# dsp.Channelizer

Polyphase FFT analysis filter bank

## Description

The `dsp.Channelizer`

System object™ separates a broadband input signal into multiple narrow subbands using a fast
Fourier transform (FFT)-based analysis filter bank. The filter bank uses a prototype lowpass
filter and is implemented using a polyphase structure. You can specify the filter coefficients
directly or through design parameters.

To separate a broadband signal into multiple narrow subbands:

Create the

`dsp.Channelizer`

object and set its properties.Call the object with arguments, as if it were a function.

To learn more about how System objects work, see What Are System Objects?

## Creation

### Syntax

### Description

creates a
polyphase FFT analysis filter bank System object that separates a broadband input signal into multiple narrowband output
signals. This object implements the inverse operation of the
`channelizer`

= dsp.Channelizer`dsp.ChannelSynthesizer`

System object.

creates an
`channelizer`

= dsp.Channelizer(M)*M*-band polyphase FFT analysis filter bank, with the NumFrequencyBands property set to *M*.

**Example:**channelizer = dsp.Channelizer(16);

creates a polyphase FFT analysis filter bank with each specified property set to the
specified value. Enclose each property name in single quotes.`channelizer`

= dsp.Channelizer(`Name,Value`

)

**Example:**channelizer = dsp.Channelizer('NumTapsPerBand',20,'StopbandAttenuation',140);

## Properties

Unless otherwise indicated, properties are *nontunable*, which means you cannot change their
values after calling the object. Objects lock when you call them, and the
`release`

function unlocks them.

If a property is *tunable*, you can change its value at
any time.

For more information on changing property values, see System Design in MATLAB Using System Objects.

`NumFrequencyBands`

— Number of frequency bands

`8`

(default) | positive integer greater than 1

Number of frequency bands *M* into which the object separates the
input broadband signal, specified as a positive integer greater than 1. This property
corresponds to the number of polyphase branches and the FFT length used in the filter
bank.

**Example: **16

**Example: **64

**Data Types: **`single`

| `double`

| `int8`

| `int16`

| `int32`

| `int64`

| `uint8`

| `uint16`

| `uint32`

| `uint64`

`DecimationFactor`

— Decimation factor

`8`

(default) | positive integer

Decimation factor *D* specified as a positive integer less than or
equal to the number of frequency bands *M*. The default value of this
property equals the number of frequency bands specified.

If the decimation factor *D* equals the number of frequency bands
*M*, then the *M*/*D* ratio equals
1, and the channelizer is known as the maximally decimated channelizer.

If the *M*/*D* ratio is greater than
`1`

, the output sample rate is different from the channel spacing,
and the channelizer is known as the non-maximally decimated channelizer. If the ratio is
an integer, the channelizer is known as the integer-oversampled channelizer. If the
ratio is not an integer, say 4/3, the channelizer is known as the rationally oversampled
channelizer. For more details, see Algorithm.

**Data Types: **`single`

| `double`

| `int8`

| `int16`

| `int32`

| `int64`

| `uint8`

| `uint16`

| `uint32`

| `uint64`

`Specification`

— Filter design parameters or coefficients

```
'Number of taps per band and stopband
attenuation'
```

(default) | `'Coefficients'`

Filter design parameters or filter coefficients, specified as one of these options:

`'Number of taps per band and stopband attenuation'`

— Specify the filter design parameters through the`NumTapsPerBand`

and Stopband attenuation (dB) properties.`'Coefficients'`

— Specify the filter coefficients directly using the LowpassCoefficients property.

`NumTapsPerBand`

— Number of filter coefficients per frequency band

`12`

(default) | positive integer

Number of filter coefficients each polyphase branch uses, specified as a positive
integer. The number of polyphase branches matches the number of frequency bands. The
total number of filter coefficients for the prototype lowpass filter is given by
`NumFrequencyBands`

× `NumTapsPerBand`

. For
a given stopband attenuation, increasing the number of taps per band narrows the
transition width of the filter. As a result, there is more usable bandwidth for each
frequency band at the expense of increased computation.

**Example: **8

**Example: **16

#### Dependencies

This property applies when you set `Specification`

to
`'Number of taps per band and stopband attenuation'`

.

**Data Types: **`single`

| `double`

| `int8`

| `int16`

| `int32`

| `int64`

| `uint8`

| `uint16`

| `uint32`

| `uint64`

`StopbandAttenuation`

— Stopband attenuation

`80`

(default) | positive real scalar

Stopband attenuation of the lowpass filter, specified as a positive real scalar in dB. This value controls the maximum amount of aliasing from one frequency band to the next. When the stopband attenuation increases, the passband ripple decreases. For a given stopband attenuation, increasing the number of taps per band narrows the transition width of the filter. As a result, there is more usable bandwidth for each frequency band at the expense of increased computation.

**Example: **80

#### Dependencies

This property applies when you set `Specification`

to
`'Number of taps per band and stopband attenuation'`

.

**Data Types: **`single`

| `double`

`LowpassCoefficients`

— Coefficients of prototype lowpass filter

`[1×49 double]`

(default) | row vector

Coefficients of the prototype lowpass filter, specified as a row vector. The default
vector of coefficients is obtained using
`rcosdesign(0.25,6,8,'sqrt')`

. There must be at least one
coefficient per frequency band. If the length of the lowpass filter is less than the
number of frequency bands, the object zero-pads the coefficients.

If you specify complex coefficients, the object designs a prototype filter that is
centered at a nonzero frequency, also known as a bandpass filter. The modulated versions
of the prototype bandpass filter appear with respect to the prototype filter and are
wrapped around the frequency range [−*F*_{s}
*F*_{s}]. For an example, see Channelizer with Complex Coefficients.

**Tunable: **Yes

#### Dependencies

This property applies when you set `Specification`

to
`'Coefficients'`

.

**Data Types: **`single`

| `double`

**Complex Number Support: **Yes

## Usage

### Description

### Input Arguments

`input`

— Data input

vector | matrix

Data input, specified as a vector or a matrix. The number of rows in the input
signal must be a multiple of the number of frequency bands of the filter bank. Each
column of the input corresponds to a separate channel. If *M* is the
number of frequency bands, and the input is an *L*-by-1 matrix, then
the output signal has dimensions *L/M*-by-*M*. Each
narrowband signal forms a column in the output. If the input has more than one
channel, that is, it has dimensions *L*-by-*N* with
*N* > 1, then the output has dimensions
*L/M*-by-*M*-by-*N*.

This object supports variable-size input signals. You can change the input frame size (number of rows) even after calling the algorithm. However, the number of channels (number of columns) must remain constant.

**Example: **randn(64,4)

**Data Types: **`single`

| `double`

**Complex Number Support: **Yes

### Output Arguments

`channOut`

— Channelizer output

matrix | 3-D array

Channelizer output, returned as a matrix or a 3-D array. If the input is an
*L*-by-1 matrix, then the output signal has dimensions
*L/M*-by-*M*, where *M* is the
number of frequency bands. Each narrowband signal forms a column in the output. If the
input has more than one channel, that is, it has dimensions
*L*-by-*N* with *N* > 1, then the
output has dimensions
*L/M*-by-*M*-by-*N*.

**Data Types: **`single`

| `double`

**Complex Number Support: **Yes

## Object Functions

To use an object function, specify the
System object as the first input argument. For
example, to release system resources of a System object named `obj`

, use
this syntax:

release(obj)

### Specific to dsp.Channelizer

`coeffs` | Coefficients of prototype lowpass filter |

`tf` | Return transfer function of overall prototype lowpass filter |

`polyphase` | Return polyphase matrix |

`freqz` | Frequency response of filters in channelizer |

`fvtool` | Visualize the filters in the channelizer |

`bandedgeFrequencies` | Compute the bandedge frequencies |

`centerFrequencies` | Compute center frequencies |

`getFilters` | Return matrix of channelizer FIR filters |

## Examples

### Channelize and Synthesize Sine Wave in MATLAB

Channelize and synthesize a sine wave signal with multiple frequencies using an *M* -channel filter bank.

The *M* -channel filter bank contains an analysis filter bank section and a synthesis filter bank section. The `dsp.Channelizer`

object implements the analysis filter bank section. The `dsp.ChannelSynthesizer`

object implements the synthesis filter bank section. These objects use an efficient polyphase structure to implement the filter bank. For more details, see **Polyphase Implementation** under **Algorithms** on the object reference pages.

**Initialization**

Initialize the `dsp.Channelizer`

and `dsp.ChannelSynthesizer`

System objects. Each object is set up with 8 frequency bands, 8 polyphase branches in each filter, 12 coefficients per polyphase branch, and a stopband attenuation of 140 dB. Use a sine wave with multiple frequencies as the input signal. View the input spectrum and the output spectrum using a spectrum analyzer.

offsets = [-40,-30,-20,10,15,25,35,-15]; sinewave = dsp.SineWave('ComplexOutput',true,'Frequency',... offsets+(-375:125:500),'SamplesPerFrame',800); channelizer = dsp.Channelizer('StopbandAttenuation',140); synthesizer = dsp.ChannelSynthesizer('StopbandAttenuation',140); spectrumAnalyzer = dsp.SpectrumAnalyzer('ShowLegend',true,... 'SampleRate',sinewave.SampleRate,... 'NumInputPorts',2,'ChannelNames',{'Input','Output'},... 'Title','Input and Output Spectra');

**Streaming**

Use the channelizer to split the broadband input signal into multiple narrow bands. Then pass the multiple narrowband signals into the synthesizer, which merges these signals to form the broadband signal. Compare the spectra of the input and output signals. The input and output spectra match very closely.

for i = 1:5000 x = sum(sinewave(),2); y = channelizer(x); v = synthesizer(y); spectrumAnalyzer(x,v) end

### Channelizer with Complex Coefficients

Create a `dsp.Channelizer`

object and set the `LowpassCoefficients `

property to a vector of complex coefficients.

**Complex Coefficients**

Using `firpm`

, determine the coefficients of a Park-McClellan's optimal equiripple FIR filter of order 30, and frequency and amplitude characteristics described by *F* = [0 0.2 0.4 1.0] and *A* = [1 1 0 0] vectors, respectively.

Create a complex version of these coefficients by multiplying with a complex exponential. The resultant frequency response is that of a bandpass filter at the specified frequency, in this case 0.4.

blowpass = firpm(30,[0 .2 .4 1],[1 1 0 0]); N = length(blowpass)-1; Fc = 0.4; j = complex(0,1); bbandpass = blowpass.*exp(j*Fc*pi*(0:N));

**Channelizer**

Create a `dsp.Channelizer`

object with 4 frequency bands and set the `Specification`

property to `'Coefficients'`

.

chann = dsp.Channelizer('NumFrequencyBands',4,... 'Specification',"Coefficients");

Pass the complex coefficients to the channelizer. The prototype filter is a bandpass filter with a center frequency of 0.4. The modulated versions of this filter appear with respect to the prototype filter and are wrapped around the frequency range [$-$*F*s *F*s].

chann.LowpassCoefficients = bbandpass

chann = dsp.Channelizer with properties: NumFrequencyBands: 4 DecimationFactor: 4 Specification: 'Coefficients' LowpassCoefficients: [0.0019 + 0.0000i 0.0005 + 0.0016i ... ]

Visualize the frequency response of the channelizer.

fvtool(chann)

## More About

### Analysis Filter Bank

The generic analysis filter bank consists of a series of parallel bandpass filters that split
an input broadband signal, *x*[*n*], into a series of
narrow subbands. Each bandpass filter retains a different portion of the input signal.
After the bandwidth is reduced by one of the bandpass filters, the signal is downsampled
to a lower sampling rate commensurate with the new bandwidth.

### Prototype Lowpass Filter

To implement the analysis filter bank efficiently, the channelizer uses a prototype lowpass filter.

The prototype lowpass filter has an impulse response of
*h*[*n*], a normalized two-sided bandwidth of
2π/*M*, and a cutoff frequency of π/*M*.
*M* is the number of frequency bands, that is, the branches of the
analysis filter bank. This value corresponds to the FFT length that the filter bank
uses. *M* can be high on the order of 2048 or more. The stopband
attenuation determines the minimum level of interference (aliasing) from one frequency
band to another. The passband ripple must be small so that the input signal is not
distorted in the passband.

The prototype lowpass filter corresponds to
*H _{0}*(

*z*) in the filter bank. The first branch of the filter bank contains

*H*(

_{0}*z*) followed by the decimator. The other

*M*– 1 branches contain filters that are modulated versions of the prototype filter. The modulation factor is given by the following equation:

$${e}^{-j{w}_{k}n},\text{\hspace{1em}}{w}_{k}=2\pi k/M,\text{\hspace{1em}}k=0,1,\mathrm{...},M-1$$

### Using the Prototype Lowpass Filter

The transfer function of the modulated *k*th
bandpass filter is given by:

$${H}_{k}(z)={H}_{0}(z{e}^{-j{w}_{k}}),\text{\hspace{1em}}{w}_{k}=2\pi k/M,\text{\hspace{1em}}k=1,2,\mathrm{...},M-1$$

This figure shows the frequency response of *M* filters.

To obtain the frequency response characteristics of the filter
*H _{k}*(

*z*), where

*k*= 1, … ,

*M*−1, uniformly shift the frequency response of the prototype filter,

*H*(

_{0}*z*), by multiples of 2π/

*M*. Each subband filter,

*H*(

_{k}*z*), {

*k*= 1, … ,

*M*– 1}, is derived from the prototype filter.

Following is an equivalent representation of the frequency response diagram with
*ω* ranging from [−π π].

### Shift Narrow Subbands to Baseband

The frequency components in the input signal, *x*[*n*], are
translated in frequency to baseband by multiplying
*x*[*n*] with the complex exponentials, $${e}^{-j{w}_{k}n},\text{\hspace{0.17em}}{w}_{k}=2\pi k/M,\text{\hspace{0.17em}}k=1,2,\mathrm{..},M-1$$ , where $${w}_{k}=2\pi k/M$$, and $$k=1,2,\mathrm{...},M-1$$. The resulting product signals are passed through the lowpass filters,
*H _{0}*(

*z*). The output of the lowpass filter is relatively narrow in bandwidth. Downsample the signal commensurate with the new bandwidth. Choose a decimation factor,

*D*≤

*M*, where

*M*is the number of branches of the analysis filter bank. When

*D*<

*M*, the channelizer is known as oversampled or non-maximally decimated channelizer.

The figure shows an analysis filter bank that uses the prototype lowpass filter.

*y _{1}*[

*m*],

*y*[

_{2}*m*], … ,

*y*[

_{M−1}*m*] are narrow subband signals translated into baseband.

## Algorithms

### Polyphase Implementation

The analysis filter bank can be implemented efficiently using the polyphase structure. For more details on the analysis filter bank, see Analysis Filter Bank.

To derive the polyphase structure, start with the transfer function of the prototype lowpass filter:

$${H}_{0}(z)={b}_{0}+{b}_{1}{z}^{-1}+\mathrm{...}+{b}_{N}{z}^{-N}$$

*N* + 1 is the length of the prototype filter.

You can rearrange this equation as follows:

$${H}_{0}(z)=\begin{array}{c}\left({b}_{0}+{b}_{M}{z}^{-M}+{b}_{2M}{z}^{-2M}+\mathrm{..}+{b}_{N-M+1}{z}^{-(N-M+1)}\right)+\\ {z}^{-1}\left({b}_{1}+{b}_{M+1}{z}^{-M}+{b}_{2M+1}{z}^{-2M}+\mathrm{..}+{b}_{N-M+2}{z}^{-(N-M+1)}\right)+\\ \begin{array}{c}\vdots \\ {z}^{-(M-1)}\left({b}_{M-1}+{b}_{2M-1}{z}^{-M}+{b}_{3M-1}{z}^{-2M}+\mathrm{..}+{b}_{N}{z}^{-(N-M+1)}\right)\end{array}\end{array}$$

*M* is
the number of polyphase components.

You can write this equation as:

$${H}_{0}(z)={E}_{0}({z}^{M})+{z}^{-1}{E}_{1}({z}^{M})+\mathrm{...}+{z}^{-(M-1)}{E}_{M-1}({z}^{M})$$

*E _{0}*(

*z*),

^{M}*E*(

_{1}*z*), … ,

^{M}*E*(

_{M−1}*z*) are polyphase components of the prototype lowpass filter

^{M}*H*(

_{0}*z*).

The other filters in the filter bank,
*H _{k}*(

*z*), where

*k*= 1, … ,

*M*−1, are modulated versions of this prototype filter.

You can write the transfer function of the *k*^{th}
modulated bandpass filter as $${H}_{k}(z)={H}_{0}(z{e}^{-j{w}_{k}})$$.

Replacing *z* with
*ze ^{-jwk}*,

$${H}_{k}(z)={h}_{0}+{h}_{1}{e}^{-jwk}{z}^{-1}+{h}_{2}{e}^{-j2wk}{z}^{-2}\mathrm{...}+{h}_{N}{e}^{-jNwk}{z}^{-N}$$

*N* + 1 is the length of the
*k*^{th} filter.

In polyphase form, the equation is as follows:

$${H}_{k}(z)=\left[\begin{array}{ccccc}1& {e}^{-j{w}_{k}}& {e}^{-j2{w}_{k}}& \cdots & {e}^{-j(M-1){w}_{k}}\end{array}\right]\left[\begin{array}{c}{E}_{0}({z}^{M})\\ {z}^{-1}{E}_{1}({z}^{M})\\ \vdots \\ {z}^{-(M-1)}{E}_{M-1}({z}^{M})\end{array}\right]$$

For all *M* channels in the filter bank, the transfer function
*H*(*z*) is given by:

$$H(z)=\left[\begin{array}{ccccc}1& 1& 1& \cdots & 1\\ 1& {e}^{-j{w}_{1}}& {e}^{-j2{w}_{1}}& \cdots & {e}^{-j(M-1){w}_{1}}\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1& {e}^{-j{w}_{M-1}}& {e}^{-j2{w}_{M-1}}& \cdots & {e}^{-j(M-1){w}_{M-1}}\end{array}\right]\left[\begin{array}{c}{E}_{0}({z}^{M})\\ {z}^{-1}{E}_{1}({z}^{M})\\ \vdots \\ {z}^{-(M-1)}{E}_{M-1}({z}^{M})\end{array}\right]$$

### Maximally decimated channelizer

When **D**** =
*** M*, the
channelizer is known as the maximally decimated channelizer or critically sampled
channelizer.

Here is the multirate noble identity for decimation, assuming that
*D* = *M*.

For example, consider the first branch of the filter bank that contains the lowpass filter.

Replace *H _{0}*(

*z*) with its polyphase representation.

After applying the noble identity for decimation, you can replace the delays and the decimation factor with a commutator switch. The switch starts on the first branch 0 and moves in the counterclockwise direction as shown in the following diagram. The accumulator at the output receives the processed input samples from each branch of the polyphase structure and accumulates these processed samples until the switch goes to branch 0. When the switch goes to branch 0, the accumulator outputs the accumulated value.

For all *M* channels in the filter bank, the transfer function
*H*(*z*) is given by:

$$H(z)=\left[\begin{array}{ccccc}1& 1& 1& \cdots & 1\\ 1& {e}^{-j{w}_{1}}& {e}^{-j2{w}_{1}}& \cdots & {e}^{-j(M-1){w}_{1}}\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1& {e}^{-j{w}_{M-1}}& {e}^{-j2{w}_{M-1}}& \cdots & {e}^{-j(M-1){w}_{M-1}}\end{array}\right]\left[\begin{array}{c}{E}_{0}(z)\\ {E}_{1}(z)\\ \vdots \\ {E}_{M-1}(z)\end{array}\right]$$

The matrix on the left is a discrete Fourier transform (DFT) matrix. With the DFT matrix, the efficient implementation of the lowpass prototype-based filter bank looks like this.

When the first input sample is delivered, the switch feeds this input to the branch 0
and the channelizer computes the first set of output values. As more input samples come
in, the switch moves in the counterclockwise direction through branches
*M*−1, *M*−2, all the way up to branch 0,
delivering one sample at a time to each branch. When the switch comes to branch 0, the
channelizer outputs the next set of output values. This process continues as the data
keeps coming in. Every time the switch comes to the first branch 0, the channelizer
outputs *y _{0}*[

*m*],

*y*[

_{1}*m*], … ,

*y*[

_{M-1}*m*]. Each branch in the channelizer effectively outputs one sample for every

*M*samples it receives. Hence, the sample rate at the output of the channelizer is

*f*/

_{s}*M*.

### Non-maximally decimated or oversampled channelizer

When **D**** <
*** M*, the
channelizer is known as the non-maximally decimated channelizer or oversampled
channelizer. In this configuration, the output sample rate is different from the channel
spacing. The non-maximally decimated channelizers offer increased design freedom, but at
the expense of increasing computational cost.

If the ratio *M*/*D* equals an integer that is
greater than 1 and is less than or equal to *M*−1, the channelizer is
known as integer-oversampled channelizer. If the ratio
*M*/*D* is not an integer, then the channelizer is
known as rationally-oversampled channelizer.

In this configuration, when the first input sample is delivered, the switch feeds this
input to branch 0 and the channelizer computes the first set of output values. As more
input samples come in, the switch moves in the counterclockwise direction through
branches *D*−1, *D*−2, all the way up to branch 0,
delivering one sample at a time to each branch. When the switch comes to branch 0, the
channelizer outputs the next set of output values. This process continues as the data
keeps coming in. Every time the switch comes to the first branch 0, the channelizer
outputs *y _{0}*[

*m*],

*y*[

_{1}*m*], … ,

*y*[

_{M-1}*m*].

As more data keeps coming in and the switch feeds these samples to the first
*D* addresses, the formal contents of these addresses are shifted
to the next set of *D* addresses, and this process of data shift
continues every time there is a new set of *D* input samples.

For every *D* input samples that are fed to the polyphase structure,
the channelizer outputs *M* samples,
*y _{0}*[

*m*],

*y*[

_{1}*m*], … ,

*y*[

_{M-1}*m*]. This process increases the output sample rate from

*f*/

_{s}*M*in the case of a maximally decimated channelizer, to

*f*/

_{s}*D*in the case of a non-maximally decimated channelizer.

For more details, see [2].

After each *D*-point data sequence is delivered to the partitioned
*M*-stage polyphase filter, the outputs of the *M*
stages are computed and conditioned for delivery to the *M*-point FFT.
The data shifting through the filter introduces frequency-dependent phase shift. To
correct for this phase shift and alias all bands to DC, a circular shift buffer is
inserted after the polyphase filters and before the *M*-point
FFT.

With the commutator switch followed by *M*-stage polyphase filter,
circular shift buffer, and a DFT matrix, the efficient implementation of the lowpass
prototype-based filter bank looks like this.

## References

[1] Harris, Fredric J, *Multirate Signal Processing for Communication Systems*, Prentice Hall PTR, 2004.

[2] Harris, F.J., Chris Dick, and Michael Rice. "Digital Receivers and Transmitters Using Polyphase Filter Banks for Wireless Communications." *IEEE ^{®} Transactions on Microwave Theory and Techniques*. 51, no. 4 (2003).

## Extended Capabilities

### C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

See System Objects in MATLAB Code Generation (MATLAB Coder).

## See Also

### Functions

`coeffs`

|`tf`

|`polyphase`

|`freqz`

|`fvtool`

|`bandedgeFrequencies`

|`centerFrequencies`

|`getFilters`

### Objects

`dsp.ChannelSynthesizer`

|`dsp.FIRHalfbandDecimator`

|`dsp.FIRHalfbandInterpolator`

|`dsp.IIRHalfbandDecimator`

|`dsp.DyadicAnalysisFilterBank`

### Blocks

- Channelizer | Channel Synthesizer | Dyadic Analysis Filter Bank | Two-Channel Analysis Subband Filter

### Functions

**Introduced in R2016b**

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