EE247
Lecture 2
•
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
© © 2004 H.K. Page 1
Nomenclature
Filter Types
Lowpass
Highpass
Bandpass
H ( jω )
H ( jω)
H ( jω)
ω
ω
Band-reject
(Notch)
Provide frequency selectivity
EECS 247 Lecture 21: Filters
H ( jω)
H ( jω)
ω
All-pass
ω
ω
Phase shaping
or equalization
© © 2004 H.K. Page 2
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
EECS 247 Lecture 2: Filters
© © 2004 H.K. Page 3
Lowpass Filter Frequency Characteristics
H ( jω)
Passband Ripple (Rpass)
f−3dB
H (0 )
3dB
Passband
Gain
Transition
Band
H ( jω)
0
H ( jω)
Passband
EECS 247 Lecture 2: Filters
fc
fs t o p
Frequency (Hz)
Stopband
Rejection
x 10
f
Stopband
Frequency
© © 2004 H.K. Page 4
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
EECS 247 Lecture 2: Filters
© © 2004 H.K. Page 5
Component Quality Factor (Q)
•
For any component with a transfer function:
H ( jω ) =
•
1
R (ω ) + jX (ω )
Quality factor is defined as:
Q=
X (ω )
Energy S t o r e d
→
A v e r a g e Power D i s s i p a t i o n
R (ω )
EECS 247 Lecture 2: Filters
perunittime
© © 2004 H.K. Page 6
Inductor & Capacitor Quality Factor
•
Inductor Q :
YL = Rs +1j ωL
•
L
QL = ω
Rs
L
Rs
Capacitor Q :
ZC =
Rp
QC = ω CRp
1
1 + jωC
Rp
C
EECS 247 Lecture 2: Filters
© © 2004 H.K. Page 7
Pole Quality Factor
jω
s-Plane
ωP
ωx
σx
QP o l e =
σ
ωx
2σ x
EECS 247 Lecture 2: Filters
© © 2004 H.K. Page 8
Bandpass Filter Quality Factor (Q)
H ( jf )
Q= fcenter /∆f
0
Magnitude (dB)
0
-5
-3dB
∆f
-10
-15
-20
1
fcenter
0.1
EECS 247 Lecture 2: Filters
10
Frequency
© © 2004 H.K. Page 9
What is Group Delay?
•
Consider a continuous time filter with s -domain transfer function G(s):
G(jω) ≡ G(jω)e
•
jθ(ω)
Let us apply a signal to the filter input composed of sum of two
sinewaves at slightly different frequencies (∆ω<<ω):
vIN (t) = A1sin(ωt) + A2 sin[(ω+∆ω) t]
•
The filter output is:
vOUT(t) = A1 G(jω) sin[ωt+θ(ω)] +
A2 G[ j(ω+∆ω)] sin[(ω+∆ω)t+ θ(ω+∆ω)]
EECS 247 Lecture 2: Filters
© © 2004 H.K. Page 10
What is Group Delay?
+ A2 G[ j(ω+∆ω)] sin
Since
∆ω
then ∆ω
ω <<1
ω
2
[ ]
θ(ω+∆ω)
ω+∆ω
≅
≅
θ(ω)
ω
{ω[ t+
vOUT(t) = A1 G(jω) sin
[ θ(ω)+
θ(ω)
+
ω
]} +
{ (ω+∆ω) [ t +
θ(ω+∆ω)
ω+∆ω
]}
à0
dθ(ω)
∆ω
dω
(
dθ(ω)
dω
][ ω1 (1 - ∆ωω ) ]
-
θ(ω)
ω
EECS 247 Lecture 2: Filters
) ∆ωω
© © 2004 H.K. Page 11
What is Group Delay?
Signal Magnitude and Phase Impairment
vOUT(t) = A1 G(jω) sin
{ ω [t +
+ A2 G[ j(ω+∆ω)]sin
•
•
•
θ(ω)
ω
]} +
θ(ω)
dθ(ω)
∆ω
{ (ω+∆ω) [ t + θ(ω)
ω +( dω - ω ) ω ]}
If the second term in the phase of the 2 nd sinwave is non-zero, then the
filter’s output at frequency ω+∆ω is time-shifted differently than the
filter’s output at frequency ω
à “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
τPD ≡ -θ(ω)/ω is called the “phase delay” and has units of time
EECS 247 Lecture 2: Filters
© © 2004 H.K. Page 12
What is Group Delay?
Signal Magnitude and Phase Impairment
•
Phase distortion is avoided only if:
dθ(ω)
dω
•
•
•
•
-
θ(ω)
ω =0
Clearly, if θ(ω)=kω, k a constant, à no phase distortion
This type of filter phase response is called “linear phase”
àPhase 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
à 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
EECS 247 Lecture 2: Filters
© © 2004 H.K. Page 13
What is Group Delay?
Signal Magnitude and Phase Impairment
•
If τGR = τPD à No phase distortion
[ (
)] +
+ A G[ j(ω+∆ω)] sin [ (ω+∆ω) ( t - τ )]
vOUT(t) = A1 G(jω) sin ω t - τGR
2
•
•
GR
If alsoG( jω)=G[ j(ω+∆ω)] 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 realizable exactly
EECS 247 Lecture 2: Filters
© © 2004 H.K. Page 14
Summary
Group Delay
• Phase delay is defined as:
τ PD ≡ -θ( ω)/ ω [ time]
• Group delay is defined as :
τGR ≡ -dθ(ω)/dω [time]
• If θ(ω)=kω, k a constant, à no phase distortion
• For a linear phase filter τ GR ≡ τ PD =k
EECS 247 Lecture 2: Filters
© © 2004 H.K. Page 15
Filter Types
Butterworth Lowpass Filter
d
•
N
H ( jω )
dω
=0
ω =0
Moderate phase distortion
- 20
- 40
- 60
0
5
-200
3
1
-400
0
1
2
Normalized Group Delay
Maximally flat amplitude
within the filter passband
Phase (degrees)
•
Magnitude (dB)
0
Normalized Frequency
Example: 5th Order Butterworth filter
EECS 247 Lecture 2: Filters
© © 2004 H.K. Page 16
Butterworth Lowpass Filter
jω
•
All poles
•
Poles located on the unit
circle with equal angles
s-plane
σ
Example: 5th Order Butterworth filter
EECS 247 Lecture 2: Filters
© © 2004 H.K. Page 17
Filter Types
Chebyshev I Lowpass Filter
-20
-40
35
0
-200
-400
0
0
1
Normalized Group Delay
Chebyshev I filter
– Equal-ripple passband
– Sharper transition band
compared to Butterworth
– Poorer group delay
Phase (degrees)
•
Magnitude (dB)
0
2
Normalized Frequency
Example: 5th Order Chebyshev filter
EECS 247 Lecture 2: Filters
© © 2004 H.K. Page 18
Chebyshev I Lowpass Filter Characteristics
jω
•
•
•
s-plane
All poles
Poles located on an ellipse
inside the unit circle
Allowing more ripple in the
passband:
σ
_Narrower transition band
_Sharper cut-off
_Higher pole Q
Chebyshev I LPF 3dB passband ripple
Chebyshev I LPF 0.1dB passband ripple
Example: 5th Order Chebyshev I Filter
EECS 247 Lecture 2: Filters
© © 2004 H.K. Page 19
Filter Types
Cheybshev II Lowpass
0
-20
-40
-60
0
Phase (deg)
Chebyshev II filter
– Ripple in stopband
– Sharper transition
band compared to
Butterworth
– Passband group
delay superior to
Chebyshev I
Magnitude (dB)
Bode Diagram
•
-90
-180
-270
-360
0
0.5
1
1.5
2
Frequency [Hz]
Example: 5th Order Chebyshev II filter
EECS 247 Lecture 2: Filters
© © 2004 H.K. Page 20
Filter Types
Cheybshev II Lowpass
•
jω
Both poles & zeros
– No. of poles n
– No. of zeros n-1
s-plane
•
Poles located both inside
& outside of the unit circle
•
•
Zeros located on jω axis
Ripple in the stopband
only
σ
Example:
5th Order
Chebyshev II Filter
poles
zeros
EECS 247 Lecture 2: Filters
© © 2004 H.K. Page 21
Filter Types
Elliptic Lowpass Filter
Elliptic filter
– Ripple in passband
– Ripple in the stopband
– Sharper transition band
compared to Butterworth &
both Chebyshevs
– Poorer group delay
-20
-40
-60
0
Phase (degrees)
•
Magnitude (dB)
0
-200
-400
0
1
2
Normalized Frequency
Example: 5th Order Elliptic filter
EECS 247 Lecture 2: Filters
© © 2004 H.K. Page 22
Filter Types
Elliptic Lowpass Filter
•
•
•
jω
Both poles & zeros
– No. of poles n
– No. of zeros n-1
s-plane
Zeros located on jω axis
Sharp cut-off
_Narrower transition
σ
band
_Pole Q higher
compared to the
previous filters
Pole
Zero
Example: 5th Order Elliptic Filter
EECS 247 Lecture 2: Filters
© © 2004 H.K. Page 23
Filter Types
Bessel Lowpass Filter
•
Bessel
jω
s-plane
– All poles
– Maximally flat group
delay
– Poor amplitude
attenuation
– Poles outside unit circle
(s-plane)
– Relatively low Q poles
σ
Pole
Example: 5th Order Bessel filter
EECS 247 Lecture 2: Filters
© © 2004 H.K. Page 24
Filter Types
Comparison of Various LPF Magnitude Response
Magnitude (dB)
0
-20
-40
0
1
Magnitude (dB)
-60
2
Normalized Frequency
All 5th order filters with same corner freq.
Bessel
Butterworth
Chebyshev I
Chebyshev II
Elliptic
EECS 247 Lecture 2: Filters
© © 2004 H.K. Page 25
Filter Types
Comparison of Various LPF Singularities
jω
Poles Bessel
Poles Butterworth
Poles Elliptic
Zeros Elliptic
Poles Chebyshev I 0.1dB
s-plane
σ
EECS 247 Lecture 2: Filters
© © 2004 H.K. Page 26
Comparison of Various LPF Groupdelay
5
28
Bessel
Chebyshev I
0.5dB Passband Ripple
1
1
12
10
Butterworth
4
1
1
Ref: A. Zverev, Handbook of filter synthesis, Wiley, 1967.
EECS 247 Lecture 2: Filters
© © 2004 H.K. Page 27
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
EECS 247 Lecture 2: Filters
© © 2004 H.K. Page 28
Magnitude Response
Bode Magnitude Diagram
0
Magnitude (dB)
-10
-20
-30
-40
-50
-60
4th Order Chebychev 1
4th Order Bessel
-70 4
10
5
6
10
Frequency [Hz]
10
EECS 247 Lecture 2: Filters
© © 2004 H.K. Page 29
Phase Response
0
-50
Phase [degrees]
-100
-150
-200
-250
-300
4th Order Chebychev 1
4th Order Bessel
-350
0
0.5
1
Frequency [Hz]
EECS 247 Lecture 2: Filters
1.5
2
5
x 10
© © 2004 H.K. Page 30
Group Delay
14
4th Ord. Chebychev 1
4th Ord. Bessel
12
Group Delay [µ s]
10
8
6
4
2
0
4
10
5
10
Frequency [Hz]
10
6
EECS 247 Lecture 2: Filters
© © 2004 H.K. Page 31
Normalized Group Delay
3
4th Ord. Chebychev 1
4th Ord. Bessel
Group Delay [normalized]
2.5
2
1.5
1
0.5
0
4
10
EECS 247 Lecture 2: Filters
5
10
Frequency [Hz]
10
6
© © 2004 H.K. Page 32
Step Response
1.4
4th Order Chebychev 1
4th Order Bessel
1.2
Amplitude
1
0.8
0.6
0.4
0.2
0
0
0.5
1
Time (sec)
EECS 247 Lecture 2: Filters
1.5
2
-5
x 10
© © 2004 H.K. Page 33
Intersymbol Interference (ISI)
ISIà Broadening of pulses resulting in interference between successive transmitted
pulses
Example: Simple RC filter
EECS 247 Lecture 2: Filters
© © 2004 H.K. Page 34
Pulse Broadening
Bessel versus Chebyshev
1.5
1.5
Input
Output
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
-1.5
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
x 10
2
-1.5
-4
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
x 10
8th order Bessel
-4
4th order Chebyshev I
Chebyshev has more pulse broadening compared to Bessel à More ISI
EECS 247 Lecture 2: Filters
© © 2004 H.K. Page 35
Response to Random Data
Chebyshev versus Bessel
1.5
1
1111011111001010000100010111101110001001
0.5
Input Signal:
130kHz max.
signal spectral density
0
-0.5
-1
-1.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
x 10
1.5
1.5
1
1
0.5
0.5
0
1111011111001010000100010111101110001001
-0.5
0
1111011111001010000100010111101110001001
-0.5
-1
-1.5
0
-4
-1
0.2
0.4
0.6
0.8
1
4th order Bessel
1.2
1.4
x 10
EECS 247 Lecture 2: Filters
-4
-1.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
x 10
-4
4th order Chebyshev I
© © 2004 H.K. Page 36
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 à immunity to noise
– Horizontal opening à timing jitter
•
EECS 247 Lecture 2: Filters
© © 2004 H.K. Page 37
Measure of Signal Degradation
Eye Diagram
3
Group Delay [normalized]
0
-1
4th Ord. Chebychev 1
4th Ord. Bessel
Magnitude (dB)
2.5
-2
-3
-4
1.5
-5
-6
-7
-8
1
0.5
-9
-10
2
Bessel
Chebychev 1
2
4
6
8
10
12
0
14
4
x 10
Frequency [Hz]
•
2
4
6
8
10
12
14
4
x 10
Frequency [Hz]
Random data with max. power spectral density of:
–
–
–
50kHz
100kHz
130kHz
EECS 247 Lecture 2: Filters
© © 2004 H.K. Page 38
Eye Diagram
Chebyshev versus Bessel
Input Signal
1
0.8
0.6
Input Signal
Random data
maximum power
spectral density à 130kHz
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0.6
0.8
1
1.2
1.4
1.6
Time
-5
x 10
4th Order Chebychev
4th Order Bessel
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
-0.8
-0.8
0.8
0.9
1
1.1
1.2
1.3
1.4
Time
1
1.5
1.1
1.2
4th order Bessel
1.3
1.4
1.5
1.6
Time
-5
x 10
1.7
-5
x 10
4th order Chebyshev I
EECS 247 Lecture 2: Filters
© © 2004 H.K. Page 39
Eye Diagrams
4th Order Bessel
1
0.8
0.6
0.6
0.4
0.4
0.2
0.2
100%
Eye opening
0
-0.2
-0.2
-0.4
-0.6
-0.6
-0.8
-0.8
2.5
3
3.5
Time
4
70%
Eye opening
0
-0.4
-1
4th Order Chebychev
1
0.8
4.5
5
-5
x 10
-1
2.5
3
3.5
Time
4
4.5
5
x 10
-5
Random data maximum power spectral density à 50kHz
EECS 247 Lecture 2: Filters
© © 2004 H.K. Page 40
Eye Diagrams
4th Order Bessel
1
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
-0.2
-0.4
40%
Eye opening
0
-0.2
80%
Eye opening
-0.4
-0.6
-0.6
-0.8
-1
4th Order Chebychev
1
0.8
-0.8
1.1
1.2
1.3
1.4
1.5
1.6
1.7
Time
1.8
1.9
2
-5
-1
1.1
x 10
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
Time
2
-5
x 10
Random data maximum power spectral density à 100kHz
Filter with constant group delay à More open eye à Lower BER (bit-error-rate)
EECS 247 Lecture 2: Filters
© © 2004 H.K. Page 41
Summary
Filter Types
– Filters with high signal attenuation per pole _
poor phase response
– For a given signal attenuation requirement of
preserving constant groupdelay àHigher 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
EECS 247 Lecture 2: Filters
© © 2004 H.K. Page 42
Summary
Filter Types
• Filters with high signal attenuation per pole à poor phase
response
• For a given signal attenuation requirement of preserving
constant groupdelay àHigher order filter
– In the case of passive filters à higher component
count
– Case of integrated active filters à higher chip area &
power dissipation
• In cases where filter is followed by ADC and DSP
àpossible to digitally correct for phase non-linearities
incurred by the analog circuitry by using digital phase
equalizers
EECS 247 Lecture 2: Filters
© © 2004 H.K. Page 43
RLC Filters
•Bandpass filter:
Vo
R
s
Vo =
RC
Vi n s2 + ωo s + ωo2
Vin
L
C
Q
ωo = 1
LC
Q = ωoRC = R
Lωo
EECS 247 Lecture 2: Filters
© © 2004 H.K. Page 44
RLC Filters
•Design a bandpass filter with:
•Center frequency of 1kHz
•Q of 20
R
Vin
Vo
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
© © 2004 H.K. Page 45
RLC Filters
Effect of Component Finite Q
1 = 1 + 1
Q filt Q ideal Qind.
filt
Q=20 (ideal L)
Q=13.3 (QL=40)
eComponent Q must be much higher compared
to desired filter Q
EECS 247 Lecture 2: Filters
© © 2004 H.K. Page 46
RLC Filters
R
Vin
Vo
L
C
Question:
Can RLC filters be integrated on-chip?
EECS 247 Lecture 2: Filters
© © 2004 H.K. Page 47
Monolithic Inductors
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
EECS 247 Lecture 2: Filters
© © 2004 H.K. Page 48
Monolithic LC Filters
• Monolithic inductor in CMOS tech.
– L<10nH with Q<7
• Max. capacitor
– C< 10pF
cLC filters in the monolithic form feasible:
- freq >500MHz
- Only low quality factor filters
Learn more in EE242
EECS 247 Lecture 2: Filters
© © 2004 H.K. Page 49
Monolithic Filters
• Desirable to integrate filters with critical frequencies
<< 500MHz
• Per previous slide LC filters not a practical option in
the integrated form
• Good alternative:
cIntegrator based filters
EECS 247 Lecture 2: Filters
© © 2004 H.K. Page 50