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Fourier Transform Analysis ofSignals and Systems
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Ideal Filters
• Filters separate what is desired from what isnot desired
• In the signals and systems context a filterseparates signals in one frequency range fromsignals in another frequency range
• An ideal filter passes all signal power in itspassband without distortion and completelyblocks signal power outside its passband
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Distortion• Distortion is construed in signal analysis to mean “changing
the shape” of a signal• Multiplication of a signal by a constant (even a negative one)
or shifting it in time do not change its shape
No Distortion Distortion
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Distortion
h t A t t( ) = −( )δ 0
Since a system can multiply by a constant or shift in time without distortion, a distortionless system would have an impulse response of the form,
h n A n n[ ] = −[ ]δ 0
or
The corresponding transfer functions are
H f Ae j ft( ) = − 2 0π
H F Ae j Fn( ) = − 2 0πor
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Filter ClassificationsThere are four commonly-used classification of filters, lowpass,highpass, bandpass and bandstop.
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Filter Classifications
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Bandwidth
• Bandwidth generally means “a range offrequencies”
• This range could be the range of frequenciesa filter passes or the range of frequenciespresent in a signal
• Bandwidth is traditionally construed to berange of frequencies in positive frequencyspace
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Bandwidth
Common Bandwidth Definitions
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Impulse Responses of Ideal Filters
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Impulse Responses of Ideal Filters
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Impulse Response and Causality
• All the impulse responses of ideal filterscontain sinc functions, alone or incombinations, which are infinite in extent
• Therefore all ideal filter impulse responsesbegin before time, t = 0
• This makes ideal filters non-causal
• Ideal filters cannot be physically realized, butthey can be closely approximated
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Examples of Impulse Responsesand Frequency Responses of Real
Causal Filters
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Examples of Impulse Responsesand Frequency Responses of Real
Causal Filters
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Examples of Causal Filter Effectson Signals
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Examples of Causal Filter Effectson Signals
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Examples of Causal Filter Effectson Signals
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Examples of Causal Filter Effectson Signals
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Two-Dimensional Filtering of Images
Causal LowpassFiltering
of Rows in an Image
Causal LowpassFiltering
of Columns in an Image
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Two-Dimensional Filtering of Images
“Non-Causal” LowpassFiltering
of Rows in an Image
“Non-Causal” LowpassFiltering
of Columns in an Image
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Two-Dimensional Filtering of Images
Causal LowpassFiltering
of Rows andColumns in
an Image
“Non-Causal” LowpassFiltering
of Rows andColumns in
an Image
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The Power Spectrum
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Noise RemovalA very common use of filters is to remove noise from a signal. Ifthe noise bandwidth is much greater than the signal bandwidth a large improvement in signal fidelity is possible.
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Practical Passive Filters
HVV
ZZ Z
jj
j
j
j j j RC
out
in
c
c R
ω ωω
ωω ω ω
( ) = ( )( )
= ( )( ) + ( ) =
+1
1
RC Lowpass Filter
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Practical Passive Filters
HVV
fff
jf
RC
j f jf
RC LC
out
in
( ) = ( )( ) =
( ) + +
2
22 12
π
π π
RLC Bandpass Filter
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Log-Magnitude Frequency-Response Plots
Consider the two (different) transfer functions,
H H1 2 2 2
12 1
3030 4 62
fj f
ff j f
( ) =+
( ) =− +π π π
and
When plotted on this scale, these magnitude frequency response plots are indistinguishable.
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Log-Magnitude Frequency-Response Plots
When the magnitude frequency responses are plotted ona logarithmic scale the difference is visible.
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Bode Diagrams
A Bode diagram is a plot of a frequency response in decibelsversus frequency on a logarithmic scale.
The Bel (B) is the common (base 10) logarithm of a power ratio and a decibel (dB) is one-tenth of a Bel.
The Bel is named in honor of Alexander Graham Bell.
A signal ratio, expressed in decibels, is 20 times the commonlogarithm of the signal ratio because signal power isproportional to the square of the signal.
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Bode Diagrams
H H1 2 2 2
12 1
3030 4 62
fj f
ff j f
( ) =+
( ) =− +π π π
and
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Bode DiagramsContinuous-time LTI systems are described by equationsof the general form,
addt
t bddt
tk
k
kk
D
k
k
kk
N
y x( ) = ( )= =∑ ∑
0 0
Fourier transforming, the transfer function is of the generalform,
HYX
jjj
b j
a j
kk
k
N
kk
k
Dω ωω
ω
ω( ) = ( )
( ) =( )
( )=
=
∑
∑0
0
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Bode Diagrams
A transfer function can be written in the form,
H j A
jz
jz
jz
jp
jp
jp
N
D
ω
ω ω ω
ω ω ω( ) =
−
−
−
−
−
−
1 1 1
1 1 1
1 2
1 2
L
L
where the “z’s” are the values of jω (not ω) at which the transfer
function goes to zero and the “p’s” are the values of jω at
which the transfer function goes to infinity. These z’s and p’s are commonly referred to as the “zeros” and “poles” of the system.
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Bode DiagramsFrom the factored form of the transfer function a system canbe conceived as the cascade of simple systems, each of whichhas only one numerator factor or one denominator factor. Sincethe Bode diagram is logarithmic, multiplied transfer functionsadd when expressed in dB.
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Bode Diagrams
System Bode diagrams are formed by adding the Bode diagrams of the simple systems which are in cascade. Each simple-system diagram is called a component diagram.
One Real Pole
H jjpk
ω ω( ) =−
1
1
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Bode Diagrams
One real zero
H jjzk
ω ω( ) = −1
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Bode Diagrams
Integrator(Pole at zero)
H jj
ωω
( ) = 1
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Bode Diagrams
Differentiator(Zero at zero)
H j jω ω( ) =
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Bode Diagrams
Frequency-IndependentGain
H j Aω( ) =
(This phase plot is for A > 0. IfA < 0, the phase would be a constantπ or - π radians.)
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Bode DiagramsComplex Pole Pair
HRe
jjp
jp
jp
p
j
p
ωω ω ω ω
( ) =−
−
=− ( ) + ( )
1
1 1
1
12
1 2
1
1
2
2
1
2
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Bode DiagramsComplex Zero Pair H
Rej
jz
jz
jz
z
j
zω ω ω ω ω( ) = −
−
= − ( ) + ( )1 1 1
2
1 2
1
1
2
2
1
2
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Practical Active Filters
The ideal operational amplifier has infinite input impedance, zero output impedance, infinite gain and infinite bandwidth.
HVV
Z
Zf
ff
f
fo
i
f
i
( ) = ( )( ) = −
( )( ) H
Z Z
Zf
f f
ff i
i
( ) =( )+ ( )
( )
Operational Amplifiers
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Practical Active FiltersActive Integrator
VV
integralof V
oi
f
fRC
fj f
i
( ) = − ( )
( )
12π123
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Practical Active FiltersActive RC Lowpass Filter
VV
0 12 1
ff
R
R j fCRi
f
s f
( )( ) = −
+π
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Practical Active Filters
An integrator with feedback is a lowpass filter.
H jj
ωω
( ) =+
11
′( )+ ( ) = ( )y y xt t t
Lowpass Filter
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Practical Active Filters
Highpass Filter
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Discrete-Time FiltersDT Lowpass Filter
H Fe j F
( ) =− −
1
145
2π
h un nn
[ ] =
[ ]4
5
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Discrete-Time FiltersComparison of a DT lowpass filter impulse response withan RC passive lowpass filter impulse response
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Discrete-Time FiltersDT Lowpass Filter Frequency Response
RC Lowpass Filter Frequency Response
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Discrete-Time FiltersMoving-Average Filter
h nn n n n N
N[ ] = [ ] + −[ ] + −[ ] + + −[ ]
+δ δ δ δ1 2
1L
H drcl ,F e F Nj NF( ) = +( )− π 1
Always Stable
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Discrete-Time Filters
Ideal DT LowpassFilter Impulse Response
Almost-Ideal DT LowpassFilter Impulse Response
Almost-Ideal DT LowpassFilter Magnitude Frequency
Response
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Discrete-Time Filters
Almost-Ideal DT LowpassFilter Magnitude Frequency
Response in dB
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Advantages of Discrete-Time Filters
• They are almost insensitive to environmentaleffects
• CT filters at low frequencies may require very largecomponents, DT filters do not
• DT filters are often programmable making themeasy to modify
• DT signals can be stored indefinitely on magneticmedia, stored CT signals degrade over time
• DT filters can handle multiple signals bymultiplexing them
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Communication Systems
A naive, absurd communication system
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Communication SystemsA better communication system using electromagnetic wavesto carry information
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Communication Systems
Problems
Antenna inefficiency at audio frequencies
All transmissions from all transmitters are in the same bandwidth, thereby interfering with each other
Solution Frequency multiplexing using modulation
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Communication SystemsDouble-Sideband Suppressed-Carrier (DSBSC) Modulation
y x cost t f tc( ) = ( ) ( )2π
Modulator
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Communication Systems
Y Xf f f f f fc c( ) = ( )∗ −( ) + +( )[ ]12
δ δ
Frequency multiplexing is using a different carrier frequency, , for each transmitter.fc
Modulator
Double-Sideband Suppressed-Carrier (DSBSC) Modulation
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Communication Systems
Typical received signal by an antenna
Synchronous Demodulation
Double-Sideband Suppressed-Carrier (DSBSC) Modulation
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Communication SystemsDouble-Sideband Transmitted-Carrier (DSBTC) Modulation
y x cost m t A f tc c( ) = + ( )[ ] ( )1 2π
Modulator
m = 1
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Communication Systems
CarrierCarrier
Modulator
Double-Sideband Transmitted-Carrier (DSBTC) Modulation
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Communication Systems
Envelope Detector
Double-Sideband Transmitted-Carrier (DSBTC) Modulation
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Communication Systems
Double-Sideband Transmitted-Carrier (DSBTC) Modulation
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Communication SystemsSingle-Sideband Suppressed-Carrier (SSBSC) Modulation
Modulator
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Communication SystemsSingle-Sideband Suppressed-Carrier (SSBSC) Modulation
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Communication SystemsQuadrature Carrier Modulation
Modulator Demodulator
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Phase and Group Delay
• Through the time- shifting property of the Fouriertransform, a linear phase shift as a function offrequency corresponds to simple delay
• Most real system transfer functions have a non-linear phase shift as a function of frequency
• Non-linear phase shift delays some frequencycomponents more than others
• This leads to the concepts of phase delay and groupdelay
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Phase and Group DelayTo illustrate phase and group delay let a system be excited by
x cos cost A t tm c( ) = ( ) ( )ω ω
X jA c m c m
c m c m
ω π δ ω ω ω δ ω ω ωδ ω ω ω δ ω ω ω
( ) =− −( ) + − +( )
+ + −( ) + + +( )
2
an amplitude-modulated carrier. To keep the analysis simple suppose that the system has a transfer function whose magnitude is the constant, 1, over the frequency range,
ω ω ω ω ωc m c m− < < +
and whose phase is φ ω( )
Modulation Carrier
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Phase and Group Delay
The system response is
Y jA
ec m c m
c m c m
jω π δ ω ω ω δ ω ω ωδ ω ω ω δ ω ω ω
φ ω( ) =− −( ) + − +( )
+ + −( ) + + +( )
( )
2
After some considerable algebra, the time-domain responsecan be written as
y cos cost A t tcc m c m
cm
c m c m
m
( ) = ++( ) + −( )
++( ) − −( )
ω φ ω ω φ ω ωω
ω φ ω ω φ ω ωω2 2
Carrier Modulation
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Phase and Group Delay
In this expression it is apparent that the carrier is shifted in time by
φ ω ω φ ω ωω
c m c m
c
+( ) + −( )2
and the modulation is shifted in time by
φ ω ω φ ω ωω
c m c m
m
+( ) − −( )2
y cos cost A t tcc m c m
cm
c m c m
m
( ) = ++( ) + −( )
++( ) − −( )
ω φ ω ω φ ω ωω
ω φ ω ω φ ω ωω2 2
Carrier Modulation
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Phase and Group DelayIf the phase function is a linear function of frequency,
the two delays are the same, -K. If the phase function is the non-linear function,
φ ω ω( ) = −K
φ ω ωω
( ) = −
−tan 1 2c
which is typical of a single-pole lowpass filter, with
ω ωc m=10
the carrier delay is 1 107.
ωc
and the modulation delay is 0 4.ωc
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Phase and Group Delay
On this scale the delays are difficult to see.
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Phase and Group Delay
In this magnified view the difference between carrier delay and modulation delay is visible. The delay of the carrier is phase delay and the delay of the modulation is group delay.
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Phase and Group Delay
φ ω ω φ ω ωω
c m c m
c
+( ) + −( )2
φ ω ω φ ω ωω
c m c m
m
+( ) − −( )2
The expression for modulation delay,
approachesddf
c
φ ωω ω
( )( )
=
as the modulation frequency approaches zero. In that samelimit the expression for carrier delay,
approaches φ ω
ωc
c
( )
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Phase and Group Delay
τ ωω
φ ω( ) = − ( )( )dd
When the modulationtime shift is negative, the group delay is positive.
Carrier time shift is proportional to phase shift at any frequency and modulation time shift is proportional to the derivative with respect to frequency of the phase shift. Group delay is defined as
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Pulse Amplitude ModulationPulse amplitude modulation is like DSBSC modulationexcept that the “carrier” is a rectangular pulse train,
p rect combttw T
tTs s
( ) =
∗
1
Modulator
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Pulse Amplitude Modulation
The response of the pulse modulator is
y x p x rect combt t t ttw T
tTs s
( ) = ( ) ( ) = ( )
∗
1
and its CTFT is
Y sinc Xf wf wkf f kfs s sk
( ) = ( ) −( )=−∞
∞
∑
where fTs
s
= 1
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Pulse Amplitude Modulation
The CTFT of the response is basically multiple replicasof the CTFT of the excitation with different amplitudes,spaced apart bythe pulse repetition rate.
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Discrete-Time ModulationDiscrete-timemodulation isanalogous tocontinuous-timemodulation. Amodulating signalmultiplies a carrier.Let the carrier be
If the modulation isx[n], the response is
c cosn F n[ ] = ( )2 0π
y x cosn n F n[ ] = [ ] ( )2 0π
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Discrete-Time Modulation
Y X C
X X
F F F
F F F F
( ) = ( ) ( )
= −( ) + +( )[ ]12 0 0
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Spectral AnalysisThe heart of a “swept-frequency” type spectrum analyzeris a multiplier, like the one introduced in DSBSC modulation, plus a lowpass filter.
Multiplying by the cosine shifts the spectrum of x(t) by and the signal power shifted into the passband of the lowpass filter is measured. Then, as the frequency, , is slowly “swept” over a range of frequencies, the spectrumanalyzer measures its signal power versus frequency.
fc
fc
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Spectral AnalysisOne benefit of spectral analysis is illustrated below.
These two signals are different but exactly how they aredifferent is difficult to see by just looking at them.
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Spectral AnalysisThe magnitude spectra of the two signals reveal immediately what the difference is. The second signal contains a sinusoid, or something close to a sinusoid, that causes the two large “spikes”.