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Design Hilbert transform IIR filter

`hilbiir`

hilbiir(ts)

hilbiir(ts,dly)

hilbiir(ts,dly,bandwidth)

hilbiir(ts,dly,bandwidth,tol)

[num,den] = hilbiir(...)

[num,den,sv] = hilbiir(...)

[a,b,c,d] = hilbiir(...)

[a,b,c,d,sv] = hilbiir(...)

The function `hilbiir`

designs a Hilbert transform
filter. The output is either

A plot of the filter's impulse response, or

A quantitative characterization of the filter, using either a transfer function model or a state-space model

An ideal Hilbert transform filter has the transfer function `H(s) = -jsgn(s)`

, where `sgn(.)`

is
the signum function (`sign`

in
MATLAB). The impulse response of the Hilbert transform filter is

$$h(t)=\frac{1}{\pi t}$$

Because the Hilbert transform filter is a noncausal filter,
the `hilbiir`

function introduces a group delay, `dly`

.
A Hilbert transform filter with this delay has the impulse response

$$h(t)=\frac{1}{\pi (t-\text{dly})}$$

The filter design is an approximation. If you provide the filter's group delay as an input argument, these two suggestions can help improve the accuracy of the results:

Choose the sample time

`ts`

and the filter's group delay`dly`

so that`dly`

is at least a few times larger than`ts`

and`rem(dly,ts) = ts/2`

. For example, you can set`ts`

to 2`*dly/N`

, where`N`

is a positive integer.At the point t =

`dly`

, the impulse response of the Hilbert transform filter can be interpreted as`0`

,`-Inf`

, or`Inf`

. If`hilbiir`

encounters this point, it sets the impulse response there to zero. To improve accuracy, avoid the point t =`dly`

.

Each of these syntaxes produces a plot of the impulse response
of the filter that the `hilbiir`

function designs,
as well as the impulse response of a corresponding ideal Hilbert transform
filter.

`hilbiir`

plots the impulse
response of a fourth-order digital Hilbert transform filter with a
one-second group delay. The sample time is 2/7 seconds. In this particular
design, the tolerance index is 0.05. The plot also displays the impulse
response of the ideal Hilbert transform filter with a one-second group
delay.

`hilbiir(ts)`

plots the
impulse response of a fourth-order Hilbert transform filter with a
sample time of `ts`

seconds and a group delay of `ts*7/2`

seconds.
The tolerance index is 0.05. The plot also displays the impulse response
of the ideal Hilbert transform filter having a sample time of `ts`

seconds
and a group delay of `ts*7/2`

seconds.

`hilbiir(ts,dly)`

is the
same as the syntax above, except that the filter's group delay is `dly`

for
both the ideal filter and the filter that `hilbiir`

designs.
See Choosing a Group Delay Parameter above for guidelines on
choosing `dly`

.

`hilbiir(ts,dly,bandwidth)`

is
the same as the syntax above, except that `bandwidth`

specifies
the assumed bandwidth of the input signal and that the filter design
might use a compensator for the input signal. If `bandwidth`

=
0 or `bandwidth`

> 1/(2`*ts`

), `hilbiir`

does
not use a compensator.

`hilbiir(ts,dly,bandwidth,tol)`

is
the same as the syntax above, except that `tol`

is
the tolerance index. If `tol`

< 1, the order of the filter is determined by

$$\frac{\text{truncated-singular-value}}{\text{maximum-singular-value}}<\text{tol}$$

If `tol`

> 1, the order of the filter is `tol`

.

Each of these syntaxes produces quantitative information about
the filter that `hilbiir`

designs, but does *not* produce
a plot. The input arguments for these syntaxes (if you provide any)
are the same as those described in Syntaxes for Plots.

`[num,den] = hilbiir(...)`

outputs
the numerator and denominator of the IIR filter's transfer function.

`[num,den,sv] = hilbiir(...)`

outputs
the numerator and denominator of the IIR filter's transfer function,
and the singular values of the Hankel matrix that `hilbiir`

uses
in the computation.

`[a,b,c,d] = hilbiir(...)`

outputs
the discrete-time state-space model of the designed Hilbert transform
filter. `a`

, `b`

, `c`

,
and `d`

are matrices.

`[a,b,c,d,sv] = hilbiir(...)`

outputs
the discrete-time state-space model of the designed Hilbert transform
filter, and the singular values of the Hankel matrix that `hilbiir`

uses
in the computation.

For an example using the function's default values, type one of the following commands at the MATLAB prompt.

hilbiir [num,den] = hilbiir

The `hilbiir`

function calculates the impulse
response of the ideal Hilbert transform filter response with a group
delay. It fits the response curve using a singular-value decomposition
method. See the book by Kailath [1].

[1] Kailath, Thomas, *Linear
Systems*, Englewood Cliffs, NJ, Prentice-Hall, 1980.

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