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# Documentation

Cascade allpass WDF filters to construct allpass WDF

## Description

hd = dfilt.cascadewdfallpass(c1,c2,...) constructs a cascade of allpass wave digital filters given the allpass coefficients in the vectors c1, c2, and so on.

Each c vector contains the coefficients for one section of the cascaded filter. C vectors must have one, two, or four elements (coefficients). Three element vectors are not supported.

When the c vector has four elements, the first and third elements of the vector must be 0. Each section of the cascade is an allpass wave digital filter, from dfilt.wdfallpass, with the coefficients given by the corresponding c vector. That is, the first section has coefficients from vector c1, the second section coefficients come from c2, and on until all of the c vectors are used.

You can mix the lengths of the c vectors. They do not need to be the same length. For example, you can cascade several fourth-order sections (length(c) = 4) with first or second-order sections.

Wave digital filters are usually used to create other filters. This toolbox uses them to implement halfband filters, which the first example in Examples demonstrates. They are most often building blocks for filters.

Generally, you do not construct these WDF allpass cascade filters directly. Instead, they result from the design process for an IIR filter. Refer to the first example in Examples for more about using dfilt.cascadewdfallpass to design an IIR filter.

For more information about the c vectors and the transfer function for the allpass filters, refer to dfilt.wdfallpass.

## Properties

In the next table, the row entries are the filter properties and a brief description of each property.

Property Name

Brief Description

AllpassCoefficients

Contains the coefficients for the allpass wave digital filter object

FilterStructure

Describes the signal flow for the filter object, including all of the active elements that perform operations during filtering — gains, delays, sums, products, and input/output.

PersistentMemory

Specifies whether to reset the filter states and memory before each filtering operation. Lets you decide whether your filter retains states from previous filtering runs. False is the default setting.

States

This property contains the filter states before, during, and after filter operations. States act as filter memory between filtering runs or sessions. They also provide linkage between the sections of a multisection filter, such as a cascade filter. For details, refer to filtstates in Signal Processing Toolbox™ documentation or in the Help system.

## Examples

To demonstrate two approaches to using dfilt.cascadewdfallpass to design a filter, these examples show both direct construction and construction as part of another filter.

The first design shown creates an IIR halfband filter that uses lattice wave digital filters. Each branch of the parallel connection in the lattice is an allpass cascade wave digital filter.

```tw = 100;  % Transition width of filter, 100 Hz.
ast = 80;  % Stopband attenuation of filter, 80 dB.
fs = 2000; % Sampling frequency of signal to filter.
% Store halfband specs.
d = fdesign.halfband('tw,ast',tw,ast,fs);```

Now perform the actual halfband design process. hd contains two dfilt.cascadewdfallpass filters.

```hd = design(d,'ellip','filterstructure','cascadewdfallpass');
```section1 = 0.8;