Filtri Analogici Generalit

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Filtri Analogici - Generalità

• Elaborazione Numerica dei Segnali
• Prof. Pasquale Daponte

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Introduzione
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Fourier Analysis
All real-world signals are sums of sinusoidal
components having various frequencies,
amplitudes, and phases.
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Filters
Filters process the sinusoid components of an
input signal differently depending of the
frequency of each component. Often, the goal of
the filter is to retain the components in certain
frequency ranges and to reject components in
other ranges.
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What is a filter?
Simply put, it retains history of the input
signal. Take this simple lowpass filter for an
example.
Vout
Vin
This simple filter is a kind of integrator, in
which the capacitor integrates the charge
provided through the resistor. As the rate of
change of Vout depends on the current through the
resistor, low frequencies will be filtered less
than high frequencies.
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In other words

• The output of a filter is a function not only of
  the input at the present time, but also of
  previous events.
• Thats what a linear filter does. No more, no
  less. We are sticking to linear filters today,
  thank you!
• There are many ways to build filters.

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How does the analog filter exhibit memory?

• In the case shown, the capacitor is the memory
  element.
• It retains previous history.
• It does so by summing the history into one value,
  the voltage across the capacitor.
• In such a way, a single component can have a long
  memory.

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Getting back to the analog filter

• The analog filters output can not change
  instantly as that would require infinite current
  in the resistor, that means that the output
  depends on the HISTORY of the signal.
• In fact, if you have an input signal, and you
  integrate the product of the time-reversed
  impulse response times the signal, you get the
  filters output.

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DID YOU SAY INTEGRATE?

• Well, it can be a sum, rather than an integral,
  and in fact in the digital domain, it is a sum,
  rather than an integral.
• In the digital domain, rather than having all
  frequencies, you have all frequencies inside the
  digital passband, which will be ½ the sample rate
  wide.

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So, what did that filter actually do?

• It stored part of the history of the signal. As
  the time went on, it forgot, exponentially, the
  contributions to its output from previous
  history.
• Not all impulse responses are so simple.
• An impulse response may be positive or negative
• An impulse response may ring or not.

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Filters
Background
. Filters may be classified as either digital
or analog.
. Digital filters are implemented using a
digital computer or special purpose digital
hardware.
. Analog filters may be classified as either
passive or active and are usually
implemented with R, L, and C components and
operational amplifiers.
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Analog vs. Digital Filters

• Digital
• Very complex filters
• Full adjustability
• Precision vs. cost
• Arbitrary magnitude
• Total linear phase
• EMI magnetic noise immunity
• Stability (temp time)
• Repeatability

• Analog
• Speed 10-100x faster
• Dynamic Range
• Amplitude 140 dB
• e.g., 12 Vrms 1 ?V noise
• Frequency 8 decades
• e.g., 0.01 Hz to 1 MHz
• Cheap, small, low power
• Precision limited by noise component tolerances

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Analogue filters

• Still needed in the world of DSP
• Also, many digital filter designs are based on
  analog filters.
• They are linear time-invariant (LTI)

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Definition of linearity

• System is LINEAR if
• for any signal x(t),
• if x(t) ? y(t) then a.x(t ) ?
  a.y(t) for any constant a.
• (2) for any signals x1(t) x2(t),
• if x1(t) ? y1(t) x2(t) ? y2(t)
• then x1(t)
  x2(t) ? y1(t) y2(t)
• (By x(t) ? y(t) we mean that applying x(t) to
  the input produces the output signal y(t). )

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Alternative definition of linearity

• System is linear if
• for any signals x1(t) x2(t), if x1(t) ? y1(t)
  x2(t) ? y2(t)
• then a1x1(t) a2 x2(t) ? a1y1(t) a2 y2(t) for
  any a1 a2

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Linearity (illustration)
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• Definition of time-invariance
• A time-invariant system must satisfy
• For any x(t), if x(t) ? y(t) then
  x(t-?) ? y1(t-?) for any ?
• Delaying input by ? seconds delays output by ?
  seconds
• Not all systems have this property.
• An LTI system is linear time invariant.
• An analogue filter is LTI.

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Analog Filters
Background
. An active filter is one that, along with R, L,
and C components, also contains an energy
source, such as that derived from an
operational amplifier.
. A passive filter is one that contains only
R, L, and C components. It is not
necessary that all three be present. L is
often omitted (on purpose) from passive
filter design because of the size and cost
of inductors and they also carry along an R
that must be included in the design.
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Se invece nel circuito è presente un
amplificatore, è possibile che il segnale di
uscita sia in rapporto di tensione o potenza gt1
con quello di ingresso in questo caso si ha un
filtro attivo.
Un circuito contenente componenti RLC, ma privo
di elementi attivi, dà luogo ad un segnale di
uscita di potenza inferiore a quella del segnale
di ingresso in questo caso si hanno filtri
passivi.
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Active vs. Passive

• Passive
• Less noise
• No power supply
• More reliable
• Less EMI susceptible
• Better at RF frequency
• No oscillations
• No on/off transients
• No hard clipping
• Handles large V I

• Active
• Gain adjustable
• No loading effects
• Parameters adjustable
• Smaller Cs
• No inductors
• Smaller, lighter cheaper
• No magnetic coupling
• High Q circuits easy

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Filters
Background
. The analysis of analog filters is well
described in filter text books. The most popular
include Butterworth, Chebyshev and elliptic
methods.
. The synthesis (realization) of analog filters,
that is, the way one builds (topological layout)
the filters, received significant attention
during 1940 thru 1960. Leading the work were
Cauer and Tuttle. Since that time, very little
effort has been directed to analog filter
realization.
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Filters
Background
. Generally speaking, digital filters have
become the focus of attention in the last 40
years. The interest in digital filters
started with the advent of the digital computer,
especially the affordable PC and special
purpose signal processing boards. People
who led the way in the work (the analysis
part) were Kaiser, Gold and Radar.
. A digital filter is simply the implementation
of an equation(s) in computer software.
There are no R, L, C components as such.
However, digital filters can also be built
directly into special purpose computers in
hardware form. But the execution is still in
software.
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Filtri analogici

• In funzione dellintervallo di frequenze del
  segnale che il filtro nominalmente non modifica
  (la cosiddetta banda passante del filtro), si
  distingue allora tra filtro passa basso,
  filtro passa altoe filtro passa banda.

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Filter types - low-pass
Low-pass with ?C 1
Low-pass
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Filter types - high-pass band-pass
G(?) 1
High-pass
?
?C
G(?) 1
Band-pass
?
?L
?U
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Filter types - band-stop
G(?) 1
?
?L
?U
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Two types of band-pass gain-responses
G(?) 1
Narrow-band (?U lt 2 ?L )
?
?L
?U
G(?) 1
Broad-band (?U gt 2 ?L )
?
?L
?U
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Three types of band-stop gain-responses
G(?) 1
Narrow-band (?U lt 2 ?L )
?
?L
?U
G(?) 1
Broad-band (?U gt 2 ?L )
?
?L
?U
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Third type of band-stop gain-response
G(?) 1
Notch
?
?N
Yet another type of gain-response
G(?) 1
All-pass
?
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DECIBELS
Transfer-Function Magnitudes and Their Decibel
Equivalents
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Logarithmic Frequency Scales
On a logarithmic scale, the variable is
multiplied by a given factor for equal increments
of length along the axis.
A decade is a range of frequencies for which the
ratio of the highest frequency to the lowest is
10.
An octave is a two-to-one change in frequency.
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PRESTAZIONI DEI FILTRI NEL TEMPO
Risposta al gradino Tempo di salita tr
Tempo di settling ts Sovraelongazione
(overshoot) Oscillazioni (ringing)
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PRESTAZIONE DEI FILTRI IN FREQUENZA
RISPOSTA IN AMPIEZZA DI UN PASSA - BASSO H ( f )
A( f )
20log10A gain in dB fc cut off
trequency GAIN AT 3dB point (at fc) A /
rad(2)
Un filtro ideale dovrebbe essere caratterizzato
da una funzione di trasferimento unitaria
allinterno della banda passante, mentre la
funzione di trasferimento dovrebbe essere
identicamente nulla allesterno della banda
passante. In altre parole il filtro non dovrebbe
attenuare le frequenze desiderate, mentre
lattenuazione dovrebbe essere infinita per
quelle indesiderate. E intuibile che una
interazione di questo tipo è impossibile per un
circuito elettronico reale (necessità di
approssimazione).
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Filter Terminology

• Corner Frequency 3 dB point half power point
• Center Frequency (any 2nd-order BP)
• fC ?fHfL
  i.e., geometric mean, where fL fH
  3 dB pts
• Q Selectivity Factor reciprocal of BW Q
  fC / fH fL fC / BW
• Group Delay rate of change of phase shift with
  respect to time, i.e., 1st derivative

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The Time and Frequency Response of the Analog
Filter
Time (impulse) Response
Frequency Response
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Ok, whats the impulse response

• Impulse response is the response of the circuit
  to a (mathematical) signal of infinite height
  (and power), and infinitely short duration.
  This unit impulse has very special
  characteristics
• It contains all frequencies
• It has energy of 1 at all frequencies.
• It describes the behavior of the filter at all
  frequencies. Completely.
• POINT 3 is what you need to remember. The impulse
  response of a filter is a complete description of
  what it does.

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Another way to look at the impulse response.

• The impulse response of a system shows how a
  filter captures the HISTORY of the signal. In
  other words
• The value of the impulse response at a time t
  demonstrates how much of the HISTORY of the
  signal is added to the output at time t later.

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FILTRI ANALOGICI
Transfer functions
Poles and zeros
?
Nel seguito esaminiamo in dettaglio queste
caratteristiche .
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Simplified Laplace Transforms

• Represents complex (frequency dependent)
  impedance, i.e., magnitude phase
• Uses the Laplace Operator, s, where
• s complex frequency variable j? j2pf
• Resistor Impedance R (freq. independent)
• Capacitor Reactance 1/sC
• Inductor Reactance sL
• Allows writing a circuits transfer function by
  summing circuit currents using Kirchoffs Law

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Transfer Functions (TF)

• Transfer functions mathematically describe the
  frequency domain behavior of filters.
• TF ratio of Laplace Transforms of a circuits
  input and output voltages
• T(s) Vout(s) / Vin(s)

The magnitude of the transfer function shows how
the amplitude of each frequency component is
affected by the filter. Similarly, the phase of
the transfer function shows how the phase of each
frequency component is affected by the filter.
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System function for analogue LTI circuits

• An analog LTI system has a system (or transfer)
  function

a0 a1s a2s2 ...
aNsN H(s) ??????????? b0
b1s b2s2 ... bMsM

• Coeffs a0, a1, ...,aN, b0, ..., bM determine its
  behaviour.
• Designer of analog lowpass filters must choose
  these carefully.
• H(s) may be evaluated for complex values of s.
• Setting s j? where ? 2?f gives a complex
  function of f.
• Modulus H(j?) is gain at ? radians/second
  (?/2? Hz)
• Argument of H(j?) is phase-lead at ? radians/s.

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Filter Transfer Functions - Poles Zeros

• General filter transfer function is the ratio of
  two polynomials

• Zeros values that make numerator equal zero,
  i.e., the roots of the numerator.
• Makes amplitude response rolloff 6 dB/oct.
• Shifts phase 90/zero (45 _at_ fc)
• Poles values that make denominator equal
  zero, i.e., the roots of the denominator.
• Makes amplitude response rise 6 dB/oct.
• Shifts phase 90/zero (45 _at_ fc)

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Filter Order

• The order or degree (equivalent terms) is the
  highest power of s in the transfer function.
• For analog circuits usually equals the number of
  capacitors (or inductors) in the circuit.
• 2nd-order most common.
• For common audio filters the order equals the
  rolloff rate divided by 6dB/oct, e.g. 24 dB/oct
  rolloff 4th order (24 /6 4)

Rule 6 dB/oct 90 per order Examples1st-order
6 dB/oct ? 90 (? 45 _at_ fc) 2nd-order
12 dB/oct ? 180 (? 90 _at_ fc) 3rd-order
18 dB/oct ? 270 (?135 _at_ fc) 4th-order 24
dB/oct ? 360 (?180 _at_ fc) etc.
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Why 6 dB/octave Slope?

• The impedance of a capacitor is half with twice
  the frequency, i.e., XC 1/sC 1/2?fC
• The impedance of an inductor is twice when
  frequency doubles, i.e., XL sL 2?fL
• Twice or Half Impedance 6 dB change
• Twice or Half Frequency One Octave change

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Why 2nd-Order?

• Maximum phase shift is 180 degrees
• Guarantees circuit is unconditionally stable
• No oscillation problems under any conditions
• Get higher order circuits by cascading 2nd-order
  sections or
• Design 4th-order section to mathematically
  emulate two cascaded 2nd-order (Ranes L-R)

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Why 1/3-Octave Centers?

• 1/3-Octave (21/3 oct x1.26) approximately
  represents the smallest region humans reliably
  detect change.
• Relates to Critical Bands a range of frequencies
  where interaction occurs an auditory filter.
• About 1/3-octave wide above 500Hz (latest info
  says more like 1/6-oct) 100 Hz below 500 Hz

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Gain phase response graphs
Gain G(?) H(j?) Phase lead
?(?) ArgH(j?)
-?(?)
G(w)
Gain
Phase-lag
?
f / (2?)
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Why Phase Shift?

• Phase shift is the flip side of time
• It takes time to build up a charge on a capacitor
  -- thats why you cannot change the voltage on a
  capacitor instantaneously.
• It takes time to build up a magnetic field (flux)
  in an inductor -- thats why you cannot change
  the current through an inductor instantaneously.
• All this time phase shift

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RISPOSTA DI FASE
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FILTRI A FASE LINEARE

• Le frequenze del segnale in ingresso sono
  ritardate della stessa entità.

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Effect of phase-response

• It may be shown that
• when input x(t) A cos(?t),
• output y(t) A . G(?) . cos( ?t ?(?)
  )
• Output is sinusoid of same frequency as input.
• Sine-wave in ? sine-wave out
• Multiplied in amplitude by G(?)
  phase-shifted by ?(?).
• Example If G(?) 3 and ?(?) ?/2 for all ?
  what is the output?
• Answer y(t) 3.A.cos(?t ?/2)
• 3.A.sin (?t)

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Phase-shift expressed as a delay

• Express y(t) A . G(?) . cos( ?t ?(?) )
• as A . G(?) . cos (?t
  ?(?)/?)
• A . G(?) . cos(? t -
  ?(?) ) where ?(?) -?(?)/?
• Cosine wave is delayed by -?(?)/? seconds.
• -?(?)/? is phase-delay in seconds
• Easier to understand than phase-shift

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Linear phase

• If -?(?)/? is constant for all ?, all frequencies
  delayed by same time.
• Then system is linear phase - this is good.
• Avoids changes in wave-shape due to phase
  distortion
• i.e different frequencies
  being delayed by differently.
• Not all LTI systems are linear phase.

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Linear phase response graph
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Examples of Low-pass analog filters Would like
ideal brick-wall gain response
linear phase response as shown below
?C cut-off frequency
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• Butterworth low-pass gain response
• Cannot realise ideal brick-wall gain response
  nor linear phase.
• Can realise Butterworth approximation of order
  n

Properties (i) G(0) 1 ( 0
dB gain at ?0) (ii) G(?C) 1/(?2) (-3dB
gain at ? ?C)
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Examples of Butterwth low-pass gain responses

• Let ?C 100 radians/second.
• G(?C) is always 1/?(2)
• Shape gets closer to ideal brick-wall response
  as n increases.

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LINEAR-LINEAR PLOT
1
G(?)
0.9
0.8
1 / ?(2)
0.7
0.6
0.5
0.4
0.3
0.2
n 2
n7
n4
0.1
0
0
50
100
150
200
250
300
350
400
radians/second
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Butterworth gain responses on dB scale

• Plot G(?) in dB, i.e. 20 log10(G(?)), against
  ?.
• With ? on linear or log scale.
• As 20 log10(1/?(2)) -3, all curves are -3dB
  when ? ?C

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dB-LINEAR PLOT
0
dB
-10
n2
-20
-30
n4
-3dB
-40
-50
-60
n7
-70
-80
-90
0
50
100
150
200
250
300
350
400
radians/second
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dB-LOG PLOT
0
dB
-10
?3 dB
-20
-30
n2
-40
-50
-60
n4
-70
-80
0
1
2
3
10
10
10
10
radians/second
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MATLAB program to plot these graphs
clear all for w 1 400 G2(w)
1/sqrt(1(w/100)4) G4(w) 1/sqrt(1(w/100)8)
G7(w) 1/sqrt(1 (w/100)14) end
plot(1400,G2,'r',1400,G4,'b',1400,G7,'k')
grid on DG220log10(G2) DG420log10(G4)
DG720log10(G7) plot(1400,DG2,'r',1400,DG
4,'b',1400,DG7,'k') grid on
semilogx(1990, DG2,'r', 1990, DG4, 'b)

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• Cut-off rate
• Best seen on a dB-Log plot
• Cut-off rate is 20n dB per decade
• or 6n dB per octave
• at frequencies
  ? much greater than ?C.
• Decade is a multiplication of frequency by 10.
• Octave is a multiplication of frequency by 2.
• So for n4, gain drops by 80 dB if frequency is
  multiplied by 10
• or by 24 dB if
  frequency is doubled.

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Normalized Transfer Function

• Low-Pass (LP) (2 poles)

2 poles -12 dB/oct
Amplitude
Frequency
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Normalized Transfer Function

• Bandpass (BP) (1 zero, 2 poles)

1 pole -6 dB/oct
1 pole -6 dB/oct
Amplitude
1 zero 6 dB/oct
Frequency
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Normalized Transfer Function

• High-Pass (HP) (2 zeros, 2 poles)

2 poles -12 dB/oct
Amplitude
2 zeros 12 dB/oct
Frequency
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Normalized Transfer Function

• Notch
• All-Pass

Poles zeros cancel amplitude but add phase
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Coefficients determine performance

• Butterworth maximally flat passband s2
  1.414s 1
• Chebyshev steeper rolloff w/magnitude ripples
  s2 1.43s 1.51
• Bessel best step response, but gentle rolloff
  s2 3s 3

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Transfer Function
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Cascaded Two-Port Networks
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BODE PLOTS
A Bode plot shows the magnitude of a network
function in decibels versus frequency using a
logarithmic scale for frequency.

• A horizontal line at zero for f lt fB /10.
• 2. A sloping line from zero phase at fB /10 to
  90 at 10fB.
• 3. A horizontal line at 90 for f gt 10fB.

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CARATTERISTICHE DI ALCUNI TIPI DI FILTRI
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CARATTERISTICHE DI ALCUNI TIPI DI FILTRI (1)
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CARATTERISTICHE E POSSIBILI REALIZZAZIONI
CIRCUITALI
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CARATTERISTICHE E POSSIBILI REALIZZAZIONI
CIRCUITALI (1)
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ALTRI ESEMPI
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Passive Analog Filters
Background
It will be shown later that the ideal filter,
sometimes called a brickwall filter, can be
approached by making the order of the filter
higher and higher.
The order here refers to the order of
the polynomial(s) that are used to define
the filter. Matlab examples will be given
later to illustrate this.
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Filtro passivo Passa Basso

• Equazione differenziale per il sistema LTI
• La funzione di trasferimento della risposta in
  frequenza H(jw) può essere determinata
  utilizzando la proprietà degli eigen systems opp.
  Utilizzando la definizione della sua risposta
  impulsiva
• Plot di ampiezza e fase mostrati a destra
• Risposta al gradino
• Elevato RC buona selezione in frequenza
• Basso RC risposta di tempo veloce
• Inevitabile compromesso di progetto
  tempo/frequenza

RC1
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FIRST-ORDER LOWPASS FILTERS
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Passive Analog Filters
Low Pass Filter
.
0 dB
Bode
-3 dB
?
1/RC
Passes low frequencies Attenuates high frequencies
1
x
Linear Plot
0.707
?
1/RC
0
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Filtro passivo passa basso RC (primo ordine)
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Passive Analog Filters
High Pass Filter
Consider the circuit below.

C

R
Vi
VO
_
_
High Pass Filter
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FIRST-ORDER HIGHPASS FILTERS
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Passive Analog Filters
High Pass Filter
0 dB
.
-3 dB
Passes high frequencies
Bode
Attenuates low frequencies
1/RC
?
1/RC
1
.
x
0.707
Linear
?
0
1/RC
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Passive Analog Filters
Bandpass Pass Filter
Consider the circuit shown below

C
L

VO
R
Vi
_
_
When studying series resonant circuit we showed
that
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Passive Analog Filters
Bandpass Pass Filter
We can make a bandpass from the previous equation
and select the poles where we like. In a typical
case we have the following shapes.
0 dB
.
.
Bode
-3 dB
?lo
?hi
?
1
.
.
0.707
Linear
?lo
?hi
?
0
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Passive Analog Filters
Bandpass Pass Filter
Example
Suppose we use the previous series RLC circuit
with output across R to design a bandpass filter.
We will place poles at 200 rad/s and 2000
rad/s hoping that our 3 dB points will be
located there and hence have a bandwidth of 1800
rad/s. To match the RLC circuit form we use
The last term on the right can be finally put in
Bode form as
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Passive Analog Filters
Bandpass Pass Filter
Example
From this last expression we notice from the part
involving the zero we have in dB form
20log(.0055) 20logw
Evaluating at w 200, the first pole break, we
get a 0.828 dB what this means is that our 3dB
point will not be at 200 because we do not have 0
dB at 200. If we could lower the gain by 0.829
dB we would have 3dB at 200 but with the RLC
circuit we are stuck with what we have. What
this means is that the 3 dB point will be at a
lower frequency. We can calculate this from

96
Passive Analog Filters
Example
Bandpass Pass Filter
This gives an wlow 182 rad/s. A similar thing
occurs at whi where the new calculated value for
whi becomes 2200. These calculations do no take
into account a 0.1 dB that one pole induces on
the other pole. This will make wlo somewhat
lower and whi somewhat higher.
One other thing that should have given us a hint
that our w1 and w2 were not going to be correct
is the following
What is the problem with this?
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Passive Analog Filters
Bandpass Pass Filter
Example
The problem is that we have
Therein lies the problem. Obviously the above
cannot be true and that is why we have aproblem
at the 3 dB points.
We can write a Matlab program and actually check
all of this. We will expect that w1 will be lower
than 200 rad/s and w2 will be higher than 2000
rad/s.
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Passive Analog Filters
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A Bandpass Digital Filter
Perhaps going in the direction to stimulate your
interest in taking a course on filtering, a 10
order analog bandpass butterworth filter will
be simulated using Matlab. The program is given
below.
N 10 10th order butterworth analog
prototype ZB, PB, KB buttap(N) numzb
poly(ZB) denpb poly(PB)   wo 600 bw
200 wo is the center freq bw
is the bandwidth numbbs,denbbs
lp2bs(numzb,denpb,wo,bw)   w 111200   Hbbs
freqs(numbbs,denbbs,w) Hb abs(Hbbs)   plot(w
,Hb) grid xlabel('Amplitude') ylabel('frequency
(rad/sec)') title('10th order Butterworth filter')
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A Bandpass Filter
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RLC Band stop Filter
Consider the circuit below

R

L
VO
Vi
_
C
_
The transfer function for VO/Vi can be expressed
as follows
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RLC Band Stop Filter
Comments
This is of the form of a band stop filter. We
see we have complex zeros on the jw axis located
From the characteristic equation we see we have
two poles. The poles an essentially be placed
anywhere in the left half of the s-plane. We
see that they will be to the left of the zeros
on the jw axis.
We now consider an example on how to use this
information.
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RLC Band Stop Filter
Example
Design a band stop filter with a center frequency
of 632.5 rad/s and having poles at 100 rad/s and
3000 rad/s.
The transfer function is
We now write a Matlab program to simulate this
transfer function.
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RLC Band Stop Filter
Example
num 1 0 300000 den 1 3100 300000 w
1 5 10000 Bode(num,den,w)
105
Computer-Generated Bode Plot
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107
SERIES RESONANCE
Resonance is a phenomenon that can be observed in
mechanical systems and electrical circuits.
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Series Resonant Circuit as a Bandpass Filter
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PARALLEL RESONANCE
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Second-Order Lowpass Filter
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Basic Active Filters
Low pass filter
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Basic Active Filters
High pass
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Basic Active Filters
Band pass filter
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Basic Active Filters
Band stop filter
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FILTRI A CAPACITA COMMUTATE
122
Analog system represented as black box
Inside we could have analogue components, or
Digital processor
Analog lowpass filter 1
Analog lowpass filter 2
x(t)
ADC
y(t)
DAC
Fs
Fs
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Analog low-pass filters

• Analog Lowpass Filter 1 is antialiasing filter
  removes any frequency components above Fs/2
  before sampling process.
• Analog Lowpass Filter 2 is reconstruction
  filter smoothes DAC output to remove all
  frequency components above Fs/2.
• Digital processor controls ADC to sample at Fs
  Hz.
• Also sends output sample to DAC at Fs samples
  per second.
• DAC produces staircase waveform smoothed by
  ALpF2.