Post on 26-May-2020
Chapter 6Frequency Response,
Bode Plots and Resonance
Signal Processing is concerned with manipulating Signals to extract Information and to use that information to
generate other useful Signals
1. Fundamental Concepts of Fourier Analysis.
2. Filter of a given input consisting of sinusoidal componentsusing the filter’s transfer function.
3. Low-pass or High-pass filter & Transfer Functions.
4. Decibels, Logarithmic Frequency Scales, and Bode plots.
5. Understand Various Filters
6. Simple Digital Signal Processing System
Goal
Fourier AnalysisAll Real-World Signals are Linear Sums of Sinusoidal
components having various frequencies, amplitudes & phases.
τπω 2
=
]sincos[2
)( tnbtnaatf nn
no ωω ++= ∑
∫+
=τ
ωτ
o
o
t
tn dttntfa cos)(2
∫+
=τ
ωτ
o
o
t
tn dttntfb sin)(2
Even Step function can be analyzed
0
0
0
Let 2 /
( ) cos sin2
2 ( ) cos
2 ( ) sin
o
o
n nn
t
n t
t
n t
af t a n t b n t
a dt f t n t
b dt f t n t
τ
τ
ω π τ
ω ω
ωτ
ωτ
+
+
=
= + +
=
=
∑
∫
∫
F(ω) or f(t)
an, bnTime t
Fourier Analysis of Music
A kind of Spectrum Analyzer
Filters
Goal : Retain & Reject the Components in Certain Frequency ranges
“Code” as Political Filter!
Filters process the Sinusoid Components of an input signal differently depending of the frequency of each component.
Filters TypesExample of Ideal Low Pass filter
Transfer FunctionsTransfer function H(f ) of Two-Port Filter :
( )in
out
VV
=fH
Phasors Output V / Phasors Input V as a function of frequency
: Amplitude Change of Each Frequency Component
: Phase Change of Each Frequency Component
( )fH( )fH∠
( )fH ( )fH∠
Filter of Multiple Components
4. Convert the phasors for each output components into time functions. Add these time functions to produce the output.
1. Determine frequency & Phasors for each input component.
2. Determine Complex Transfer function for each component.
3. Obtain the phasors for each output component by multiplying the phasors for each input component by transfer-function
Experimental Determination of H(f)
- Application of Sinusoidal Source at fixed frequency - Measurement of Output Signal Amplitude, Phase - Procedure is repeated for each frequency of interest
Active Noise Canceller
Noise Reduction in Airplane or Car- Passive : Absorber- Active : Canceller
Measurement of Signal - Source : Engine
Inverse Signal to Loud Speaker- Receiver : Seat
Best Cancellation to Passenger
Low Pass Filter
First-Order Low Pass Filters
fCjR
CjRZZZ CRt πω 2
11+=+=+=Total Impedance : Zt
fCjR
in
t
in
π21
+==
VZVI IIZV
fCjCout π21
==
fCjRfCj
inout
ππ
212
1
+×=
VVfRCj
fHin
out
π211)(
+==
VV
RCfB π2
1=( ) ( )Bffj
fH+
=1
1
( )( )21
1
BfffH
+= ( )
−=∠
BfffH arctan
First-Order Low Pass Filters
Low Pass RC & RL Filters
in
out
VV
Bff
RCfB π2
1=
( ) ( )BffjfH
+=
11
LRfB π2
=
I
Decibels
( ) ( )fHfH log20dB=
Clear Description of Frequency of Certain Range of freqeuncy
Cascaded Two-Port Networks
( ) ( ) ( )fHfHfH 21 ×=( ) ( ) ( )
dB2dB1dBfHfHfH +=
2
2
1
1
1
2)(in
out
in
out
in
out
in
outfHVV
VV
VV
VV
×===
Logarithmic Frequency ScalesOn a logarithmic scale, the variable is multiplied by a given factor for equal increments of length along the axis.
Decade : a range of frequencies for which the ratio of the highestfrequency to the lowest is 10.
=
1
2log Decades ofNumber ff
Octave : a two-to-one change in frequency for clear description oflow range frequency such as 12 to 20 MHz.
( )( )
=
=
2logloglog Octaves ofNumber 12
1
22
ffff
Bode PlotsMagnitude in Decibels versus a logarithmic scale frequency.
( ) ( )BffjfH
+=
11
( ) ( )fHfH log20dB=
( )
+−=
2
dB1log10
BfffH
vs log f
In Low-Pass filter
( )
−=
BfffH log20
dB
( ) 0dB≅fH
fB=Corner frequencyBreaking Frequency( ) BffatdBfH =−−= 3
dB
1. A horizontal line at zero for f < fB /10. 2. A sloping line from zero phase at fB /10 to –90° at 10fB.3. A horizontal line at –90° for f > 10fB.
Phase Bode Plot
Example of Bode Plot
First-Order High-Pass Filters
fCjR
CjRZZZ CRt πω 2
11+=+=+=Total Impedance : Zt
fCjR
in
t
in
π21
+==
VZVI IIZV RRout ==
fCjR
R inout
π21
+×=
VVfRCj
fRCjfHin
out
ππ21
2)(+
==VV
High Pass RC Filters
( ) ( )( )B
B
ffjffjfH
+==
1in
out
VV
RCfB π2
1=
( )( )21
1
BfffH
+= ( )
−=∠
BfffH arctan90o
in
out
VV
Bff
RCfB π2
1=
LRfB π2
=
High Pass RC & RL Filters
j2πfL
Bode Plot of High Pass filter
( )[ ] 2/12/1
1
Cin
out
ffVV
+=
( )( )[ ] 2/12/1
/
C
C
in
out
ff
ffVV
+=
( )RCfc π2/1=
in
out
VV
Cff
Summary High Pass Filter
Low Pass Filter
Red Line : Multiplication
( )( )[ ]2/1
/2
C
C
in
out
ff
ffVV
+=
Red Line shows Band Selection!
( ) ( )[ ] 2222 1 ffrffff
ffVV
HLLH
H
in
out
+++−=
LLH CR
fπ2
1=
HHL CR
fπ2
1=
Response of a band-pass filter
Band Pass RC Filters with Cascade Network
L
H
RRr =
in
out
VV
BandPassfff LL :≤≤Low Pass Filter High Pass Filter
Band Reject RC Filters
LLH CR
fπ2
1=
HHL CR
fπ2
1=
in
out
VV
Cff
Not effective combination→Use Active filter!
RC Circuit as Differentiator
RVVV
dtdCI out
outin =−= ][
RV
dtdVC
dtdV
dtdVVV
dtd
RCV outinoutin
outinout =→>>→−= ][
dtdVRCV in
out =
out
outin
VacrossVoltageVVCacrossVoltage
:R: −
For small RCFor small RC
Differentiation of Input Voltage
Differentiation of Square Wave
Vout
Leading Edge Detector
RC Circuit as Integrator
RVV
dtdVCI outinout −
==
RV
dtdVCVVVV
dtd
RCV inout
outinoutinout =→>>→−= ][
∫ += constdtVRC
V inout1
out
outin
VacrossVoltageVVRacrossVoltage
:C: −
For Large RCFor Large RC
Integration of Input Voltage
Mechanical Resonance Electrical Resonance
Wind & Bridge Electron & RLC
Series Resonance
LCf
π21
0 =
CRfRLfQs
0
0
212
ππ
==
( )
−+=
ff
ffjQRfZ ss
0
0
1
( )fC
jRfLjfZs ππ
212 −+=
Resonance Frequency
Quality factor
Total Impedance
R C L
Impedance Characteristics
Filter Case of Series ResonanceFilter Case of Series Resonance
Low Pass
High Pass
Band Pass
Band Notch
R
C
L
Second-Order Low Pass Filter
( ) ( )( )ffffjQ
ffjQfHs
s
00
0
in
out
1 −+−
==VV
LCf
π21
0 =
( )
−+=
ff
ffjQRfZ ss
0
0
1
CRfRLfQs
0
0
212
ππ
==
( )fC
jRfLjfZs ππ
212 −+=
Capacitor Component
First & Second Order Low Pass Filters
Second-Order High Pass Filter
( ) ( )( )ffffjQ
ffjQfHs
s
00
0
in
out
1 −+==
VV
LCf
π21
0 =CRfRLfQs
0
0
212
ππ
==
( )
−+=
ff
ffjQRfZ ss
0
0
1
( )fC
jRfLjfZs ππ
212 −+=
Inductor Component
Bode Plot of Second-Order High Pass Filter
Series Resonant Circuit as a Band-Pass Filter
( )ffffjQsin
out
0011
−+=
VV
( )
−+=
ff
ffjQRfZ ss
0
0
1
( )fC
jRfLjfZs ππ
212 −+=
LCf
π21
0 =CRfRLfQs
0
0
212
ππ
==
Resistor Component
Vout
+
-
Vin
sLH Q
fffB 0=−=
20BffH +≅
20BffL −≅1>>sQ
Bode Plot of Second-Order Band Pass Filter
( )
−+=
ff
ffjQRfZ ss
0
0
1
]2
12[fC
fLjZZZ CLLC ππ −=+=
( )( )ffffjQ
ffffjQ
s
s
in
out
00
00
1 −+−
=VV
Series Resonant Circuit as Band-Reject Filter
LCf
π21
0 =CRfRLfQs
0
0
212
ππ
==
Bode Plot of Second-Order Band Reject Filter
PARALLEL RESONANCE
( ) ( )fLjfCjRZ p ππ 2121
1−+
=
LCf
π21
0 =CRfLf
RQp 00
22
ππ
==
( )ffffjQRZ
pp
001 −+=
Qualityfactor
Resonancefrequency
Total Impedance
( )ffffjQR
pout
001 −+=
IV
Parallel Resonant Circuit as Band-Pass Filter
pLH Q
fffB 0=−=
Resonance Filter
LCfo π2/1=
]2
12[221111
fLfCj
jfC
fLjCLLC πππ
π−=−=+=
ZZZLCRtotal ZZZ +=
RCfQ oπ2=
fCfLjRtotal ππ 2)2/1( −
+=Z
( )ffffjQsLCR
LC
in
out
0011
−−=
+=
ZZZ
VV
Photograph of Various Noise Filter
SAW Filter
fo
SAW : Surface Acoustic Wave
DIGITAL SIGNAL PROCESSING
Analog-DigitalConverter
Digital-AnalogConverter
DSP
If a signal contains no components with frequencies higher than fH, the signal can be exactly reconstructed from its samples, provided that the sampling rate fs is selected to be more than twice fH.
ADC : Conversion of Signals from Analog to Digital Form
Hs ff 2> : Sampling Rate (frequency)
sfT 1=
kN 2=
Quantization Error
Sampling Error
Example of ADC (3bit ADC)
Digital Low Pass Filter
( ) ( ) ( ) ( )nxanayny −+−= 11 TTaτ
τ+
=1
( ) 0)()(=+
−dt
tdyCR
txty
KCL at Top Node
( )
RCwithdt
tdytxty
=
=+−
τ
τ 0)()( ( )T
nynytty
dttdy )1()()( −−
=≅δ
δ
( ) ( ) ( ) ( )nxanayny −+−= 11 TTaτ
τ+
=1
Application of Digital Low Pass filter to Step Signal
Step Signal (Red Line)
o : Digitized Signal x : Filtered Signal
Step function must be smoothed by Low Pass filter, leading to Constant value
Signal with Typical Noise
Filtered Signal