: (ε = 1)

1/τk = (1/τ)(sin( (2k-1)π/(2N) ) + j cos( (2k-1)π/(2N) ) ), k=1…N

T(jω) = 1

(1 + s τ1 ) (1 + s τ2 ) … (1 + s τN )

Butterworth Filter Design

102 103 10410-1

100

101

Frequency (Hz)

Sub

-Filt

er g

ain

τ = 1.5915e-005

Q = 3.83, 1.31, 0.821, 0.630, 0.541, 0.504

Specs: ε = 1, fpb =1e4, fsb =1.5e4, Tsb = 1e-2

102 103 10410-6

10-5

10-4

10-3

10-2

10-1

100

Frequency (Hz)

Filte

r gai

n

N = 12

Chebyshev Filter Design

Tpb = √

1

1 + ε2

Definition of ε:

Tpb = 1/ √2 for ε = 1

(fpb τ = 2π )

Transfer Function(low-pass)

√1

1 + ε2 cos2( N cos-1( ωτ ) )|T(jω)| = ωτ >1

√ 1 + ε2 cosh2( N cosh-1( ωτ ) )= ωτ <1

Chebyshev Filter Design

Tpb = √

1

1 + ε2

Definition of ε:

Tpb = 1/ √2 for ε = 1

(fsb τ = 2π )

Transfer Function(low-pass)

Need to solve to meet the specification of Tsb at fsb : Filter Order (N)

√1

1 + ε2 cos2( N cos-1( ωτ ) )|T(jω)| = ωτ >1

√ 1 + ε2 cosh2( N cosh-1( ωτ ) )= ωτ <1

( 1 + ε2 cosh2( N cosh-1(fsb / fpb ) ) ) Tsb = 1

Chebyshev Filter Design

Tpb = √

1

1 + ε2

Definition of ε:

Tpb = 1/ √2 for ε = 1

(fsb τ = 2π )

Transfer Function(low-pass)

Need to solve to meet the specification of Tsb at fsb : Filter Order (N)

Pole locations:

Where k goes from 1, 2, ….N

√1

1 + ε2 cos2( N cos-1( ωτ ) )|T(jω)| = ωτ >1

√ 1 + ε2 cosh2( N cosh-1( ωτ ) )= ωτ <1

( 1 + ε2 cosh2( N cosh-1(fsb / fpb ) ) ) Tsb2 = 1

1/τk = (1/τ)(sin( (2k-1)π/(2N) ) sinh( (1/N) sinh-1(1/ ε) )

+ j cos( (2k-1)π/(2N) ) cosh( (1/N) sinh-1(1/ ε) ) )

Chebyshev Filter Design

102 103 10410-3

10-2

10-1

100

Frequency (Hz)

Filte

r gai

n

102 103 10410-2

10-1

100

101

Frequency (Hz)

Sub

-Filt

er g

ain

Specs: ε = 1, fpb =1e4, fsb =1.5e4, Tsb = 1e-2

N = 6

τ = 22.034µs, Q = 3.4645

τ = 16.288µs, Q = 12.804

τ = 53.433µs, Q = 1.0459

Fewer stages, higher Qs….

Transformations between s and z

z-1 = e-sTSimple Transformation

s ~

z-1 ~ 1 - sT

(1 - z-1 ) T

1

T = sampling period

Transformations between s and z

Bilinear transform

s ~ ~

z-1 ~

1 - z-1

1 + z-1

1 – s T 1 + s T

T

1

z-1 = e-sT

~ s

1 - e-sT

1 + e-sTT

1

1 - ( 1 – sT + 0.5*(sT)2 -…)

1 + 1 – sT + 0.5*(sT)2 -… T

1

Simple Transformation

s ~

z-1 ~ 1 - sT

(1 - z-1 ) T

1

T = sampling period

~

General Filter Topology

1st OrderΣ 1st OrderΣ 1st OrderΣ 1st OrderΣ… VoutVin

General Filter Topology

N*N parameters

N poles, N zeros for N-th order filter

Problem is underspecified…therefore can optimizefor SNR, complexity, etc…

Partitioning H(s) for Circuit Implementation

“Partition” the transfer function into simple parts

• Factorization into first order and second order terms• Nearest neighbor feedback (~LC ladder filter network)• Addition of factors

(maybe out of an approximation of an FIR filter)• Additional feedback / feedforward terms

Similar for either H(s) or H(z)

Typical Filter Topologies

2nd Order 2nd Order 2nd Order 2nd Order 1st OrderVin Vout

Cascade of First and Second-Order Sections

Typical Filter Topologies

2nd Order 2nd Order 2nd Order 2nd Order 1st Order

Vout

Vin

1stO

rder

1st OrderΣ 1st OrderΣ 1st OrderΣ 1st OrderΣ

Σ

Σ

1stO

rder

1stO

rder

Σ

Σ

1stO

rder

1stO

rder

Σ

Σ

1stO

rder

…

…

Vout

Vin

Nearest Neighbor Feedback (Inspired by LC ladder network filters)

VinVout

Cascade of First and Second-Order Sections

Conclusions

Basic directions for integrated circuit filters

• Continuous or Discrete: Time and/or Amplitude

• High level specifications of filters

• Obtaining a filter function (H(s) or H(z))

• Implementing the filter function into basic blocks(first and second-order filter sections, integrators, delays, etc.)