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