EE247 - Lecture 2 Filters

• Filters: – Nomenclature – Specifications • Quality factor • 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 ()jω H ()jω H ()jω H ()jω H ()jω

ω ω ω ω ω

Provide frequency selectivity Phase shaping or equalization

• 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 Frequency Characteristics Example: Lowpass H ()j[dB]ω Passband Ripple (Rpass) f−3dB

H ()0 Passband 3dB Transition Gain H ()jω Band Stopband Rejection

0 H ()jω Frequency (Hz) f fc f x 10 stop Stopband Passband Frequency

• Filters: – Nomenclature – Specifications • 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()ω = 1 RjX()ωω+ ()

• Quality factor is defined as:

X ()ω Energy Stored Q = → per unit time R()ω Average Power Dissipation

Component Quality Factor (Q) Inductor & Capacitor Quality Factor

• Inductor Q :

==1Lω YQLL+ ω L Rs jL Rs Rs

• Capacitor Q : Rp 1 ZCC==QCRω p 1 + jωC Rp C

EECS 247 Lecture 2: Filters © 2009 H.K. Page 8 Pole Quality Factor jω • Typically filter s-Plane singularities include pairs of complex conjugate poles. ω ωx P • Quality factor of complex conjugate σ poles are defined as: σ x

ω p Q = Pole σ 2 x

Bandpass Filter Quality Factor (Q) Hjf() = /Δ Q fcenter f 0

-3dB Δ f = f2 -f1

Magnitude [dB] Magnitude

0.1f1 fcenter 1f2 Frequency 10

What is Group Delay?

• Consider a continuous time filter with s-domain transfer function G(s): jθ(ω) G(jω) ≡⏐G(jω)⏐e

• Let us apply a signal to the filter input composed of sum of two sinewaves at slightly different frequencies (Δω<<ω):

ω ω Δω vIN(t) = A1sin( t) + A2sin[( + ) t]

• The filter output is:

⏐ ω ⏐ ω θ ω vOUT(t) = A1 G(j ) sin[ t+ ( )] +

⏐ ω Δω ⏐ ω Δω θ ω Δω A2 G[ j( + )] sin[( + )t+ ( + )]

θ(ω) v (t) = A ⏐G(jω)⏐ sin ω OUT 1 {]}[ t + ω +

⏐ ω Δω ⏐ θ(ω+Δω) + A2 G[ j( + )] sin (ω+Δω) t + {]}[ ω+Δω Δω Since then Δω 2 ω <<1 []ω 0

θ(ω+Δω) dθ(ω) Δω ≅ θ(ω)+ Δω 1 ω+Δω [][dω ω (]1 - ω ) ≅ θ(ω) dθ(ω) θ(ω) Δω ω + ( dω - ω ) ω

What is Group Delay? Signal Magnitude and Phase Impairment

θ(ω) v (t) = A ⏐G(jω)⏐ sin ω OUT 1 {]}[t + ω +

θ(ω) dθ(ω) θ(ω) Δω + A ⏐G[ j(ω+Δω)]⏐sin (ω+Δω) t + 2 {]}[ ω +( dω - ω ) ω

τ ≡ θ ω ω • PD - ( )/ is called the “phase delay” and has units of time

• If the second delay term is zero, then the filter’s output at frequency ω+Δω and the output at frequency ω are each delayed in time by -θ(ω)/ω

• If the second term in the phase of the 2nd sin wave is non-zero, then the filter’s output at frequency ω+Δω is time-shifted differently than the filter’s output at frequency ω “Phase distortion”

• Phase distortion is avoided only if:

dθ(ω) θ(ω) dω - ω = 0 • Clearly, if θ(ω)=kω, k a constant, no phase distortion • This type of filter phase response is called “linear phase” Phase shift varies linearly with frequency τ ≡ θ ω ω • GR -d ( )/d is called the “group delay” and also has units of τ ≡τ time. For a linear phase filter GR PD =k τ τ GR= PD implies linear phase • Note: Filters with θ(ω)=kω+c are also called linear phase filters, but they’re not free of phase distortion

What is Group Delay? Signal Magnitude and Phase Impairment

τ τ •If GR= PD No phase distortion

⏐ ω ⏐ ω t - τ vOUT(t) = A1 G(j ) sin [)]( GR + + A ⏐G[ j(ω+Δω)]⏐ sin (ω+Δω) t - τ 2 [ ( GR )]

• If also⏐G( jω)⏐=⏐G[ j(ω+Δω)]⏐ 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: τ ≡ θ ω ω PD - ( )/ [ time] • Group delay is defined as : τ ≡ θ ω ω GR -d ( )/d [time]

•If θ(ω)=kω, k a constant, no phase distortion

τ ≡τ • For a linear phase filter GR PD =k

Filters

• Maximally flat amplitude within -20 the filter passband

-40 N dH(j)ω (dB) Magnitude = 0 -60 dω 0 5 ω=0

-200 3 Delay Group

• Moderate phase distortion

Phase (degrees) -400 1

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

Lowpass Butterworth Filter

• All poles jω

s-plane • Number of poles equal to filter order

• Poles located on the unit σ 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

jω s-plane • All poles

• Poles located on an ellipse inside the unit circle σ

• 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) – 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 jω • 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 σ

• Complex conjugate zeros located on jω axis

• Zeros create nulls in stopband Example: 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

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

• Zeros located on jω axis σ • Sharp cut-off Narrower transition band Pole Q higher compared to the previous filter types Pole Zero Example: 5th Order Elliptic Filter

• Bessel jω s-plane

–All poles

–Poles outside unit circle σ –Relatively low Q poles

–Maximally flat group delay

–Poor out-of-band attenuation Pole

Example: 5th Order Bessel filter

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

0 d -10 se ea -20 cr In n=1 r -30 e rd O -40 r te il n Filter order -50 F 2

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

-20

-40 Magnitude (dB) Magnitude

-60

0 12 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 jω 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)

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

-50

-100

-150 4th order Bessel

-200

Phase [degrees] -250 4th order Chebyshev 1 -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 [usec] Group Delay 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 -4 x 10 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 Psuedo-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 • Possible to digitally correct for phase impairments incurred by the analog circuitry by using digital phase equalizers & thus reducing the required filter order

Filters

R Vo • Bandpass filter (2nd order): V s in L C VoRC = ω Vin 22++o ω ssQ o jω ωo =1LC R s-Plane QRC==ωo Lωo σ

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

RLC Filters Example

• Design a bandpass filter with: R Vo Center frequency of 1kHz Filter quality factor of 20 Vin 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) 111≈+ 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

= (ω QL L/R)

QL measured at frequencies of operation ( >1GHz)

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 © 2009 H.K. 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:

Need to build active filters built without inductor

Filters 2nd Order Transfer Functions (Biquads) • Biquadratic (2nd order) transfer function: 1 H(s)= ss2 1++ ωω2 PPQ P H( jω) = 1 ω =0 1 H( jω ) = H( jω) = 0 2 ω→∞ ⎛⎞ωω2 2 −+⎛⎞ ω = ⎜⎟1 H( j ) ω=ω QP ⎜⎟ωω2 ⎜⎟ P ⎝⎠P ⎝⎠PPQ

ω =−P ⎛⎞ ± − 2 Biquad poles @: s⎜⎟ 1 1 4QP 2QP ⎝⎠

Note: for Q≤ 1 poles are real, complex otherwise P 2

Q >→1 Complex conjugate poles: P 2

ω S-plane =−P ⎛⎞ ±2 − jω s141⎜⎟jQP 2QP ⎝⎠ poles d Distance from origin in s-plane:

2 σ ⎛ ω ⎞ 2 = ⎜ P ⎟ ()+ 2 − d ⎜ ⎟ 1 4QP 1 ⎝ 2QP ⎠ = ω 2 P

s-Plane jω

= ω radius P poles 1 arccos 2QP σ

ω = P real part - ω =−P ⎛⎞ ±2 − 2QP s141⎜⎟jQP 2QP ⎝⎠

• Passive RC: only real poles can’t implement complex conjugate poles

• Terminated LC – Low power, since it is passive – Only fundamental noise sources load and source resistance – As previously analyzed, not feasible in the monolithic form for f <350MHz

• Active Biquads – Many topologies can be found in filter textbooks! – Widely used topologies: • Single-opamp biquad: Sallen-Key • Multi-opamp biquad: Tow-Thomas • Integrator based biquads

Active Biquad Sallen-Key Low-Pass Filter

G Hs()= C1 ss2 1++ ω Q ω2 R1 R2 PP P G ω = 1 V V P in C2 out RCR112 C 2 ω = P QP − 111++G • Single gain element R11CRCRC 21 2 2 • Can be implemented both in discrete & monolithic form • “Parasitic sensitive” • Versions for LPF, HPF, BP, … Advantage: Only one opamp used Disadvantage: Sensitive to parasitic – all pole no finite zeros

2 ⎛⎞s • Sharpen transition band 1+⎜⎟ ω • Can “notch out” interference H(s)= K ⎝⎠Z 2 • High-pass filter (HPF) ss⎛⎞ 1++⎜⎟ ωω • Band-reject filter PPQ ⎝⎠ P 2 ⎛⎞ω ω = P H( j )ω→∞ K⎜⎟ ⎝⎠ωZ

Note: Always represent transfer functions as a product of a gain term, poles, and zeros (pairs if complex). Then all coefficients have a physical meaning, and readily identifiable units.

Imaginary Zeros = • Zeros substantially sharpen transition band fP 100kHz = • At the expense of reduced stop-band attenuation at QP 2 high frequencies = fZ 3 fP 6 x 10 2 Pole-Zero Map 10 1.5 With zeros No zeros 1 0 0.5 -10 0 -20 -0.5Imag Axis

Magnitude [dB] Magnitude -30 -1 -40 -1.5 -50 4 5 6 7 -2 10 10 10 10 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 6 Frequency [Hz] Real Axis x 10

= fP 100kHz Q = 2 P 5 Pole-Zero Map = x10 fZ fP 6 4 20

10 2

[dB] 0 0

-10 Imag Axis -2 -20

Magnitude -4 -30 -6 -40 -6 -4 -2 0 2 4 6 -50 5 104 105 106 107 x10 Frequency [Hz] Real Axis

Tow-Thomas Active Biquad

• Parasitic insensitive • Multiple outputs

Ref: P. E. Fleischer and J. Tow, “Design Formulas for biquad active filters using three operational amplifiers,” Proc. IEEE, vol. 61, pp. 662-3, May 1973.

V ()()b a − b s + b a − b o1 = −k 2 1 1 2 0 0 2 2 + + Vin s a1s a0 2 + + Vo2 = b2s b1s b0 2 + + Vin s a1s a0 ()()− + − Vo3 = − 1 b0 b2a0 s a1b0 a0b1 2 + + Vin k1 a0 s a1s a0

• Vo2 implements a general biquad section with arbitrary poles and zeros

•Vo1 and Vo3 realize the same poles but are limited to at most one finite zero

Component Values

R given ai ,bi ,ki , C1, C2 and R8 b = 8 0 1 R3R5R7C1C2 R = 1 a C 1 ⎛ R R R ⎞ 1 1 = ⎜ 8 − 1 8 ⎟ b1 ⎜ ⎟ k R C R R R R = 1 1 1 ⎝ 6 4 7 ⎠ 2 it follows that a0 C2 = R8 b2 R R = 1 1 ω = 8 6 R3 P k k a C R2R3R7C1C2 R 1 2 0 1 a = 8 = ω 0 1 1 1 QP P R1C1 R2R3R7C1C2 R = 4 k a b − b C 1 2 1 2 1 1 a = 1 k a R1C1 = 1 0 R5 b0C2 = R2R8C2 k1 R8 R3R7C1 = R6 b2 = R7 k2 = R7 k2R8 R8

• One way of building higher-order filters (n>2) is via cascade of 2nd order biquads & 1st order , e.g. Sallen-Key,or Tow-Thomas

1st or 2nd order 2nd order 2nd order Filter Filter …… Filter 1 2 Ν

Nx 2nd order sections Filter order: n=2N

Cascade of 1st and 2nd order filters: ☺ Easy to implement Highly sensitive to component mismatch -good for low Q filters only For high Q applications good alternative: Integrator-based ladder filters

Integrator Based Filters

• Main building block for this category of filters Integrator • By using signal flowgraph techniques Conventional RLC filter topologies can be converted to integrator based type filters

• Next lecture: – Introduction to signal flowgraph techniques – 1st order integrator based filter – 2nd order integrator based filter – High order and high Q filters