Fundamental Concepts - NCKU

Prof. Tai-Haur Kuo, EE, NCKU, 2000 Advanced Analog IC Design for Communications

Fundamental Concepts• Filter

Filtery(t)x(t)

input (or excitation) output (or response)

• Fourier series expansion:– periodic signal x(t) of period

∑

∑

∑

∞

−∞=

∞

=

∞

=

=

++=

++=

k

tjkk

kkk

kkk

eX

tkAA

tkbtkaatx

0

00

000

1

1

)cos(

)sincos()(

ω

φω

ωωωπ

0

2

where Xk represents the discrete spectrum of x(t)5-1


Fundamental Concepts(Cont.)– x(t) is not periodic

• Physical meaning

∫∞∞−= ωω

πdejXtx jwt)(

21)(

where represents the continuous spectrum of x(t))( ωjX

1. Spectrum-shaping where number Xk or function X(jw)are altered in certain way in order to produce desiredform of output signal y(t).

2. If the filter is linear, the harmonic content can not be richerthan that of the input signal

3. Desired filter operation can be performed by the appropriateinterconnection of elements with chosen values.

5-2


Historical Review

Example : Bell system progress in filter technology ofvoice-frequency ( f < 4kHz ) application over a nearly 60-years span(1920~1980)

1920 - Passive LC (1)1969 - Discrete active RC (1)1973 - Thin film active RC (1)1975 - Active RC DIP (1)1980 - Switched-capacitor building block (11)

Digital signal processor (37)

(N) No. of biquardratic section

5-3


System classification• Black box representation:

– single-input, single-output system is a special case. For a filter, its enough.

input ( or excitation ) x(t) ; output ( or response ) y(t)

Systemx(t) y(t) = f X(t)

Black box– For a filter, x(t) and y(t) are electrical signals, e.g. voltage,

current, or charge.– Filter is composed of lumped active and passive elements.– A lumped element is defined as one having physical

dimensions small compared to the wavelength of the applied signals.

5-4


System classification(Cont.)• System classification

1. linear and nonlinear systems.2. continuous-time and discrete-time (or sampled-data)

systems.3. time-invariant and time-varying systems.

( Linear time-invariant is abbreviated as LTI )• A system is linear if superposition principle is satisfied.

– superposition

– A linear system can be described by a linear differential or difference equation .

– For a filter , nonlinearity must be eliminated or minimized.e.g. overdrive an amplifier => nonlinearity occurs

)( 11 xfy = )( 22 xfy =; ; 21 xxx βα +=

212121 )()()()( yyxfxfxxfxfy βαβαβα +=+=+==

5-5


Continuous-Time and Discrete-Time• Continuous-time

Input and output are continuous functions of thecontinuous variable time.

t is time• Discrete-time

;)(&)( tyytxx ==

Input and output change at only discrete instants of time. (e.g. sampling instants)

;)(&)( kTyykTxx == where k is an integer and T is the time interval between samples

• Mathematical distinction– Continuous-time systems are characterized by differential

equations.– Discrete-time systems are characterized by difference

equations.5-6


Continuous-Time and Discrete-Time ( cont.)

Amplitude

Time

τ D

Loss

Input

Output

Continuous analog system,e.g., active RC filter

5-7


Continuous-Time and Discrete-Time ( cont.)Amplitude

Time

τ D

Loss

Input

Output

Discrete time or sampled-data system

5-8


Continuous-Time and Discrete-Time( cont.)

Amplitude

Time

τ D

Loss

Input

Output

Sampled-data system with sample-and-hold(S/H),e.g.,active switched-capacitor filter

5-9


• Time-invariant– Mathematical characteristic

Time-Invariant and Time-Varying Systems

A. Continuous-time systems x(t) => y(t)x(t - τ ) => y(t - τ )for all x(t) and all τ

B. Discrete-time systemsx(kT) => y(kT)x[(k-n)T] => y[(k-n)T]for any x(kT) and n

– Physical meaning

System response depends only on the shape of input and not on the time of application .

5-10


Causal System• Response can’t precede the excitation

if x(t)=0 for t < t0 or mT for all t0 or mT

then y(t)=0 for t < t0 or mT

5-11


Representations of Continuous-LTI Systems• Single-input, single-output, continuous LTI systems

– Input/output relationship ( linear differential equation )

– If x(t) and initial conditions are known, then y(t) is completely determined.

.......)()(.......)()(111

1

1 ++=++ −−−

−

− m

m

mm

m

mn

n

nn

n

n dttxda

dttxda

dttydb

dttydb

where ( i ) y(t) is output( ii ) ai and bj are real and depend on the network

elements(a) LTI: ai and bj are constant.(b) nonlinear: ai and bj are functions

of x and/or y.(c) time-dependent: ai and bj are functions

of time.

1

1 )0(,...,)0(),0( −

−

n

n

dtdy

dtdyy

5-12


Representations of Continuous-LTI Systems(Cont.)• Zero-input response

Response obtained when the input is zero.( Response is not necessarily zero because initial conditionsmay not be zeros )

• Zero-state response

• For a linear system, the complete response is equal to the sum or superposition of the zero-input and zero-state responses.

Response obtained for any arbitrary input when all initial conditions are zero .

5-13


Frequency-Domain Concepts• Laplace transform techniques can be used to transfer time-

domain differential equations into frequency domain equations, e.g.

1

121 )0(.......)0()0()()(

−

−−− −−−=

n

nnnn

n

n

dtyd

dtdySySsYS

dttydL

Hence, )(......)()(

001

1

1 ttyty ybdt

dbdtdb n

n

nn

n

n +++ −

−

−

)(......)(

00 ttx

xadtda n

n

m ++= ( for a LTI system)

can be transformed into

)()()......(

)()()......(

02

21

1

02

21

1

sICsXbSaSaSa

sICsYbSbSbSb

xm

mm

mm

m

yn

nn

nn

n

++++=

++++−

−−

−

−−

−−

where ICy(s) and ICx(s) are from initial conditions of y and x. X(s) and Y(s) are excitation and zero-state response.

5-14


Frequency-Domain Concepts(Cont.)• Transfer function H(s) of network

)()(

....

....)()(

))((

))(()(SDSN

Sb

SaSXSY

txexcitationL

tyresponsestatezeroLSH nn

mm =

+

+==

−=

where for any realizable practical network.nm ≤– Transfer function

; voltage transfer function

; impedance transfer function

; admittance transfer function

)()(sVsV

in

out

)()(sIsV

in

out

)()(sVsI

in

out

– Driving-point impedance and admittance functions

)( Z 1)(;

)()()(Y

)()(V )(

inin

ins

sYwhereSVSIs

SISsZ in

inin

inin === &

5-15


• System analysisFrequency-Domain Concepts(Cont.)

1. Time-domain differential equation.2. Frequency-domain equation( i.e. S-domain equation).

( 2 is proved to be a more convenient method from experience).

• Transfer function of continuous LTI– A ratio of two polynomials in S with real coefficients.– Can be factored as

))...()(())...()((

)()()(

21

21

nn

nm

PSPSPSbZSZSZSa

SDSNSH

−−−−−−

==

where Zi are zeros ( H(S)=0 when S = Zi )Pi are poles ( D(S)=0 when S = Pi )

– Zi , Pi =σ+jw on complex S-plane.

5-16


Frequency-Domain Concepts(Cont.)– Transfer function with real coefficient

– Poles lie in the left planezero input response decays with time

– Poles lie on the jw-axisthe network oscillates

– Poles lie in the right planeresponses grow exponentially with time

– when zero of N(S) lie on or to the left of the jw-axis.( i.e. there are no right-plane zeros), H(S) is referredto as a minimum-phase function.

Poles and zeros are real or conjugate pairs(complex or imaginary).

– For stability, all poles must lie in the left plane.(i.e. D(s) is a Hurwitz polynomial)

5-17


Time-Domain Concepts• Continuous LTI systems ( system representation )

– convolute excitation x(t) with impulse response h(t) to obtain y(t)

A. differential equation

B. convolution or superposition integral

....)()(....)()(1

1

11

1

1 ++=++ −

−

−−

−

− m

m

mm

m

mn

n

nn

n

n dttxda

dttxda

dttydb

dttydb

∫ −=t

dtxhty0

)()()( λλλ

where λ is a dummy integral variable and we assume that the system is causal and x(t)=0 for t<0.

5-18


Time-Domain Concepts(Cont.)– h(t) is impulse response

– convolution v.s. frequency-domain representation

Assume , then)()( ttx δ=)()()()()()(

00thdtthdthty

tt=−=−= ∫∫ λλδλλδλ

(definition : with for )

1)(0

=−∫t

dt λλδ 0)( =−λδ tt≠λ

∫ ∫ =−== −t tst SXSHdtdhtxetyLSY0 0

)()(])()([)()( λλλ

∫∞ −==

0)()()( dtethtyLSH stwhere transfer function

5-19


Ideal Distortionless Transmission• y(t) is perfect replica of x(t); may be with amplification

k and delay .

– Where

0τ)()( 0τ−= tkxty

frequency domain

0)( τSKeSH −=

)()()( 0 SXketyLSY Sτ−==

input

output

0τ 0τ1. H(S) has constant magnitude K

linear phase 0ωτφ −=2. H(S) is not a real rational function

not realizable as a lumped network with a finite number of elements.

3. group delay tconstan)( 0 ==τωτ5-20


Ideal Distortionless Transmission(Cont.)– From 2, method for approximating the above

by a rational function must be developed such that it becomes realizable physically.

0)( τSKeSH −=

– Approximation method results in transmission errors since any physical network has in practice frequency-dependent magnitude and delay .

– Two method to define deviations from an ideal transmission:

1. step response.

2. impulse response.

5-21


Step Response• Ideal transmission

where H(S) is a physically unrealizable transfer function

0)( τSKeSH −=

step input : )()( tutx =S

SX 1)( =

output step function(step responses): a(t)

)()()()()( 01 τtkuSXSHLtaty −=== −

: time required for step response to rise for 10% to 90%

• Delay time & rise time :dτ rτdτ

rτ

: time required for step response to reach 50% of its final value

γ : overshoot5-22


Step Response(Cont.)• Example: one-pole case

– no over shoot

• In general for overshoot 2.23 ≈− dBrωτ 35.03 ≈− dBr fτ(i.e. )

%5≤γ

1

1)(aS

SH+

= dBa 31 −=ω;

IF x(t)=u(t)

)()1()( 1 tuety ta−−=

0=γ

1

69.0ad ≈τ

1

2.2ar ≈τ&

x(t)

ty(t)/k

rτdτ

γ

t20τ0.1

0.5

0.91.0

real

ideal

t

5-23


Impulse ResponseIdeal transmission

– Impulse response– step response

One can obtain impulse response

)()( ttX δ=oSkesH τ−=)(

)()()( 0tkthty τ−δ==

)()()()(

0

0

tkutatkth

τ−=

τ−δ=

; unrealizable

dttdath )()( ==>

Area=1

Dτ

Rτ

t

y(t)=h(t)

t

X(t)

∆ 0→∆

from step response, and vice versa .

5-24


Calculation of and Precise calculation of and are usually time-consuming Convenient method resulting in considerable simplification isproposed by Elmore(Assuming negligible overshoot or none)

Dτ

Dτ Rτ

∫ ∞=τ 0D dttht )(∫∞ τ−π=τ 0

212

DR dttht2 ])()([

- Elmore’s definition

21

022 ])([2 ∫

∞ −= Ddttht τπ

Rτ

Consider the normalized transfer function ( H(0)=1)- H(s)=- by direct division- from impulse response

∫∫∞∞ − −+−== 0

22

0 ....)!2

1)(()()( dttsstthdtethsH st

..)(!

−τ+π

τ+τ−= 2

D

2R

2

D 22ss1

mnsbsbsb1sasasa1

nn

221

mm

221 ≥

++++++++

..........

(2)

...)()()( +−+−+−−= 22211

2111 sbababsab1SH (1)

s)definitionsElmore'ingincorporatby(

5-25


Calculation of and (Cont.)Dτ Rτ

21

2221

21R

11D

ba2ab2

ab

)]([ −+−π=τ

−=τ

- Equating (1) and (2) yields

(3)

(4)

Ease of computation using Elmore’s definition- For higher-order systems with no overshoot (i.e.monotonic)can be decomposed to K monotonic cascaded stages.

∑

∑

=

=

τ=τ

τ=τ

k

1i2

12iRtotalR

k

1iiDtotalD

])([)(

)()(Area=1

Dτ

Rτ

t

h(t)monotonic

Dτ

Rτ

a(t)

t

monotonic

Example 1-2

5-26


NormalizationAdvantages

Normalization

1. avoid the tedium of having to manipulate large power of 102. minimize the effect of roundoff errors.

0Ω

1. Frequency normalization-Frequency scale changed by dividing the frequency variable by a conveniently chosen normalization frequency

0n

SSΩ

=Normalization equation

2. Impedance normalization-by dividing all impedances in the circuit by a normalizationresistance .0R o

noono

n RLLRCC

RRR === ,,

-Normalization equation5-27


Normalization(Cont.)

0n R

RR =0

00

0nn R

LSRSLLS ΩΩ

==)/(

0000 )/(111,CRSSCRCS nn ΩΩ

==

0

0n R

LL Ω= 00n RCC Ω=

0n R

RR =

,

=>

The actual unnormalized physical parameters R, L, and C are obtained by inverting the normalization equations.

Comments(practical concerns):

),,( ensionlessdimareCandLRS nnnn

=>Easy remember.

1. normalization is to remove dimensions

2. Dimensionless network

=>designer can choose convenient and practical element values.

, ,

Example 1-35-28


1. Lowpass filter2. Highpass filter3. Bandpass filter4. Bandreject filter5. Allpass filter--phase or delay specs are of primary concern.

Type of FiltersFive major types

Filter magnitude specifications

Magnitude spec. are of primary concern

-Lowpass filter -Highpass filterdBjH |,)(| ω

ripplePB

nattenuatioSB

ωpω sω

PBTB

SB

pωω

dBjH |,)(| ω

sω

ripplePB

SB TBPB

5-29


Type of Filters(Cont.)-Bandpass filter -Bandreject filter

-All pass filter

ω

dBjH |,)(| ω

LPB

HPBL

TBH

TBSB

nattenuatioSBrippleHPB

rippleLPB

SLω PLω PHω SHωω

dBjH |,)(| ω

nattenuatioSBL

LSB H

SB

PLω PHωSLω SHω

PBLTB H

TB

ripplePB

nattenuatioSBH

phasegain

pω

dB0

ο−180

dBjwH |)(| reesdegw)(φ

ω360−

5-30


1. Realization of nonminimum phase function

(i) Let

Filter Phase or Delay Specs.Frequency dependent delay

Usually not important for voice or audio.(Human ear is

Can cause intolerable distortion in video or digital

Examples :

)s(H)s(H)s(H APMN =where

N: nonminimum phase

M: minimum phase

AP: all-pass

|||| NM HH =

5-31

=>Nonminimum phase function may be needed

-

-

very insensitive to phase change with frequency.)

- Minimum phase function：with only left half-phase zeros

transmission


Filter Phase or Delay Specs.(Cont.)

)()(

sDsNH

AP

APAP = is a allpass function

where is formed by all right-plane zerosAPN

APD

(ii) Augment pole-zero

(iii)

Hence ,

)()()(

sDsDH

sDN

AP

APAP

APAP

−±=

−±=

)()(tan wD

wD1AP

R

I2phase −−=φ

5-32

is formed by all left-plane poles which are mirrorimages of the right-plane zeros

)](Im[)( jwDwDwhere API =)](Re[)( jwDwD ApR =

dwwdwDelay AP )()( φ

−=τ


Filter Phase or Delay Specs.(Cont.)-Using , a desirable delay function without any effect on

the magnitude can be achieved.

)(sHAP

|)(||)(|)()()(

sHsHwheresHsHsH

T

APT

==

-Example

)()()()()()(

wwwwww

APT

APT

τ+τ=τφ+φ=φ

The cascaded allpass can, of course, only increase the phase and delay of H(s); this is normaly no problem, because fordistortionless transmission, only the linearity of i.e., the constancy of , in the frequency range of interest is important, not its actual size.

Example 1-4

Tφ

Tτ

-

-5-33


(i) make total delay as flat as possible in the frequency range of interest.

(ii) obtain prescribed delay-Precision design requires computer aids.-Uncritical design of low order( can be performed manually with the aid of the curves below.

%)~ 2010≈ττ∆

normalized delay,

110−

210

110

210−

)(5.0PAPP ω

ωτω ×

Pωω

010

31

1.0

02.050

02.0=pQ

1.03

1

50

5

Filter Phase or Delay Specs.(Cont.)2. Phase or delay equalization

5-34


Second Order Filters• 2nd-order transfer function

))(())(()(

21212

012

012

2PSPSZSZSa

bSbS

aSaSasH++++

=++

++=

2p

p

p2

z2

z

z2

21

211

2

21

211

2

SQ

S

SQ

SK

PPSP2SZZSZ2SK

ω+ω

+

ω+ω+=

++++++

=)(

)(

)Im()Re()]Re([)Im()Re()]Re([

; where P&Z are real or

)Re()][Im()][Re(

)Re( 1

21

21

1

pp P2

PPP2

Q +=

ω= is the pole quality

factor

)Re()][Im()][Re(

)Re( 1

21

21

1

zz Z2

ZZZ2

Q +=

ω=

is the zero quality factor

)Im( z

)Re( z

1PZω

where

complex pairs

)Im(1P

)Re(1P*1P

1Ppω

5-35


Second Order Filters(Cont.)

2p

p

p2

2

HP2p

p

p2

2p

LPS

QS

SKsHS

QS

KsHω+

ω+

=ω+

ω+

ω=

)()(;

)()(

2p

p

p2

2z

2

BR2p

p

p2

p

p

BPS

QS

SKsHS

QS

SQ

KsHω+

ω+

ω+=

ω+ω

+

ω

=)(

)(;)(

)()(

1QSS

1QSS

KS

QS

SQ

SKsH

p

n2n

p

n2n

2p

p

p2

2p

p

p2

AP++

+−=

ω+ω

+

ω+ω

−=

)(

)(

)(

)()(

where ; n means normalized p

np

nSS

ωω

=ωω

= &

5-36


Second Order Filters(Cont.)

– Normalized phase

– normalized delay

)(tan)( 2n

pn

1nAP 1

Q2

ω−

ω

−=ωφ −

2p

n22n

2n

pnAPpnAPn

Q1

1Q2

)()()()(, ω+ω−

ω+=ωτω=ωτ

phasegain

pω

dB0

ο180−

dBjwH |)(| reesdegw )(φ

110−

210

110

210−

normalized delay, )(5.0PAPP ω

ωτω ×

Pωω

010

31

1.0

02.050

02.0=pQ

1.03

1

50

5

5-37


Approximation MethodsDistortionless filter is not realizable as discussed before.

(If phase or delay performance is important , allpass filter can be used to achieve the necessary phase correction.

1. H(s) must be a real rational function such that it can be realized by lumped circuits as discussed before.

2. H(s) is only a approximation of ideal characteristics on bothmagnitude and phase/delay.

1. Butterworth response : maximally flat magnitude in the passband.

2. Chebyshev response : equal ripple in the passband.3. Elliptic response : equal ripple in both the passband and

stopband.(Low order and the most economical realization)

5-38

(i.e. Ideal transfer characteristic are not realizable)Practical realization

Magnitude approximation


Approximation Methods(Cont.)4. Gaussian response :

(a) freedom from ringing or overshoot.(b) symmetry about the time for which the response is a

maximum.5. And many othersPhase or delay approximation Bessel-Thomson response : maximally flat delay.

5-39


Example

n222

11jwHωε+

=|)(|

Assume ε=1n2

0

n22

js1

11

1sH)(

|)(|

Ω+

=ω+

=

Assume )..( normalizedei10 =Ω

n22

S11sH

−=|)(| n2

2

S11SHSHsH

−=−= )()(|)(|

1. For n=1 ))(()()(S1

1s1

1S1

1SHSH 2 −+=

−=−

)()(s1

1SH+

= (Only the left-plane pole is selected)

Butterworth

,

=>The denominator of realizable filter function must be Hurwitz polynomials

5-40


Example(Cont.)

2. For n=2

]][][][[)()(

2j

21S

2j

21S

2j

21S

2j

21S

1SHSH−−−++−++

=−

]][[)()(

2j

21S

2j

21S

1SHSH−+++

=−=>

• Similarly transfer functions of other types can be derived.

• Refer to appendix III

5-41


Frequency Transformations

• Lowpass prototype =>

– Lowpass prototype

the edge of the lowpass passband – Target filter S=a+jw– Frequency transformation such that

highpass

wjaS +=s 1w =

)(sFs =

======>≤≤ 1||0 w passband of target filter

======>> 1|| w

mapped to

mapped tostopband of target filter

bandpassbandreject…….

frequency variable which is normalized such that at

5-42


Frequency Transformations(Cont.)Examples

(ii) LP=>BP

(iii) LP=>two-passband BP

=>

(i) LP=>HP

5-43

0 0

=>00


Frequency Transformations(Cont.)Lowpass to highpass

(i)

ω

1

-1

)(ωω f=

1

1S2531S5351S7160S7160H 23

3

HP+++

==>...

.

0.5dB1

ω

2LP |)(H| wj=

1w

2|)(| jwHHP

0.5dB1

ww

ss js 11

= →= = ω

716.0535.1253.1716.0)( .. 23 +++

=sss

sHge LP

5-44


Frequency Transformations(Cont.)

(ii) RC : CR transformation

5-45

baSSSHsHLP ++

∞= 2

2)()(

K outViVK outViV

bassbHsHLP ++

= 2

)0()(


Frequency Transformations(Cont.)LP=>BP

SSQ

SB

SS 11 22

0 +=

+Ω=

where luul0 B Ω−Ω=ΩΩ=Ω &

- Example 1-14LP=>BR

1SS

Q1S 2 +

=S=jw

1Q1

2 +ωω

−=ω

+1

-1 1

w

uw

lw

)(wf=ω

ωωω 12 −

= → = Qwjs

- Example 1-155-46

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Fundamental Concepts - NCKU