Fourier Transform Analysis of
Signals and Systems
Ideal Filters
• Filters separate what is desired from what is
not desired
• In the signals and systems context a filter
separates signals in one frequency range from
signals in another frequency range
• An ideal filter passes all signal power in its
passband without distortion and completely
blocks 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
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Distortion
3
Distortion
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(t ) = Aδ (t − t 0 )
or
h[n ] = Aδ [ n − n0 ]
The corresponding
transfer functions are
H( f ) = Ae − j 2πft0
or
H( F ) = Ae − j 2πFn0
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Filter Classifications
There 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 of
frequencies”
• This range could be the range of frequencies
a filter passes or the range of frequencies
present in a signal
• Bandwidth is traditionally construed to be
range of frequencies in positive frequency
space
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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 filters
contain sinc functions, alone or in
combinations, which are infinite in extent
• Therefore all ideal filter impulse responses
begin before time, t = 0
• This makes ideal filters non-causal
• Ideal filters cannot be physically realized, but
they can be closely approximated
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Examples of Impulse Responses
and Frequency Responses of Real
Causal Filters
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Examples of Impulse Responses
and Frequency Responses of Real
Causal Filters
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Examples of Causal Filter Effects
on Signals
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Examples of Causal Filter Effects
on Signals
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Examples of Causal Filter Effects
on Signals
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Examples of Causal Filter Effects
on Signals
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Two-Dimensional Filtering of Images
Causal Lowpass
Filtering
of Rows in
an Image
Causal Lowpass
Filtering
of Columns in
an Image
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Two-Dimensional Filtering of Images
“Non-Causal”
Lowpass
Filtering
of Rows in
an Image
“Non-Causal”
Lowpass
Filtering
of Columns in
an Image
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Two-Dimensional Filtering of Images
Causal
Lowpass
Filtering
of Rows and
Columns in
an Image
“Non-Causal”
Lowpass
Filtering
of Rows and
Columns in
an Image
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The Power Spectrum
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Noise Removal
A very common use of filters is to remove noise from a signal. If
the noise bandwidth is much greater than the signal bandwidth a
large improvement in signal fidelity is possible.
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Practical Passive Filters
Vout ( jω )
H( jω ) =
Vin ( jω )
RC Lowpass Filter
Z c ( jω )
1
=
=
Z c ( jω ) + Z R ( jω ) jωRC + 1
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Practical Passive Filters
RLC Bandpass Filter
2πf
Vout ( f )
RC
H( f ) =
=
Vin ( f ) ( j 2πf )2 + j 2πf + 1
RC LC
j
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Log-Magnitude FrequencyResponse Plots
Consider the two (different) transfer functions,
1
30
H1 ( f ) =
and H 2 ( f ) =
j 2πf + 1
30 − 4π 2 f 2 + j 62πf
When plotted on this scale, these magnitude frequency response
plots are indistinguishable.
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Log-Magnitude FrequencyResponse Plots
When the magnitude frequency responses are plotted on
a logarithmic scale the difference is visible.
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Bode Diagrams
A Bode diagram is a plot of a frequency response in decibels
versus 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 common
logarithm of the signal ratio because signal power is
proportional to the square of the signal.
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Bode Diagrams
1
30
H1 ( f ) =
and H 2 ( f ) =
j 2πf + 1
30 − 4π 2 f 2 + j 62πf
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Bode Diagrams
Continuous-time LTI systems are described by equations
of the general form,
N
dk
dk
ak k y(t ) = ∑ bk k x(t )
∑
dt
dt
k =0
k =0
D
Fourier transforming, the transfer function is of the general
form,
N
Y( jω )
H( jω ) =
=
X( jω )
∑ bk ( jω )
k
k =0
D
∑ ak ( jω )
k
k =0
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Bode Diagrams
A transfer function can be written in the form,

1 −

H( jω ) = A

1 −

jω  
 1 −
z1  
jω  
 1 −
p1  
jω  
 L 1 −
z2  
jω  
 L 1 −
p2  
jω 

zN 
jω 

pD 
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 Diagrams
From the factored form of the transfer function a system can
be conceived as the cascade of simple systems, each of which
has only one numerator factor or one denominator factor. Since
the Bode diagram is logarithmic, multiplied transfer functions
add 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
1
H( jω ) =
jω
1−
pk
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Bode Diagrams
One real zero
jω
H( jω ) = 1 −
zk
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Bode Diagrams
Integrator
(Pole at zero)
1
H( jω ) =
jω
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Bode Diagrams
Differentiator
(Zero at zero)
H( jω ) = jω
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Bode Diagrams
Frequency-Independent
Gain
H( jω ) = A
(This phase plot is for A > 0. If
A < 0, the phase would be a constant
π or - π radians.)
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Bode Diagrams
Complex
Pole Pair
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1
1
=
H( jω ) =

2 Re( p1 ) ( jω )2
jω  
jω 
+
 1 −   1 −  1 − jω
2
2

p1  
p2 
p1
p1
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Bode Diagrams
Complex
Zero Pair
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2 Re( z1 ) ( jω )

jω  
jω 
H( jω ) =  1 −   1 −  = 1 − jω
+
2
2

z1  
z2 
z1
z1
2
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Practical Active Filters
Operational Amplifiers
The ideal operational amplifier has infinite input impedance,
zero output impedance, infinite gain and infinite bandwidth.
Zf( f )
Vo ( f )
H( f ) =
=−
Vi ( f )
Zi ( f )
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Z f ( f ) + Zi ( f )
H( f ) =
Zi ( f )
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Practical Active Filters
Active Integrator
Vo ( f ) = −
1 Vi ( f )
2π3
RC 1
j2
f
integral
of Vi ( f )
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Practical Active Filters
Active RC Lowpass Filter
Rf
V0 ( f )
1
=−
Vi ( f )
Rs j 2πfCR f + 1
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Practical Active Filters
Lowpass Filter
An integrator with feedback is a lowpass filter.
y′(t ) + y(t ) = x(t )
H( jω ) =
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Practical Active Filters
Highpass Filter
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Discrete-Time Filters
DT Lowpass Filter
H( F ) =
1
4
1 − e − j 2πF
5
n
 4
h[n ] =
u[n ]
 5
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Discrete-Time Filters
Comparison of a DT lowpass filter impulse response with
an RC passive lowpass filter impulse response
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Discrete-Time Filters
DT Lowpass Filter
Frequency Response
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RC Lowpass Filter
Frequency Response
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Discrete-Time Filters
Moving-Average Filter
H( F ) = e − jπNF drcl( F , N + 1)
h[n ] =
δ [n ] + δ [n − 1] + δ [n − 2 ] + L + δ [n − N ]
N +1
Always Stable
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Discrete-Time Filters
Ideal DT Lowpass
Filter Impulse Response
Almost-Ideal DT Lowpass
Filter Impulse Response
Almost-Ideal DT Lowpass
Filter Magnitude Frequency
Response
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Discrete-Time Filters
Almost-Ideal DT Lowpass
Filter Magnitude Frequency
Response in dB
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Advantages of Discrete-Time Filters
• They are almost insensitive to environmental
effects
• CT filters at low frequencies may require very large
components, DT filters do not
• DT filters are often programmable making them
easy to modify
• DT signals can be stored indefinitely on magnetic
media, stored CT signals degrade over time
• DT filters can handle multiple signals by
multiplexing them
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Communication Systems
A naive, absurd communication system
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Communication Systems
A better communication system using electromagnetic waves
to 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
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Frequency multiplexing using modulation
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Communication Systems
Double-Sideband Suppressed-Carrier (DSBSC) Modulation
y(t ) = x(t ) cos( 2πfct )
Modulator
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Communication Systems
Double-Sideband Suppressed-Carrier (DSBSC) Modulation
1
Y( f ) = X( f ) ∗ [δ ( f − fc ) + δ ( f + fc )]
2
Modulator
Frequency multiplexing is using a different carrier frequency,
fc , for each transmitter.
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Communication Systems
Double-Sideband Suppressed-Carrier (DSBSC) Modulation
Typical received signal by an antenna
Synchronous Demodulation
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Communication Systems
Double-Sideband Transmitted-Carrier (DSBTC) Modulation
y(t ) = [1 + m x(t )]Ac cos( 2πfct )
Modulator
m=1
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Communication Systems
Double-Sideband Transmitted-Carrier (DSBTC) Modulation
Modulator
Carrier
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Carrier
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Communication Systems
Double-Sideband Transmitted-Carrier (DSBTC) Modulation
Envelope Detector
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Communication Systems
Double-Sideband Transmitted-Carrier (DSBTC) Modulation
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Communication Systems
Single-Sideband Suppressed-Carrier (SSBSC) Modulation
Modulator
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Communication Systems
Single-Sideband Suppressed-Carrier (SSBSC) Modulation
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Communication Systems
Quadrature Carrier Modulation
Modulator
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Demodulator
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Phase and Group Delay
• Through the time- shifting property of the Fourier
transform, a linear phase shift as a function of
frequency corresponds to simple delay
• Most real system transfer functions have a nonlinear phase shift as a function of frequency
• Non-linear phase shift delays some frequency
components more than others
• This leads to the concepts of phase delay and group
delay
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Phase and Group Delay
To illustrate phase and group delay let a system be excited
by
x(t ) = A cos(ω m t ) cos(ω ct )
Modulation
Carrier
Aπ δ (ω − ω c − ω m ) + δ (ω − ω c + ω m ) 
X( jω ) =


2 +δ (ω + ω c − ω m ) + δ (ω + ω c + ω m )
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,
ωc − ωm < ω < ωc + ωm
and whose phase is
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φ(ω )
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Phase and Group Delay
The system response is
Aπ δ (ω − ω c − ω m ) + δ (ω − ω c + ω m )  jφ ( ω )
Y( jω ) =

e
2 +δ (ω + ω c − ω m ) + δ (ω + ω c + ω m )
After some considerable algebra, the time-domain response
can be written as
  φ (ω c + ω m ) + φ (ω c − ω m ) 
  φ (ω c + ω m ) − φ (ω c − ω m ) 
y(t ) = A cos ω c  t +
cos


ω m  t +


2ω c
2ω m
 
 
Carrier
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Modulation
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Phase and Group Delay
  φ (ω c + ω m ) + φ (ω c − ω m ) 
  φ (ω c + ω m ) − φ (ω c − ω m ) 
y(t ) = A cos ω c  t +
cos


ω m  t +




2
2
ω
ω



m
c
Carrier
Modulation
In this expression it is apparent that the carrier is shifted in time by
φ (ω c + ω m ) + φ (ω c − ω m )
2ω c
and the modulation is shifted in time by
φ (ω c + ω m ) − φ (ω c − ω m )
2ω m
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Phase and Group Delay
If the phase function is a linear function of frequency,
φ(ω ) = −Kω
the two delays are the same, -K. If the phase function is the
non-linear function,
ω
−1 
φ(ω ) = − tan  2 
 ωc 
which is typical of a single-pole lowpass filter, with
ω c = 10ω m
1.107
the carrier delay is
and the modulation delay is
ωc
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0.4
ωc
68
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
The expression for modulation delay,
approaches
φ (ω c + ω m ) − φ (ω c − ω m )
2ω m
d

 df (φ(ω ))

ω =ω c
as the modulation frequency approaches zero. In that same
limit the expression for carrier delay,
φ (ω c + ω m ) + φ (ω c − ω m )
2ω c
approaches
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φ (ω c )
ωc
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Phase and Group Delay
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
d
τ (ω ) = −
(φ(ω ))
dω
When the modulation
time shift is negative,
the group delay is
positive.
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Pulse Amplitude Modulation
Pulse amplitude modulation is like DSBSC modulation
except that the “carrier” is a rectangular pulse train,
Modulator
 t
t 1
p(t ) = rect
∗ comb 
 w  Ts
 Ts 
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Pulse Amplitude Modulation
The response of the pulse modulator is

 t 
t 1
y(t ) = x(t ) p(t ) = x(t )rect
∗ comb  


w Ts
 Ts  

and its CTFT is
Y( f ) = wfs
∞
∑ sinc(wkf ) X( f − kf )
s
s
k =−∞
1
where fs =
Ts
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Pulse Amplitude Modulation
The CTFT of the
response is basically
multiple replicas
of the CTFT of the
excitation with
different amplitudes,
spaced apart by
the pulse repetition
rate.
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Discrete-Time Modulation
Discrete-time
modulation is
analogous to
continuous-time
modulation. A
modulating signal
multiplies a carrier.
Let the carrier be
c[n ] = cos( 2πF0 n )
If the modulation is
x[n], the response is
y[n ] = x[n ] cos( 2πF0 n )
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Discrete-Time Modulation
Y( F ) = X( F ) C( F )
1
= [ X( F − F0 ) + X( F + F0 )]
2
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Spectral Analysis
The heart of a “swept-frequency” type spectrum analyzer
is a multiplier, like the one introduced in DSBSC modulation,
plus a lowpass filter.
Multiplying by the cosine shifts the spectrum of x(t) by
fc and the signal power shifted into the passband of the
lowpass filter is measured. Then, as the frequency, fc , is
slowly “swept” over a range of frequencies, the spectrum
analyzer measures its signal power versus frequency.
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Spectral Analysis
One benefit of spectral analysis is illustrated below.
These two signals are different but exactly how they are
different is difficult to see by just looking at them.
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Spectral Analysis
The 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”.
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