Analog Filters
For our purposes, an
analog filter is any filter which operates
on
continuoustime signals. In other respects, they are just
like
digital filters. In particular, linear, timeinvariant (
LTI)
analog filters can be
characterized by their (continuous)
impulse response , where
is time in seconds. Instead of a
difference equation, analog filters
may be described by a
differential equation. Instead of using
the
z transform to compute the
transfer function, we use the
Laplace
transform (introduced in Appendix
D). Every aspect of the theory
of digital filters has its counterpart in that of analog filters. In
fact, one can think of analog filters as simply the limiting case of
digital filters as the
samplingrate is allowed to go to infinity.
In the real world, analog filters are often electrical models, or
``analogues'', of mechanical systems working in continuous time. If
the physical system is LTI (
e.g.,
consisting of elastic
springs and
masses which are constant over
time), an LTI analog filter can be used to model it. Before the
widespread use of digital computers, physical systems were simulated
on socalled ``analog computers.'' An analog computer was much like
an analog synthesizer providing modular buildingblocks (such as
``integrators'') that could be patched together to build models of
dynamic systems.
Example Analog Filter
Figure E.1:
Simple RC lowpass.

Figure
E.1 shows a simple analog filter consisting of one resistor
(
Ohms) and one capacitor (
Farads). The voltages across these
elements are
and
, respectively, where
denotes
time in seconds. The filter input is the externally applied voltage
, and the filter output is taken to be
. By
Kirchoff's loop constraints [
20], we have

(E.1) 
and the loop current is
.
Capacitors
A
capacitor can be made physically using two parallel conducting
plates which are held close together (but not touching). Electric
charge can be stored in a capacitor by applying a voltage across the
plates.
The defining equation of a capacitor
is

(E.2) 
where
denotes the capacitor's
charge in
Coulombs,
is the
capacitance in
Farads, and
is the
voltage drop across the capacitor in volts. Differentiating with
respect to time gives
where
is now the
current in
Amperes. Note that, by convention, the current is taken to be
positive when flowing from plus to minus across the capacitor (see the
arrow in Fig.
E.1 which indicates the direction of current
flowthere is only one current
flowing clockwise around the
loop formed by the voltage source, resistor, and capacitor when an
external voltage
is applied).
Taking the
Laplace transform of both sides gives
by the
differentiation theorem for Laplace transforms (§
D.4.2).
Assuming a zero initial voltage across the capacitor at time 0, we have
We call this the
drivingpoint impedance of the capacitor. The
drivingpoint
impedance facilitates
steady state analysis (zero
initial conditions) by allowing the capacitor to be analyzed like a
simple resistor, with value
Ohms.
Mechanical Equivalent of a Capacitor is a Spring
The mechanical analog of a capacitor is the
compliance of a
spring. The voltage
across a capacitor
corresponds to the
force used to
displace a spring. The charge
stored in
the capacitor corresponds to the
displacement of the spring.
Thus, Eq.
(
E.2) corresponds to
Hooke's law for ideal springs:
where
is called the
spring constant or
spring stiffness.
Note that
Hooke's law is usually written as
. The quantity
is called the
spring compliance.
Figure E.2:
An RLC filter,
input , output
.

An
inductor can be made physically using a coil of wire, and it
stores magnetic flux when a current flows through it. Figure
E.2
shows a circuit in which a resistor
is in series with the
parallel
combination of a capacitor
and inductor
.
The defining equation of an inductor
is

(E.3) 
where
denotes the inductor's stored magnetic flux at time
,
is the
inductance in
Henrys (H), and
is the
current through the inductor coil in
Amperes (A), where
an Ampere is a Coulomb (of electric charge) per second.
Differentiating with respect to time gives

(E.4) 
where
is the voltage across the inductor in
volts. Again, the current
is taken to be positive when flowing
from plus to minus through the inductor.
Taking the
Laplace transform of both sides gives
by the
differentiation theorem for Laplace transforms.
Assuming a zero initial current in the inductor at time 0, we have
Thus, the
drivingpoint impedance of the inductor is
.
Like the capacitor, it can be analyzed in steady state (
initial
conditions neglected) as a simple resistor with value
Ohms.
Mechanical Equivalent of an Inductor is a Mass
The mechanical analog of an inductor is a
mass. The voltage
across an inductor
corresponds to the
force used to
accelerate a mass
. The current
through in the inductor
corresponds to the
velocity
of the mass. Thus,
Eq.
(
E.4) corresponds to
Newton's second law for an
ideal mass:
where
denotes the
acceleration of the mass
.
From the defining equation
for an inductor [Eq.
(
E.3)], we
see that the stored magnetic flux in an inductor is analogous to mass
times velocity, or
momentum. In other words, magnetic flux may
be regarded as electriccharge momentum.
Referring again to Fig.
E.1, let's perform an
impedance
analysis of the simple RC
lowpass filter.
Taking the
Laplace transform of both sides of Eq.
(
E.1) gives
where we made use of the fact that the
impedance of a capacitor is
, as derived above. The driving point impedance of the whole
RC
filter is thus
Alternatively, we could simply note that impedances always sum in
series and write down this result directly.
Since the input and output
signals are defined as
and
, respectively, the
transfer function of this analog
filter is given by, using
voltage divider rule,
The parameter
is called the
RC time constant,
for reasons we will soon see.
In the same way that the
impulse response of a
digital filter is given
by the inverse
z transform of its
transfer function, the impulse response
of an
analog filter is given by the inverse
Laplace
transform of its transfer function,
viz.,
where
denotes the
Heaviside unit step function
This result is most easily checked by taking the
Laplace transform of
an
exponential decay with
timeconstant :
In more complicated situations, any rational
(ratio of
polynomials in
) may be expanded into firstorder terms by means of
a
partial fraction expansion (see §
6.8) and each term in
the expansion inverted by inspection as above.
The ContinuousTime Impulse
The continuoustime
impulse response was derived above as the
inverse
Laplace transform of the
transfer function. In this section,
we look at how the
impulse itself must be defined in the
continuoustime case.
An
impulse in continuous time may be loosely defined as any
``
generalized function'' having
``zero width'' and
unit
area under it. A simple valid definition is

(E.5) 
More generally, an impulse can be defined as the limit of
any pulse shape
which maintains unit area and approaches zero width at time 0. As a
result, the impulse under every definition has the socalled
sifting property under integration,

(E.6) 
provided
is continuous at
. This is often taken as the
defining property of an impulse, allowing it to be defined in terms
of nonvanishing function limits such as
An impulse is not a function in the usual sense, so it is called
instead a
distribution or
generalized function
[
13,
44]. (It is still commonly called a ``delta function'',
however, despite the misnomer.)
In the simple RC
filter example of §
E.4.3, the
transfer function is
Thus, there is a single
pole at
, and we can say
there is one
zero at infinity as well. Since resistors and
capacitors always have positive values, the
time constant
is always nonnegative. This means the
impulse response is always an
exponential
decaynever a growth. Since the pole is at
, we find that it is
always in the lefthalf
plane. This turns out to be the case also for any
complex
analog onepole filter. By consideration of the
partial fraction
expansion of any
, it is clear that, for
stability of an analog
filter,
all poles must lie in the left half of the complex
plane. This is the analog counterpart of the requirement for
digital
filters that all poles lie inside the unit circle.
RLC Filter Analysis
Referring now to Fig.
E.2, let's perform an
impedance
analysis of that RLC network.
By inspection, we can write
where
denotes ``in parallel with,'' and we used the general formula,
memorized by any electrical engineering student,
That is, the
impedance of the
parallel combination of
impedances
and
is given by the product divided by the sum
of the impedances.
The transfer function in this example can similarly be found using
voltage divider rule:
From the
quadratic formula, the two
poles are located at
and there is a zero at
and another at
. If the
damping
is sufficienly small so that
, then
the poles form a complexconjugate pair:
Since
, the poles are always in the lefthalf
plane, and hence the analog RLC
filter is always stable. When the
damping is zero, the poles go to the
axis:
The
impulse response is again the inverse
Laplace transform of the
transfer function. Expanding
into a sum of complex one
pole
sections,
where
.
Equating numerator coefficients gives
This pair of equations in two unknowns may be solved for
and
.
The impulse response is then
Consider the
continuoustime complex onepole
resonator with
plane
transfer function
where
is the
Laplacetransform variable, and
is the single complex pole. The numerator scaling
has been set to
so that the
frequency response is normalized to
unity gain at resonance:
The
amplitude response at all frequencies is given by
Without loss of generality, we may set
, since changing
merely translates the amplitude response with respect to
.
(We could alternatively define the translated frequency variable
to get the same simplification.) The squared amplitude response is now
Note that
This shows that the
3dB bandwidth of the resonator in radians
per second is
, or twice the absolute value of the real
part of the pole. Denoting the 3
dB bandwidth in Hz by
, we have
derived the relation
, or
Since a
dB attenuation is the same thing as a power scaling by
, the 3dB bandwidth is also called the
halfpower
bandwidth.
It now remains to ``digitize'' the continuoustime resonator and show
that relation Eq.
(
8.7) follows. The most natural mapping of the
plane to the
plane is
where
is the
sampling period. This mapping follows directly from
sampling the Laplace transform to obtain the
z transform. It is
also called the
impulse invariant transformation [
68, pp.
216219], and for digital poles it is the same as the
matched z transformation [
68, pp. 224226].
Applying the matched
z transformation to the pole
in the
plane gives the
digital pole
from which we identify
and the relation between pole radius
and analog 3dB bandwidth
(in Hz) is now shown. Since the mapping
becomes
exact as
, we have that
is also the 3dB bandwidth of the
digital resonator in the limit as the
sampling rate approaches
infinity. In practice, it is a good approximate relation whenever the
digital pole is close to the unit circle (
).
Quality Factor (Q)
The
quality factor (Q) of a two
pole resonator is defined by
[
20, p. 184]

(E.7) 
where
and
are parameters of the resonator
transfer
function

(E.8) 
Note that Q is defined in the context of
continuoustime
resonators, so the transfer function
is the
Laplace transform
(instead of the
z transform) of the
continuous (instead of
discretetime)
impulseresponse . An introduction to
Laplacetransform analysis appears in Appendix
D. The parameter
is called the
damping constant (or ``damping factor'')
of the secondorder transfer function, and
is called the
resonant frequency [
20, p. 179].
The resonant frequency
coincides with the physical
oscillation frequency of the resonator
impulse response when the
damping constant
is zero. For light damping,
is
approximately the physical frequency of impulseresponse oscillation
(
times the zerocrossing rate of
sinusoidal oscillation under
an
exponential decay). For larger damping constants, it is better to
use the imaginary part of the pole location as a definition of
resonance frequency (which is exact in the case of a single complex
pole). (See §
B.6 for a more complete discussion of resonators,
in the discretetime case.)
By the
quadratic formula, the poles of the transfer function
are given by

(E.9) 
Therefore, the poles are complex only when
. Since real poles
do not resonate, we have
for any resonator. The case
is called
critically damped, while
is called
overdamped. A resonator (
) is said to be
underdamped, and the limiting case
is simply
undamped.
Relating to the notation of the previous section, in which we defined
one of the complex poles as
, we have
For resonators,
coincides with the classically defined
quantity [
20, p. 624]
Since the imaginary parts of the complex resonator poles are
, the zerocrossing rate of the resonator impulse
response is
crossings per second. Moreover,
is very close to the peakmagnitude frequency in the resonator
amplitude response. If we eliminate the
negativefrequency pole,
becomes
exactly the peak frequency. In other
words, as a measure of resonance peak frequency,
only
neglects the interaction of the positive and negativefrequency
resonance peaks in the
frequency response, which is usually negligible
except for highly damped, lowfrequency resonators. For any amount of
damping
gives the impulseresponse zerocrossing rate
exactly, as is immediately seen from the derivation in the next
section.
Another well known rule of thumb is that the
of a
resonator is the
number of ``periods'' under the
exponential decay of its
impulse
response. More precisely, we will show that, for
, the
impulse response decays by the factor
in
cycles, which
is about 96 percent decay, or 27
dB.
The impulse response corresponding to Eq.
(
E.8) is found by
inverting the
Laplace transform of the
transfer function . Since it
is only second order, the solution can be found in many tables of
Laplace transforms. Alternatively, we can break it up into a sum of
firstorder terms which are invertible by inspection (possibly after
rederiving the Laplace transform of an
exponential decay, which is
very simple). Thus we perform the
partial fraction expansion of
Eq.
(
E.8) to obtain
where
are given by Eq.
(
E.9), and some algebra gives
as the respective residues of the
poles .
The impulse response is thus
Assuming a resonator,
, we have
, where
(using notation of the
preceding section), and the impulse response reduces to
where
and
are overall amplitude and phase constants,
respectively.
^{E.1}
We have shown so far that the impulse response
decays as
with a
sinusoidal radian frequency
under the
exponential envelope. After
Q periods at frequency
, time has advanced to
where we have used the definition Eq.
(
E.7)
.
Thus, after
periods, the
amplitude envelope has decayed to
which is about 96 percent decay. The only approximation in this derivation
was
which holds whenever
, or
.
Yet another meaning for
is as follows [
20, p. 326]
where the
resonator is freely decaying (unexcited).
Proof. The total stored energy at time
is
equal to the total energy of the remaining response. After an
impulse
at time 0, the stored energy in a secondorder resonator is
The energy dissipated in the first
period
is
, where
Assuming
as before,
so that
Assuming further that
, we obtain
This is the energy dissipated in one cycle. Dividing this into the
total stored energy at time zero,
, gives
whence
as claimed. Note that this rule of thumb requires
, while
the one of the previous section only required
.
It turns out that analog
allpass filters are considerably simpler
mathematically than digital allpass filters (discussed in
§
B.2). In fact, when working with digital allpass filters,
it can be fruitful to convert to the analog case using the
bilinear
transform (§
I.3.1), so that the filter may be manipulated in the
analog
plane rather than the digital
plane. The analog case
is simpler because analog allpass filters may be described as having a
zero at
for every
pole at
, while digital allpass
filters must have a zero at
for every pole at
.
In particular, the
transfer function of every firstorder analog
allpass filter can be written as
where
is any constant phase offset.
To see why
must be allpass, note that
its
frequency response is given by
which clearly has modulus 1 for all
(since
). For real allpass filters,
complex poles must occur in conjugate pairs, so that the ``allpass
rule'' for
poles and zeros may be simplified to state that a zero is
required at
minus the location of every pole,
i.e., every
real firstorder allpass filter is of the form
and, more generally, every real allpass transfer function can be factored as

(E.14) 
This simplified rule works because every complex pole
is
accompanied by its conjugate
for some
.
Multiplying out the terms in Eq.
(
E.14), we find that the numerator
polynomial
is simply related to the denominator polynomial
:
Since the roots of
must be in the lefthalf
plane for
stability,
must be a
Hurwitz polynomial, which implies
that all of its coefficients are nonnegative. The polynomial
can be seen as a
rotation of
in the
plane; therefore,
its roots must have nonpositive real parts, and its coefficients form
an alternating sequence.
As an example of the greater simplicity of analog allpass filters
relative to the discretetime case, the
graphical method for computing
phase response from poles and zeros (§
8.3) gives immediately
that the phase response of every real analog allpass filter is equal
to
twice the phase response of its numerator (plus
when
the frequency response is negative at
dc). This is because the angle
of a vector from a pole at
to the point
along the
frequency axis is
minus the angle of the vector from a zero at
to the point
.
Lossless Analog Filters
As discussed in §
B.2, the an
allpass filter can be defined
as any filter that
preserves signal energy for every input
signal . In the continuoustime case, this means
where
denotes the output signal, and
denotes the
L2 norm of
. Using the
Rayleigh energy theorem
(
Parseval's theorem) for
Fourier transforms [
87],
energy preservation can be expressed in the
frequency domain by
where
and
denote the Fourier transforms of
and
, respectively,
and frequencydomain L2
norms are defined by
If
denotes the
impulse response of the
allpass
filter, then its
transfer function
is given by the
Laplace transform of
,
and we have the requirement
Since this equality must hold for every input signal
, it must be
true in particular for complex
sinusoidal inputs of the form
, in which case [
87]
where
denotes the Dirac ``delta function'' or continuous
impulse function (§
E.4.3). Thus, the allpass condition becomes
which implies

(E.15) 
Suppose
is a rational analog filter, so that
where
and
are polynomials in
:
(We have normalized
so that
is monic (
) without
loss of generality.) Equation (
E.15) implies
If
, then the allpass condition reduces to
,
which implies
where
is any real phase constant. In other words,
can be any unitmodulus
complex number. If
, then the
filter is allpass provided
Since this must hold for all
, there are only two solutions:
 and , in which case
for all .

and , i.e.,
Case (1) is trivially allpass, while case (2) is the one discussed above
in the introduction to this section.
By analytic continuation, we have
If
is real, then
, and we can write
To have
, every
pole at
in
must be canceled
by a zero at
in
, which is a zero at
in
.
Thus, we have derived the simplified ``allpass rule'' for real analog
filters.
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