EE247 - Lecture 2
Filters
• Material covered today: – Nomenclature
– Filter specifications
• Quality factor
• Frequency characteristics
• Group delay
– Filter types
• Butterworth
• Chebyshev I
• Chebyshev II
• Elliptic
• Bessel
– Group delay comparison example
EECS 247 Lecture 2: Filters © © 2005 H.K. Page 1
EECS 247 Lecture 2: Filters ©
Filter Specifications
Frequency characteristics (lowpass filter):
– 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 outputreferred
• third order intercept point)
• Power/pole & Area/pole
• © 2005 H.K. Page 4
Nomenclature
Filter Types
Provide frequency selectivity Phase shaping
or equalization
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EECS 247 Lecture 2: Filters ©
•
Quality Factor (Q)
The term Quality Factor (Q) has different definitions:
– Component quality factor (inductor & capacitor
Q)
• – Pole quality factor
– Bandpass filter quality factor
Next 3 slides clarifies each
© 2005 H.K. Page 6
Lowpass Filter Frequency Characteristics
H ( jw )
Passband Ripple (Rpass) f−3dB
H (0)
Passband 3dB
Transition
Gain H ( jw ) Band
Stopband
Rejection
H ( jw )
fc f
0
fstop
Frequency (Hz)
x 10
Passband Stopband
Frequency
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EECS 247 Lecture 2: Filters ©
Component Quality Factor (Q)
• For any component with a transfer function:
H j( w)= R(w)+1jX (w)
• Inductor
Inductor Q: & Capacitor Quality Factor
YL= Rs+1j Lw
•
Capacitor Q : QL=wRsL Rs L
ZC = 1 +1jwC
Rp
Rp
C
QC =wCRp
© 2005 H.K. Page 8
• Quality factor is defined as:
Q= XR((ww)) → Average Power
DissipationEnergy Stored
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EECS 247 Lecture 2: Filters ©
Pole Quality Factor
jw
s-Plane
wx wP
sx s
wx
QPole =
2s x
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EECS 247 Lecture 2: Filters ©
What is Group Delay?
• Consider a continuous time filter with s-domain transfer function
G(s): jq(w) G(jw) ” ‰G(jw)‰e
• Let us apply a signal to the filter input composed of sum of two
sinewaves at slightly different frequencies (∆ω<<ω):
vIN(t) = A1sin(wt) + A2sin[(w+Dw) t]
• The filter output is:
vOUT(t) = A1 ‰G(jw)‰sin[wt+q(w)] +
A2 ‰G[ j(w+Dw)]‰sin[(w+Dw)t+ q(w+Dw)]
EECS 247 Lecture 2: Filters © © 2005 H.K. Page 11
EECS 247 Lecture 2: Filters ©
What is Group Delay?
Signal Magnitude and Phase Impairment
vOUT(t) = A1 ‰G(jw)‰sin {w [t + q(ww) ]} +
{
+ A2 ‰G[ j(w+Dw)]‰sin (w+Dw) t + q(ww) + ddq(ww) - [ (
q(ww) Dww) ]}
What is Group Delay?
Signal Magnitude and Phase Impairment
Phase distortion is avoided only if:
dq(w) q(w)
-
dw w =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 τGR ≡ -
• dθ(ω)/dω is called the “group delay” and also has units
• of time. For a linear phase filter τGR ≡τPD =k τGR=τPD
implies linear phase
•
Note: Filters with θ(ω)=kω+c are also called linear phase filters, but
they’re not free of phase distortion
• © 2005 H.K. Page 14
• If the second term in the phase of the 2nd sin wave is non-zero, then
the filter’s output at frequency ω+∆ω is time-shifted differently than
the filter’s output at frequency ω
“Phase distortion”
• If the second term is zero, then the filter’s output at frequency
ω+∆ω and the output at frequency ω are each delayed in time by -
θ(ω)/ω
EECS 247 Lecture 2: Filters ©
• τPD ≡ -θ(ω)/ω is called the “phase delay” and has units of time EECS
247 Lecture 2: Filters © © 2005 H.K. Page 13
EECS 247 Lecture 2: Filters ©
What is Group Delay?
Signal Magnitude and Phase Impairment
• If τGR=τPD No phase distortion vOUT(t) = A1 ‰G(jw)‰sin
[w(t - t )] +
GR
Summary
Group Delay
• Phase delay is defined as:
τPD ≡ -θ(ω)/ω [ time]
•
Group delay is defined as : τGR ≡ -
dθ(ω)/dω [time]
•
If θ(ω)=kω, k a constant, no phase distortion
•
For a linear phase filter τGR ≡τPD =k
© 2005 H.K. Page 16
+ A2 ‰G[ j(w+Dw)]‰
sin
[(w+Dw) (t - t )]
GR
• If alsoG( jw)=G[ j(w+Dw)] for all input frequencies
within the signal-band, vOUT is a scaled, time-shifted
replica of the input, with no “signal magnitude distortion” :
EECS 247 Lecture 2: Filters ©
• In most cases neither of these conditions are realizable
exactly EECS 247 Lecture 2: Filters © © 2005 H.K. Page 15
EECS 247 Lecture 2: Filters ©
Filter Types
Lowpass Butterworth
Filter 0
Magnitude (dB)
-20
• Maximally flat amplitude within
the filter passband -40
-60
NormalizedGroup Delay
dN H( jw) 0 5
Phase (degrees)
=0
-200 3
dw w=0
-400 1
0 1 2
• Moderate phase distortion
Normalized Frequency
Example: 5th Order Butterworth filter
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EECS 247 Lecture 2: Filters ©
Filter Types
Chebyshev I Lowpass Filter
• Chebyshev I filter
– Ripple in the passband
– Sharper transition band
compared to Butterworth
– Poorer group delay
– As more ripple is allowed in
the passband:
• Sharper transition band
• Poorer phase response
Normalized Frequency
Example: 5th Order Chebyshev filter
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EECS 247 Lecture 2: Filters ©
Filter Types
Chebyshev II Lowpass
• Chebyshev II filter Bode
Diagram
Magnitude (dB)
0
– Ripple in stopband
-20
– Sharper transition
band compared to -40
Butterworth -60
– Passband phase 0
more linear
Phase (deg)
-90
compared to -180
Chebyshev I
-270
-360
0 0.5 1 1.5 2
Frequency [Hz]
Example: 5th Order Chebyshev II filter
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EECS 247 Lecture 2: Filters ©
Filter Types
Elliptic Lowpass Filter
0
• Elliptic filter
Magnitude (dB)
-20
– Ripple in passband
– Ripple in the stopband -40
– Sharper transition band
-60
compared to Butterworth & both
0
Chebyshevs
Phase (degrees)
– Poorest phase response
-200
-400
0 1 2
Normalized Frequency
Example: 5th Order Elliptic filter
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EECS 247 Lecture 2: Filters ©
Filter Types
Bessel Lowpass Filter
• Bessel
– All poles
– Maximally flat group delay
– Poor amplitude attenuation
– Poles outside unit circle
(s-plane)
– Relatively low Q poles
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EECS 247 Lecture 2: Filters ©
Filter Types
Comparison of Various Type LPF Magnitude Response
Magnitude dB
)
-20
(
-40
-60
0 1 2
Magnitude (dB)
Normalized Frequency
Bessel
All 5th order filters with
same corner freq. Butterworth
Chebyshev I
Chebyshev II
Elliptic
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EECS 247 Lecture 2: Filters ©
•
• Group Delay Comparison
Example
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
© 2005 H.K. Page 30
Comparison of Various LPF Groupdelay
EECS 247 Lecture 2: Filters ©
5
28
Bessel
Chebyshev I
0.5dB Passband Ripple
1
12
10 Butterworth
4
1
Ref: A. Zverev, Handbook of filter synthesis, Wiley, 1967.
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EECS 247 Lecture 2: Filters ©
Magnitude Response
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Phase Response
-50
-100
Phase [degrees]
-150
-200
-250
-300
4th Order Chebychev1
4th Order Bessel
-350
0 0.5 1 1.5 2
5
Frequency [Hz] x 10
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Group Delay
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EECS 247 Lecture 2: Filters ©
Step Response
1.4
4th Order Chebychev1
4th Order Bessel
1.2
Amplitude
0.8
0.6
0.4
0.2
0
0 0.5 1 1.5 2
-5
Time (sec) x 10
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EECS 247 Lecture 2: Filters ©
1.5
Input
1 Output
0.5
1.5
-0.5
0.5
-1
-1.5
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
x 10
-4
-0.5
Pulse Broadening -1
Bessel versus -1.5
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
Chebyshev
-4
x 10
8th order Bessel 4th order Chebyshev I
Chebyshev has more pulse broadening compared to Bessel More ISI
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EECS 247 Lecture 2: Filters ©
Measure of Signal Degradation
Eye Diagram
• Eye diagram is a useful graphical illustration for signal
degradation
• Consists of many overlaid traces of a signal using an oscilloscope
where the symbol timing serves as the scope trigger
• It is a visual summary of all possible intersymbol interference
waveforms
– The vertical opening immunity to noise
– Horizontal opening timing jitter
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EECS 247 Lecture 2: Filters ©
Eye Diagram
EECS 247 Lecture 2: Filters ©
Chebyshev versus Bessel
EECS 247 Lecture 2: Filters ©
0.8
0.6
0.4
0.2
-0.2
-0.4
-0.6
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EECS 247 Lecture 2: Filters ©
Summary
Filter Types
– Filters 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
• Case of integrated active filters _ higher chip area &
power dissipation
– In cases where filter is followed by ADC and DSP
• Possible to digitally correct for phase non-linearities
incurred by the analog circuitry by using phase equalizers
© 2005 H.K. Page 44
Eye Diagrams
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
Time x 10-5 Time x 10-5
Random data maximum symbol rate 1/100kHz Filter with
constant group delay More open eye Lower BER (bit-error-rate)
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EECS 247 Lecture 2: Filters ©
RLC Filters
•Bandpass filter: R Vo
Vin L C
VVino =s2+wQRCoss+wo2
wo =1 LC
Q=woRC=LwRo
Singularities: Complex conjugate poles + zeros and zero & infinity
EECS 247 Lecture 2: Filters © © 2005 H.K. Page 45
RLC Filters
•Design a bandpass filter with:
R Vo
•Center frequency of 1kHz
•Q of 20 V in L C
•Assume that the inductor has series R resulting in an
inductor Q of 40
•What is the effect of finite inductor Q on the overall Q?
EECS 247 Lecture 2: Filters © Page 46
© 2005 H.K.
RLC Filters
Effect of Finite Component Q
1 1 1
= +
Q filt Qideal Qind.
filt Q=20 (ideal L)
Q=13.3 (Qind.=40)
eComponent Q must be much higher compared to
desired filter Q
EECS 247 Lecture 2: Filters © © 2005 H.K. Page 47
RLC Filters
R Vo
Vin L C
Question:
Can RLC filters be integrated on-chip?
EECS 247 Lecture 2: Filters © 2005©
H.K. Page 48
Monolithic Inductors
•
Monolithic LC Filters
Monolithic inductor in CMOS tech.
– L<10nH with Q<7
Max. capacitor size (based on realistic chip area)
– C< 10pF
cLC filters in the monolithic form feasible:
- Frequency >500MHz
- Only low quality factor filters
Learn more in EE242
© 2005 H.K. Page 50
Feasible Quality Factor & Value
c Feasible monolithic inductor in CMOS tech. <10nH with Q <7
vRef: “Radio Frequency Filters”, Lawrence Larson; Mead workshop presentation 1999
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EECS 247 Lecture 2: Filters ©
Monolithic Filters
• Desirable to integrate filters with critical frequencies <<
500MHz • Per previous slide LC filters not a practical
option in the integrated form for non-RF frequencies
• Good alternative:
cActive filters built without the need for inductors
EECS 247 Lecture 2: Filters © © 2005 H.K. Page 51
