## Digitization of Lumped Models

Since lumped models are described by *differential equations*,
they are digitized (brought into the digital-signal domain) by
converting them to corresponding *finite-difference equations*
(or simply ``*difference equations*''). General aspects of
finite difference schemes are discussed in Appendix D. This
chapter introduces a couple of elementary methods in common use:

### Finite Difference Approximation

A *finite difference approximation* (FDA) approximates derivatives with
finite differences, *i.e.*,

for sufficiently small .

^{8.5}

Equation (7.2) is also known as the *backward difference*
approximation of differentiation.

See §C.2.1 for a discussion of using the FDA to model ideal vibrating strings.

#### FDA in the Frequency Domain

Viewing Eq.(7.2) in the frequency domain, the ideal
differentiator transfer-function is , which can be viewed as
the Laplace transform of the operator (left-hand side of
Eq.(7.2)). Moving to the right-hand side, the *z* transform of the
first-order difference operator is
. Thus, in the
frequency domain, the finite-difference approximation may be performed
by making the substitution

in any continuous-time transfer function (Laplace transform of an integro-differential operator) to obtain a discrete-time transfer function (

*z*transform of a finite-difference operator).

The inverse of substitution Eq.(7.3) is

As discussed in §8.3.1, the FDA is a special case of the matched transformation applied to the point .

Note that the FDA does not alias, since the conformal mapping is one to one. However, it does warp the poles and zeros in a way which may not be desirable, as discussed further on p. below.

#### Delay Operator Notation

It is convenient to think of the FDA in terms of *time-domain
difference operators* using a *delay operator notation*. The
*delay operator* is defined by

*shift theorem*for transforms, is the transform of delayed (right shifted) by samples.

The obvious definition for the second derivative is

However, a better definition is the

*centered finite difference*

where denotes a unit-sample

*advance.*This definition is preferable as long as one sample of look-ahead is available, since it avoids an operator delay of one sample. Equation (7.5) is a

*zero phase filter,*meaning it has no delay at any frequency, while (7.4) is a

*linear phase filter*having a delay of sample at all frequencies.

### Bilinear Transformation

The bilinear transform is defined by the substitution

where is some positive constant [83,326]. That is, given a continuous-time transfer function , we apply the bilinear transform by defining

It can be seen that analog dc () maps to digital dc () and
the highest analog frequency () maps to the highest digital
frequency (). It is easy to show that the entire axis
in the plane (where
) is mapped exactly
*once* around the unit circle in the plane (rather than
summing around it infinitely many times, or ``aliasing'' as it does in
ordinary sampling). With real and positive, the left-half
plane maps to the interior of the unit circle, and the right-half
plane maps outside the unit circle. This means *stability is
preserved* when mapping a continuous-time transfer function to
discrete time.

Another valuable property of the bilinear transform is that
*order is preserved*. That is, an th-order -plane transfer
function carries over to an th-order -plane transfer function.
(*Order* in both cases equals the maximum of the degrees of the
numerator and denominator polynomials [449]).^{8.6}

The constant provides one remaining degree of freedom which can be used
to map any particular finite frequency from the axis in the
plane to a particular desired location on the unit circle
in the plane. All other frequencies will be *warped.* In
particular, approaching half the sampling rate, the frequency axis
compresses more and more. Note that at most one resonant frequency can be
preserved under the bilinear transformation of a mass-spring-dashpot
system. On the other hand, filters having a single transition frequency,
such as lowpass or highpass filters, map beautifully under the bilinear
transform; one simply uses to map the cut-off frequency where it
belongs, and the response looks great. In particular, ``equal ripple'' is
preserved for optimal filters of the elliptic and Chebyshev types because
the values taken on by the frequency response are identical in both cases;
only the frequency axis is warped.

The bilinear transform is often used to design digital filters from analog prototype filters [343]. An on-line introduction is given in [449].

#### Finite Difference Approximation vs. Bilinear Transform

Recall that the Finite Difference Approximation (FDA) defines the
elementary differentiator by
(ignoring the
scale factor for now), and this approximates the ideal transfer
function by
. The bilinear transform
calls instead for the transfer function
(again
dropping the scale factor) which introduces a pole at and gives
us the recursion
.
Note that this new pole is right on the unit circle and is therefore
undamped. Any signal energy at half the sampling rate will circulate
forever in the recursion, and due to round-off error, it will tend to
grow. This is therefore a potentially problematic revision of the
differentiator. To get something more practical, we need to specify
that the filter frequency response approximate
over a
*finite range* of frequencies
, where
, above which we allow the response to ``roll off''
to zero. This is how we pose the differentiator problem in terms of
general purpose filter design (see §8.6) [362].

To understand the properties of the finite difference approximation in the frequency domain, we may look at the properties of its -plane to -plane mapping

Setting to 1 for simplicity and solving the FDA mapping for z gives

*inside*the unit circle rather than onto the unit circle in the plane. Solving for the image in the z plane of the axis in the s plane gives

Under the FDA, analog and digital frequency axes coincide well enough at
very low frequencies (high sampling rates), but at high frequencies
relative to the sampling rate, *artificial damping* is introduced as
the image of the axis diverges away from the unit circle.

While the bilinear transform ``warps'' the frequency axis, we can say the
FDA ``doubly warps'' the frequency axis: It has a progressive, compressive
warping in the direction of increasing frequency, like the bilinear
transform, but unlike the bilinear transform, it also warps *normal*
to the frequency axis.

Consider a point traversing the upper half of the unit circle in the z plane, starting at and ending at . At dc, the FDA is perfect, but as we proceed out along the unit circle, we diverge from the axis image and carve an arc somewhere out in the image of the right-half plane. This has the effect of introducing an artificial damping.

Consider, for example, an undamped mass-spring system. There will be a complex conjugate pair of poles on the axis in the plane. After the FDA, those poles will be inside the unit circle, and therefore damped in the digital counterpart. The higher the resonance frequency, the larger the damping. It is even possible for unstable -plane poles to be mapped to stable -plane poles.

In summary, both the bilinear transform and the FDA preserve order, stability, and positive realness. They are both free of aliasing, high frequencies are compressively warped, and both become ideal at dc, or as approaches . However, at frequencies significantly above zero relative to the sampling rate, only the FDA introduces artificial damping. The bilinear transform maps the continuous-time frequency axis in the (the axis) plane precisely to the discrete-time frequency axis in the plane (the unit circle).

### Application of the Bilinear Transform

The impedance of a mass in the frequency domain is

*i.e.*, maps to which implies . Then the impedance relation maps across as

*difference equation*

This difference equation is diagrammed in Fig. 7.16.
We recognize this recursive digital filter as the *direct form I*
structure. The direct-form II structure is obtained by commuting the
feedforward and feedback portions and noting that the two delay
elements contain the same value and can therefore be shared [449].
The two other major
filter-section forms are obtained by *transposing* the two direct
forms by exchanging the input and output, and reversing all
arrows. (This is a special case of Mason's Gain Formula which works
for the single-input, single-output case.) When a filter structure is
transposed, its summers become branching nodes and vice versa.
Further discussion of the four basic filter section forms can be found
in [333].

#### Practical Considerations

While the digital mass simulator has the desirable properties of the bilinear transform, it is also not perfect from a practical point of view: (1) There is a pole at half the sampling rate (). (2) There is a delay-free path from input to output.

The first problem can easily be circumvented by introducing a loss factor , moving the pole from to , where and . This is sometimes called the ``leaky integrator''.

The second problem arises when making loops of elements (*e.g.*, a mass-spring
chain which forms a loop). Since the individual elements have no delay
from input to output, a loop of elements is not computable using standard
signal processing methods. The solution proposed by Alfred Fettweis is
known as ``wave digital filters,'' a topic taken up in §F.1.

### Limitations of Lumped Element Digitization

Model discretization by the FDA (§7.3.1) and bilinear transform
(§7.3.2)
methods are *order preserving*. As a result, they suffer from
significant approximation error, especially at high frequencies
relative to half the sampling rate. By allowing a *larger order*
in the digital model, we may obtain arbitrarily accurate
transfer-function models of LTI subsystems, as discussed
in Chapter 8. Of course, in higher-order approximations, the state
variables of the simulation no longer have a direct physical
intepretation, and this can have implications, particularly when
trying to extend to the nonlinear case. The benefits of a physical
interpretation should not be given up lightly. For example, one may
consider *oversampling* in place of going to higher-order element
approximations.

**Next Section:**

More General Finite-Difference Methods

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