# Recursive Digital Filter Design

The subject of *digital filter design* is enormous--much larger
than we can hope to address in this book. However, a surprisingly
large number of applications can be addressed using small filter
sections which are easily designed by hand, as exemplified in
Appendix B. This appendix describes some of the ``classic''
methods for IIR filter design based on the
*bilinear transformation of prototype analog filters*, followed by the simple but
powerful *weighted equation error method* for general purpose IIR
design. For further information on digital filter design, see the
documentation for the Matlab Toolboxes for Signal Processing and
Filter Design, and/or
[64,68,60,78].

## Lowpass Filter Design

We have discussed in detail (Chapter 1) the simplest lowpass filter, having the transfer function with one zero at and one pole at . From the graphical method for visualizing the amplitude response (§8.2), we see that this filter totally rejects signal energy at half the sampling rate, while lower frequencies experience higher gains, reaching a maximum at . We also see that the pole at has no effect on the amplitude response.

A *high quality* lowpass filter should look more like the ``box
car'' amplitude response shown in Fig.1.1. While it is
impossible to achieve this ideal response exactly using a finite-order
filter, we can come arbitrarily close. We can expect the amplitude
response to improve if we add another pole or zero to the
implementation.

Perhaps the best known ``classical'' methods for lowpass filter
designs are those derived from analog *Butterworth*,
*Chebyshev*, and *Elliptic Function* filters
[64]. These generally yield IIR filters with the same number
of poles as zeros. When an *FIR* lowpass filter is desired, different
design methods are used, such as the
*window method*
[68, p. 88]
(Matlab functions `fir1` and `fir2`),
*Remez exchange algorithm*
[68, pp. 136-140], [64, pp. 89-106]
(Matlab functions `remez` and `cremez`),
*linear programming*
[93], [68, p. 140],
and *convex optimization* [67]. This
section will describe only Butterworth IIR lowpass design in some detail.
For the remaining classical cases (Chebyshev, Inverse Chebyshev, and
Elliptic), see, *e.g.*, [64, Chapter 7] and/or Matlab/Octave functions
`butter`,
`cheby1`,
`cheby2`, and
`ellip`.

## Butterworth Lowpass Design

Almost all methods for filter design are *optimal* in some sense,
and the choice of optimality determines nature of the design.
*Butterworth filters* are optimal in the sense of having a
*maximally flat amplitude response*, as measured using a Taylor
series expansion about dc [64, p. 162]. Of course,
the trivial filter has a perfectly flat amplitude response,
but that's an allpass, not a lowpass filter. Therefore, to constrain the
optimization to the space of lowpass filters, we need
*constraints* on the design, such as and .
That is, we may require the dc gain to be 1, and the gain at half the
sampling rate to be 0.

It turns out Butterworth filters (as well as Chebyshev and Elliptic
Function filter types) are much easier to design as *analog
filters* which are then converted to digital filters. This means
carrying out the design over the plane instead of the plane,
where the plane is the complex plane over which analog filter
transfer functions are defined. The analog transfer function
is very much like the digital transfer function , except that it
is interpreted relative to the analog frequency axis
(the `` axis'') instead of the digital frequency axis
(the ``unit circle''). In particular, analog filter poles
are stable if and only if they are all in the *left-half* of the
plane, *i.e.*, their real parts are *negative*. An
introduction to Laplace transforms is given in Appendix D, and an
introduction to converting analog transfer functions to digital
transfer functions using the bilinear transform appears in
§I.3.

#### Butterworth Lowpass Poles and Zeros

When the maximally flat optimality criterion is applied to the general (analog) squared amplitude response , a surprisingly simple result is obtained [64]:

where is the desired order (number of poles). This simple result is obtained when the response is taken to be maximally flat at as well as dc (

*i.e.*, when both and are maximally flat at dc).

^{I.1}Also, an arbitrary scale factor for has been set such that the cut-off frequency (-3dB frequency) is rad/sec.

The *analytic continuation*
(§D.2)
of
to the whole
-plane may be obtained by substituting
to obtain

with

A Butterworth lowpass filter additionally has zeros at . Under the bilinear transform , these all map to the point , which determines the numerator of the digital filter as .

Given the poles and zeros of the analog prototype, it is straightforward to convert to digital form by means of the bilinear transformation.

#### Example: Second-Order Butterworth Lowpass

In the second-order case, we have, for the analog prototype,

To convert this to digital form, we apply the bilinear transform

(I.4) | |||

(I.5) | |||

(I.6) | |||

(I.7) |

Note that the numerator is , as predicted earlier. As a check, we can verify that the dc gain is 1:

*i.e.*, that there is a (double) notch at half the sampling rate.

In the analog prototype, the cut-off frequency is rad/sec, where, from Eq.(I.1), the amplitude response is . Since we mapped the cut-off frequency precisely under the bilinear transform, we expect the digital filter to have precisely this gain. The digital frequency response at one-fourth the sampling rate is

and dB as expected.

Note from Eq.(I.8) that the phase at cut-off is exactly -90 degrees in the digital filter. This can be verified against the pole-zero diagram in the plane, which has two zeros at , each contributing +45 degrees, and two poles at , each contributing -90 degrees. Thus, the calculated phase-response at the cut-off frequency agrees with what we expect from the digital pole-zero diagram.

In the plane, it is not as easy to use the pole-zero diagram to calculate the phase at , but using Eq.(I.3), we quickly obtain

A related example appears in §9.2.4.

##
Digitizing Analog Filters with the

Bilinear Transformation

The desirable properties of many filter types (such as lowpass,
highpass, and bandpass) are preserved very well by the
mapping called the *bilinear transform*.

### Bilinear Transformation

The *bilinear transform* may be defined by

where is an arbitrary positive constant that we may set to map one analog frequency precisely to one digital frequency. In the case of a lowpass or highpass filter, is typically used to set the

*cut-off frequency*to be identical in the analog and digital cases.

### Frequency Warping

It is easy to check that the bilinear transform gives a one-to-one,
order-preserving, *conformal map* [57] between the
analog frequency axis
and the digital frequency axis
, where is the sampling interval. Therefore, the
amplitude response takes on exactly the same values over both axes,
with the only defect being a
*frequency warping* such
that equal increments along the unit circle in the plane
correspond to larger and larger bandwidths along the axis in
the plane [88]. Some kind of frequency warping
is obviously unavoidable in any one-to-one map because the analog
frequency axis is infinite while the digital frequency axis is finite.
The relation between the analog and digital frequency axes may be
derived immediately from Eq.(I.9) as

Given an analog cut-off frequency , to obtain the same cut-off frequency in the digital filter, we set

### Analog Prototype Filter

Since the digital cut-off frequency may be set to any value, irrespective of the analog cut-off frequency, it is convenient to set the analog cut-off frequency to . In this case, the bilinear-transform constant is simply set to

### Examples

Examples of using the bilinear transform to ``digitize'' analog
filters may be found in §I.2 and
[64,5,6,103,86].
Bilinear transform design is also inherent in the construction of
*wave digital filters* [25,86].

##
Filter Design by Minimizing the

L2 Equation-Error Norm

One of the simplest formulations of recursive digital filter design is
based on minimizing the *equation error*. This method allows matching
of both spectral phase and magnitude. Equation-error methods can be
classified as variations of *Prony's method* [48]. Equation error
minimization is used very often in the field of *system identification*
[46,30,78].

The problem of fitting a digital filter to a given spectrum may be formulated as follows:

Given a continuous complex function
,
corresponding to a causal^{I.2} desired
frequency-response, find a stable digital filter of the form

with given, such that some norm of the error

The approximate filter is typically *constrained* to be stable,
and since positive powers of do not appear in
, stability
implies causality. Consequently, the impulse response of the filter
is zero for . If were noncausal, all impulse-response components
for would be approximated by zero.

### Equation Error Formulation

The *equation error* is defined (in the frequency domain) as

By comparison, the more natural frequency-domain error
is the so-called *output error*:

The names of these errors make the most sense in the time domain. Let
and denote the filter input and output, respectively, at time
. Then the equation error is the error in the *difference equation*:

while the output error is the difference between the ideal and approximate
filter *outputs*:

Denote the norm of the equation error by

where is the vector of unknown filter coefficients. Then the problem is to minimize this norm with respect to . What makes the equation-error so easy to minimize is that it is

*linear in the parameters*. In the time-domain form, it is clear that the equation error is linear in the unknowns . When the error is linear in the parameters, the sum of squared errors is a

*quadratic form*which can be minimized using one iteration of Newton's method. In other words, minimizing the norm of any error which is linear in the parameters results in a set of linear equations to solve. In the case of the equation-error minimization at hand, we will obtain linear equations in as many unknowns.

Note that (I.11) can be expressed as

*weighted output error*in which the frequency weighting function on the unit circle is given by . Thus, the weighting function is determined by the filter

*poles*, and the error is weighted

*less*near the poles. Since the poles of a good filter-design tend toward regions of high spectral energy, or toward ``irregularities'' in the spectrum, it is evident that the equation-error criterion assigns less importance to the most prominent or structured spectral regions. On the other hand, far away from the roots of , good fits to

*both phase and magnitude*can be expected. The weighting effect can be eliminated through use of the

*Steiglitz-McBride algorithm*[45,78] which iteratively solves the weighted equation-error solution, using the canceling weight function from the previous iteration. When it converges (which is typical in practice), it must converge to the output error minimizer.

### Error Weighting and Frequency Warping

Audio filter designs typically benefit from an *error weighting
function* that weights frequencies according
to their audibility. An oversimplified but useful weighting function
is simply , in which low frequencies are deemed generally
more important than high frequencies. Audio filter designs also
typically improve when using a *frequency warping*, such as
described in [88,78] (and similar to that
in §I.3.2). In principle, the effect of a frequency-warping can
be achieved using a weighting function, but in practice, the numerical
performance of a frequency warping is often much better.

### Stability of Equation Error Designs

A problem with equation-error methods is that *stability* of the filter
design is *not guaranteed*. When an unstable design is encountered,
one common remedy is to reflect unstable poles inside the unit circle,
leaving the magnitude response unchanged while modifying the phase of the
approximation in an ad hoc manner. This requires polynomial factorization
of
to find the filter poles, which is typically more work
than the filter design itself.

A better way to address the instability problem is to repeat the
filter design employing a *bulk delay*. This amounts to
replacing
by

*delays*the desired impulse response,

*i.e.*, . As the bulk delay is increased, the likelihood of obtaining an unstable design decreases, for reasons discussed in the next paragraph.

Unstable equation-error designs are especially likely when
is
*noncausal*. Since there are no constraints on where the poles of
can be, one can expect unstable designs for desired
frequency-response functions having a linear phase trend with positive
slope.

In the other direction, experience has shown that best results are obtained
when is *minimum phase*, *i.e.*, when all the zeros of are
inside the unit circle. For a given magnitude,
,
minimum phase gives the maximum concentration of impulse-response energy
near the time origin . Consequently, the impulse-response tends to start
large and decay immediately. For non-minimum phase , the
impulse-response may be small for the first samples, and the
equation error method can yield very poor filters in these cases. To see
why this is so, consider a desired impulse-response which is zero
for
, and arbitrary thereafter. Transforming into the
time domain yields

where ``'' denotes convolution, and the additive decomposition is due the fact that for . In this case the minimum occurs for ! Clearly this is not a particularly good fit. Thus, the introduction of bulk-delay to guard against unstable designs is limited by this phenomenon.

It should be emphasized that for minimum-phase , equation-error methods are very effective. It is simple to convert a desired magnitude response into a minimum-phase frequency-response by use of cepstral techniques [22,60] (see also the appendix below), and this is highly recommended when minimizing equation error. Finally, the error weighting by can usually be removed by a few iterations of the Steiglitz-McBride algorithm.

### An FFT-Based Equation-Error Method

The algorithm below minimizes the equation error in the frequency-domain.
As a result, it can make use of the FFT for speed. This algorithm is
implemented in Matlab's `invfreqz()` function when no iteration-count
is specified. (The iteration count gives that many iterations of the
Steiglitz-McBride algorithm, thus transforming equation error to output
error after a few iterations. There is also a time-domain implementation of
the Steiglitz-McBride algorithm called `stmcb()` in the Matlab Signal
Processing Toolbox, which takes the desired impulse response as input.)

Given a desired spectrum at equally spaced frequencies , with a power of , it is desired to find a rational digital filter with zeros and poles,

normalized by , such that
Since is a quadratic form, the solution is readily obtained by
equating the gradient to zero. An easier derivation follows from minimizing
equation error variance in the time domain and making use of the orthogonality
principle [36].
This may be viewed as a system identification problem where the
known input signal is an impulse, and the known output is the desired
impulse response. A formulation employing an *arbitrary* known input
is valuable for introducing complex weighting across the frequency grid,
and this general form is presented. A detailed derivation appears in
[78, Chapter 2], and here only the final algorithm is given:

Given spectral output samples and input samples , we minimize

Let : denote the column vector determined by , for filled in from top to bottom, and let : denote the size symmetric Toeplitz matrix consisting of : in its first column. A nonsymmetric Toeplitz matrix may be specified by its first column and row, and we use the notation :: to denote the by Toeplitz matrix with left-most column : and top row :. The inverse Fourier transform of is defined as

where the overbar denotes complex conjugation, and four corresponding Toeplitz matrices,

where negative indices are to be interpreted mod , *e.g.*,
.

The solution is then

### Prony's Method

There are several variations on equation-error minimization, and some
confusion in terminology exists. We use the definition of *Prony's
method* given by Markel and Gray [48]. It is equivalent to ``Shank's
method'' [9]. In this method, one first computes the
denominator
by minimizing

This step is equivalent to minimization of *ratio error*
(as used in *linear prediction*) for the
all-pole part
, with the first terms of the time-domain
error sum discarded (to get past the influence of the zeros
on the impulse response). When
, it coincides with the
covariance method of linear prediction [48,47]. This idea for
finding the poles by ``skipping'' the influence of the zeros on the
impulse-response shows up in the stochastic case under the name of *modified Yule-Walker equations* [11].

Now, Prony's method consists of next minimizing output error with the pre-assigned poles given by . In other words, the numerator is found by minimizing

### The Padé-Prony Method

Another variation of Prony's method, described by Burrus and Parks
[9] consists of using *Padé* approximation to
find the numerator
after the denominator
has been found
as before. Thus,
is found by matching the first
samples of , *viz.*,
. This method is faster, but does not generally give
as good results as the previous version. In particular, the degenerate
example
gives
here as did pure
equation error. This method has been applied also in the stochastic
case [11].

On the whole, when is causal and minimum phase (the ideal situation for just about any stable filter-design method), the variants on equation-error minimization described in this section perform very similarly. They are all quite fast, relative to algorithms which iteratively minimize output error, and the equation-error method based on the FFT above is generally fastest.

**Next Section:**

Matlab Utilities

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A View of Linear Time Varying Digital Filters