EE247 - Lecture 2 Filters

• Filters: – Nomenclature – Specifications • Quality factor • Magnitude/phase response versus frequency characteristics • Group delay – Filter types • Butterworth • Chebyshev I & II • Elliptic • Bessel – Group delay comparison example – Biquads

Nomenclature Filter Types wrt Frequency Range Selectivity

Lowpass Highpass Bandpass Band-reject All-pass (Notch) H j H j

 

Provide frequency selectivity Phase shaping or equalization

• Magnitude response versus frequency characteristics: – Passband ripple (Rpass) – Cutoff frequency or -3dB frequency – Stopband rejection – Passband gain • Phase characteristics: – Group delay • SNR (Dynamic range) • SNDR (Signal to Noise+Distortion ratio) • Linearity measures: IM3 (intermodulation distortion), HD3 (harmonic distortion), IIP3 or OIP3 (Input-referred or output- referred third order intercept point) • Area/pole & Power/pole

Filter Magnitude versus Frequency Characteristics Example: Lowpass H j [dB] Passband Ripple (Rpass) f3dB

H0 3dB Passband Transition Gain H j Band Stopband Rejection

0 H j Frequency (Hz) f fc f x 10 stop Stopband Passband Frequency

• Filters: – Nomenclature – Specifications • Magnitude/phase response versus frequency characteristics • Quality factor • Group delay – Filter types • Butterworth • Chebyshev I & II • Elliptic • Bessel – Group delay comparison example – Biquads

Quality Factor (Q)

• The term quality factor (Q) has different definitions in different contexts: –Component quality factor (inductor & capacitor Q) –Pole quality factor –Bandpass filter quality factor

• Next 3 slides clarifies each

• For any component with a transfer function:

Hj   1 R  jX  

• Quality factor is defined as:

X  Energy Stored Q   per unit time R Average Power Dissipation

Component Quality Factor (Q) Inductor & Capacitor Quality Factor

• Inductor Q :

Rs  series parasitic resistance

1L R L YQLLs Rss j L R

• Capacitor Q :

Rp  parallel parasitic resistance Rp 1 ZQCRCC p 1  jC Rp C

EECS 247 Lecture 2: Filters © 2010 Page 8 Pole Quality Factor j • Typically filter s-Plane singularities include pairs of complex conjugate poles. x P • Quality factor of complex conjugate poles are defined as: x 

p QPole  2 x

Bandpass Filter Quality Factor (Q) H jf  Q fcenter /Df

 0

 -3dB

Df = f2 - f1 

 Magnitude [dB] Magnitude

  0.1 1 10 f1 fcenter f2 Frequency

What is Group Delay?

• Consider a continuous-time filter with s-domain transfer function G(s): j() G(j)  G(j)e

• Let us apply a signal to the filter input composed of sum of two sine waves at slightly different frequencies (D):

vIN(t) = A1sin(t) + A2sin[(+D) t]

• The filter output is:

vOUT(t) = A1 G(j) sin[t+()] +

A2 G[ j(+D)] sin[(+D)t+ (+D)]

() v (t) = A G(j) sin  OUT 1 { [ t +  ]} +

(+D) + A2 G[ j(+D)] sin (+D) t + { [ +D ]}

D D 2 Since <<1 then  [  ] 0

(+D) d() 1 D  ()+ D 1 - +D [ d ][  (  )]

() d() () D  -  + ( d  ) 

What is Group Delay? Signal Magnitude and Phase Impairment

() v (t) = A G(j) sin  OUT 1 { [t +  ]} +

() d() () D + A2 G[ j(+D)]sin (+D) t + - { [  +( d  )  ]}

• PD  -()/ is called the “phase delay” and has units of time

• If the delay term d is zero the filter’s output at frequency +D and the output at frequency  are each delayed in time by -()/

• If the term d is non-zerothe filter’s output at frequency +D is time- shifted differently than the filter’s output at frequency 

 “Phase distortion”

• Phase distortion is avoided only if:

d() () d -  = 0

• Clearly, if ()=k, k a constant,  no phase distortion • This type of filter phase response is called “linear phase” Phase shift varies linearly with frequency

• GR  -d()/d is called the “group delay” and also has units of time. For a linear phase filter GR  PD =-k

 GR= PD implies linear phase • Note: Filters with ()=k+c are also called linear phase filters, but they’re not free of phase distortion

What is Group Delay? Signal Magnitude and Phase Impairment

• If GR= PD  No phase distortion

t -  vOUT(t) = A1 G(j) sin [ ( GR )] +

+ A G[ j(+D)] sin (+D) t -  2 [ ( GR )]

• If alsoG( j)=G[ j(+D)] for all input frequencies within

the signal-band, vOUT is a scaled, time-shifted replica of the input, with no “signal magnitude distortion”

• In most cases neither of these conditions are exactly realizable

• Phase delay is defined as:

PD  -()/ [ time] • Group delay is defined as :

GR  -d()/d [time]

• If ()=k, k a constant,  no phase distortion

• For a linear phase filter GR  PD =-k

Filters

• Filters: – Nomenclature – Specifications • Magnitude/phase response versus frequency characteristics • Quality factor • Group delay – Filter types (examples considered all lowpass, the highpass and bandpass versions similar characteristics) • Butterworth • Chebyshev I & II • Elliptic • Bessel – Group delay comparison example – Biquads

• Maximally flat amplitude within -20 the filter passband

-40 N d H( j ) Magnitude (dB)  0 -60 d 0 5 0

-200 3 Delay Group

• Moderate phase distortion

Phase (degrees)Phase -400 1

0 1 2 Normalized Normalized Frequency Example: 5th Order Butterworth filter

Lowpass Butterworth Filter

• All poles j

s-plane • Number of poles equal to filter order

• Poles located on the unit  circle with equal angles

pole

Example: 5th Order Butterworth Filter

0 • Chebyshev I filter – Ripple in the passband -20 – Sharper transition band

compared to Butterworth (for [dB] Magnitude -40 the same number of poles) 0 35 – Poorer group delay

compared to Butterworth -200

– More ripple in passband  [degrees]Phase poorer phase response -400

0 Group Delay Normalized 0 1 2 Normalized Frequency Example: 5th Order Chebyshev filter

Chebyshev I Lowpass Filter Characteristics

j s-plane • All poles

• Poles located on an ellipse inside the unit circle 

• Allowing more ripple in the passband: _Narrower transition band _Sharper cut-off _Higher pole Q Chebyshev I LPF 3dB passband ripple _Poorer phase response Chebyshev I LPF 0.1dB passband ripple Example: 5th Order Chebyshev I Filter

• Chebyshev II filter 0 – No ripple in passband -20

– Nulls or notches in -40 stopband

-60 Magnitude (dB) Magnitude

– Sharper transition band 0 compared to -90 Butterworth -180 – Passband phase more -270

linear compared to (deg) Phase -360 Chebyshev I 0 0.5 1 1.5 2 Normalized Frequency Example: 5th Order Chebyshev II filter

Filter Types Chebyshev II Lowpass j • Poles & finite zeros – No. of poles n (n  filter order) s-plane – No. of finite zeros: n-1

• Poles located both inside & outside of the unit circle 

• Complex conjugate zeros located on j axis

• Zeros create nulls in Example: stopband 5th Order pole Chebyshev II Filter zero

0 • Elliptic filter -20

– Ripple in passband -40

Magnitude (dB) Magnitude -60

– Nulls in the stopband 0

– Sharper transition band compared to Butterworth & -200

both Chebyshevs Phase (degrees)Phase – Poorest phase response -400 0 1 2 Normalized Frequency Example: 5th Order Elliptic filter

Filter Types Elliptic Lowpass Filter

• Poles & finite zeros j – No. of poles: n s-plane – No. of finite zeros: n-1

• Zeros located on j axis  • Sharp cut-off _Narrower transition band _Pole Q higher compared to the previous filter types Pole Zero Example: 5th Order Elliptic Filter

• Bessel j s-plane

–All poles

–Poles outside unit circle  –Relatively low Q poles

–Maximally flat group delay

Pole –Poor out-of-band attenuation

Example: 5th Order Bessel filter

Magnitude Response Behavior as a Function of Filter Order Example: Bessel Filter

-10 -20 n=1 -30

-40 n Filter order -50 2

-60 3 Magnitude [dB] Magnitude -70 4 -80 5 6 -90 7 -100 0.1 1 10 100 Normalized Frequency

-20

-40 Magnitude (dB) Magnitude

-60

0 1 2

Normalized Frequency Magnitude(dB)

Bessel All 5th order filters with same corner freq. Butterworth Chebyshev I Chebyshev II Elliptic

Filter Types Comparison of Various LPF Singularities j Poles Bessel Poles Butterworth Poles Elliptic s-plane Zeros Elliptic Poles Chebyshev I 0.1dB



1 12 10 Butterworth

4 1

1 Ref: A. Zverev, Handbook of filter synthesis, Wiley, 1967.

Filters

• Lowpass filter with 100kHz corner frequency

• Chebyshev I versus Bessel – Both filters 4th order- same -3dB point

– Passband ripple of 1dB allowed for Chebyshev I

Magnitude Response 4th Order Chebyshev I versus Bessel

-20

-40 Magnitude (dB) Magnitude

-60 4th Order Chebyshev 1 4th Order Bessel 104 105 106 Frequency [Hz]

-50

-100

-150 4th order Bessel

-200

Phase [degrees] Phase -250 4th order Chebyshev I -300

-350 0 50 100 150 200 Frequency [kHz]

Group Delay 4th Order Chebyshev I versus Bessel

8 4th order Chebyshev 1 6

Group Delay Delay [usec] Group 4 4th order Bessel

0 10 100 1000 Frequency [kHz]

1.4

1.2 4th order Bessel 1

0.8 4th order Chebyshev 1

0.6 Amplitude 0.4

0.2

0 0 5 10 15 20 Time (usec)

Intersymbol Interference (ISI)

ISI Broadening of pulses resulting in interference between successive transmitted pulses Example: Simple RC filter

1.5 1.5 Input

1 Output 1

0.5 0.5

0 0

-0.5 -0.5

-1 -1

-1.5 -1.5 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 -4 x 10-4 x 10 4th order Bessel 4th order Chebyshev I

Note that in the case of the Chebyshev filter not only the pulse has broadened but it also has a long tail  More ISI for Chebyshev compared to Bessel

Response to Pseudo-Random Data Chebyshev versus Bessel

1.5 1111011111001010000100010111101110001001 1 Input Signal: 0.5 Symbol rate 1/130kHz 0

-0.5

-1

-1.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 -4 x 10

1.5 1.5

1 1

0.5 0.5

1111011111001010000100010111101110001001 0 1111011111001010000100010111101110001001 0

-0.5 -0.5

-1 -1

-1.5 -1.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 -4 -4 x 10 x 10 4th order Bessel 4th order Chebyshev I

– Filter types with high signal attenuation per pole _ poor phase response

– For a given signal attenuation, requirement of preserving constant groupdelay Higher order filter • In the case of passive filters _ higher component count • For integrated active filters _ higher chip area & power dissipation

– In cases where filter is followed by ADC and DSP • In some cases possible to digitally correct for phase impairments incurred by the analog circuitry by using digital phase equalizers & thus possible to reduce the required analog filter order

Filters

R Vo • Bandpass filter (2nd order):

s Vin L C Vo RC Vin 22o ssQ o j o 1 LC R s-Plane QRCo Lo 

Singularities: Pair of complex conjugate poles Zeros @ f=0 & f=inf.

RLC Filters Example

• Design a bandpass filter with: R . Center frequency of 1kHz . Filter quality factor of 20 L C

• First assume the inductor is ideal • Next consider the case where the inductor has series R resulting in a finite inductor Q of 40 • What is the effect of finite inductor Q on the overall filter Q?

Q =20 (ideal L) 1 1 1 filt. QQfiltideal ind. Q filt Qfilt. =13.3 (QL.=40)

Need to have component Q much higher compared to desired filter Q

RLC Filters

R Vo

Vin L C

Question: Can RLC filters be integrated on-chip?

Top View

Monolithic Inductors Feasible Quality Factor & Value

Typically, on-chip inductors built as spiral structures out of metal/s layers

QL  L/R)

QL measured at frequencies of operation ( >1GHz)

c Feasible monolithic inductor in CMOS tech. <10nH with Q <7

Ref: “Radio Frequency Filters”, Lawrence Larson; Mead workshop presentation 1999

EECS 247 Lecture 2: Filters © 2010 Page 48 Integrated Filters • Implementation of RLC filters in CMOS technologies requires on- chip inductors – Integrated L<10nH with Q<10 – Combined with max. cap. 20pF  LC filters in the monolithic form feasible: freq>350MHz  (Learn more in EE242 & RF circuit courses)

• Analog/Digital interface circuitry require fully integrated filters with critical frequencies << 350MHz

• Hence:

c Need to build active filters without using inductors

Filters 2nd Order Transfer Functions (Biquads) • Biquadratic (2nd order) transfer function: 1 H( s )  ss2 1 2 PPQ P H( j)  1 0 1 H( j )  H( j)  0  2 2 2  1 H( j)  QP 2  P P PPQ

P 2 Biquad poles @: s  1  1  4QP 2QP 

Note: for Q 1 poles are real, comple x otherwise P 2

Q 1 Complex conjugate poles: P 2 S-plane P 2 j s  1 jQ 4P  1 2QP  poles d Distance from origin in s-plane:

2     2  P  2 d    1 4QP 1  2QP  2  P

s-Plane

radius  P poles 1 arccos 2QP

 real part  - P P 2 2QP s  1 jQ 4P  1 2QP 