dsp.ChannelSynthesizer

Polyphase FFT synthesis filter bank

Description

The dsp.ChannelSynthesizer System object™ merges multiple narrowband signals into a broadband signal by using an FFT based synthesis 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 merge multiple narrowband signals into a broadband signal:

Create the dsp.ChannelSynthesizer 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

synthesizer = dsp.ChannelSynthesizer

synthesizer = dsp.ChannelSynthesizer(Name,Value)

Description

example

synthesizer = dsp.ChannelSynthesizer creates a synthesizer object, using the default properties.

synthesizer = dsp.ChannelSynthesizer(Name,Value) specifies additional properties using Name,Value pairs. Unspecified properties have default values.

Example:

synthesizer =
            dsp.ChannelSynthesizer('NumTapsPerBand',20,'StopbandAttenuation',140)

Properties

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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.

`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 StopbandAttenuation properties.
'Coefficients' — Specify the filter coefficients directly using LowpassCoefficients.

`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 product of the number of frequency bands and 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.

Dependencies

This property applies when you set Specification to 'Number of taps per band and stopband attenuation'.

`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. Larger is the stopband attenuation, smaller is the passband ripple.

Dependencies

This property applies when you set Specification to 'Number of taps per band and stopband attenuation'.

Data Types: single | double

`LowpassCoefficients` — Coefficients of the 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].

Tunable: Yes

Dependencies

This property applies when you set Specification to 'Coefficients'.

Data Types: single | double
Complex Number Support: Yes

Usage

Syntax

synthOut = synthesizer(input)

Description

example

synthOut = synthesizer(input) merges the narrowband input signals contained as columns in input into broadband signal, synthOut.

Input Arguments

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`input` — Narrowband signals
matrix | 3-D array

Narrowband signals, specified as a matrix or a 3-D array. Each narrowband signal is stored as a column in the input signal. The number of columns in input corresponds to the number of frequency bands of the filter bank. If input is three-dimensional, each matrix corresponds to a separate channel. If M is the number of frequency bands, and input is an L-by-M matrix, then the output signal, synthOut, has dimensions L×M-by-1. If input has more than one channel, that is, it has dimensions L-by-M-by-N, then synthOut has dimensions L×M-by-N.

This object also accepts variable-size inputs. That is, once the object is locked, you can change the size of each input channel. The number of channels cannot change.

Data Types: single | double
Complex Number Support: Yes

Output Arguments

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`synthOut` — Merged broadband signal
matrix | 3-D array

Merged broadband signal, returned as a matrix or a 3-D array. If M is the number of frequency bands, and input is an L-by-M matrix, then the output signal, synthOut, has dimensions L×M-by-1. If input has more than one channel, that is, it has dimensions L-by-M-by-N, then synthOut has dimensions L×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)

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Specific to dsp.ChannelSynthesizer

`coeffs`	Coefficients of prototype lowpass filter
`tf`	Return transfer function of overall prototype lowpass filter
`polyphase`	Return polyphase matrix

Common to All System Objects

`step`	Run System object algorithm
`release`	Release resources and allow changes to System object property values and input characteristics
`reset`	Reset internal states of System object

Examples

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Quadrature Mirror Filter Bank

Open Script

The quadrature mirror filter bank (QMF) contains an analysis filter bank section and a synthesis filter bank section. dsp.Channelizer implements the analysis filter bank. dsp.ChannelSynthesizer implements the synthesis filter bank using the efficient polyphase implementation based on a prototype lowpass filter.

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,'NumInputPorts',...
    2,'ChannelNames',{'Input','Output'},'Title','Input and Output of QMF');

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

More About

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Synthesis Filter Bank

The synthesis filter bank consists of a set of parallel bandpass filters that merge multiple input narrowband signals, y₀(n), y₁(n),..., y_M-1(n) into a single broadband signal, v(n). The input narrowband signals are in the baseband. Each narrowband signal is interpolated to a higher sampling rate by using the upsampler, and then filtered by the lowpass filter. A complex exponential that follows the lowpass filter centers the baseband signal around w_k.

Prototype Lowpass Filter

To implement the synthesis filter bank efficiently, the synthesizer uses a prototype lowpass filter. This 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 synthesis filter bank. This value corresponds to the FFT length that the filter bank uses. M can be high, in 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 models the first branch of the filter bank. The other M – 1 branches are modeled by filters that are modulated versions of the prototype filter. The modulation factor is given by $e^{j w_{k} n}, w_{k} = 2 π k / M, k = 0, 1, ..., M - 1$ .

The output of each bandpass filter forms a specific portion of the broadband signal. The output of all the branches are added to form the broadband signal, v(n).

Algorithms

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Polyphase Implementation

The synthesis filter bank can be implemented efficiently using the polyphase structure.

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

$H_{0} (z) = b_{0} + b_{1} z^{- 1} + ... + b_{N} z^{- N}$

N+1 is the length of the prototype filter.

You can rearrange this equation as follows:

$H_{0} (z) = \begin{matrix} (b_{0} + b_{M} z^{- M} + b_{2 M} z^{- 2 M} + .. + b_{N - M + 1} z^{- (N - M + 1)}) + \\ z^{- 1} (b_{1} + b_{M + 1} z^{- M} + b_{2 M + 1} z^{- 2 M} + .. + b_{N - M + 2} z^{- (N - M + 1)}) + \\ \begin{matrix} ⋮ \\ z^{- (M - 1)} (b_{M - 1} + b_{2 M - 1} z^{- M} + b_{3 M - 1} z^{- 2 M} + .. + b_{N} z^{- (N - M + 1)}) \end{matrix} \end{matrix}$

M is the number of polyphase components.

You can write this equation as:

$H_{0} (z) = E_{0} (z^{M}) + z^{- 1} E_{1} (z^{M}) + ... + z^{- (M - 1)} E_{M - 1} (z^{M})$

E₀(z^M), E₁(z^M),..., E_M-1(z^M) are polyphase components of the prototype lowpass filter, H₀(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 kth modulated bandpass filter as $H_{k} (z) = H_{0} (z e^{j w_{k}})$ . Replacing z with ze^jw_k,

$H_{k} (z) = h_{0} + h_{1} e^{j w k} z^{- 1} + h_{2} e^{j 2 w k} z^{- 2} ... + h_{N} e^{j N w k} z^{- N}$

N+1 is the length of the kth filter.

In polyphase form, the equation is as follows:

$H_{k} (z) = [1 e^{j w_{k}} e^{j 2 w_{k}} ... e^{j (M - 1) w_{k}}] [\begin{matrix} E_{0} (z^{M}) \\ z^{- 1} E_{1} (z^{M}) \\ ⋮ \\ z^{- (M - 1)} E_{M - 1} (z^{M}) \end{matrix}]$

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

$H (z) = [\begin{matrix} \begin{array}{l} 1 & 1 & 1 & ... & 1 \end{array} \\ \begin{array}{l} 1 & e^{j w_{1}} & e^{j 2 w_{1}} & ... & e^{j (M - 1) w_{1}} \end{array} \\ ⋮ \\ \begin{array}{l} 1 & e^{j w_{M - 1}} & e^{j 2 w_{M - 1}} & ... & e^{j (M - 1) w_{M - 1}} \end{array} \end{matrix}] [\begin{matrix} E_{0} (z^{M}) \\ z^{- 1} E_{1} (z^{M}) \\ ⋮ \\ z^{- (M - 1)} E_{M - 1} (z^{M}) \end{matrix}]$

Here is the multirate noble identity for interpolation, assuming that D = M:

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

Replace H₀(z) with its polyphase representation.

After applying the noble identity for interpolation, you can replace the delays, interpolation factor, and the adder with a commutator switch.

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

$H (z) = [\begin{matrix} \begin{array}{l} 1 & 1 & 1 & ... & 1 \end{array} \\ \begin{array}{l} 1 & e^{j w_{1}} & e^{j 2 w_{1}} & ... & e^{j (M - 1) w_{1}} \end{array} \\ ⋮ \\ \begin{array}{l} 1 & e^{j w_{M - 1}} & e^{j 2 w_{M - 1}} & ... & e^{j (M - 1) w_{M - 1}} \end{array} \end{matrix}] [\begin{matrix} E_{0} (z) \\ E_{1} (z) \\ ⋮ \\ E_{M - 1} (z) \end{matrix}]$

The matrix on the left is an IDFT matrix. With the IDFT matrix, the efficient implementation of the lowpass prototype based filter bank looks like the following.

References

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

[2] Harris, F.J., Chris Dick, Michael Rice. "Digital Receivers and Transmitters Using Polyphase Filter Banks for Wireless Communications." IEEE Transactions on microwave theory and techniques. Vol. 51, Number 4, April 2003.

Documentation

dsp.ChannelSynthesizer

Description

Creation

Syntax

Description

Properties

Specification — Filter design parameters or coefficients 'Number of taps per band and stopband attenuation' (default) | 'Coefficients'

NumTapsPerBand — Number of filter coefficients per frequency band 12 (default) | positive integer

Dependencies

StopbandAttenuation — Stopband attenuation 80 (default) | positive real scalar

Dependencies

LowpassCoefficients — Coefficients of the prototype lowpass filter [1×49 double] (default) | row vector

Dependencies

Usage

Syntax

Description

Input Arguments

input — Narrowband signals matrix | 3-D array

Output Arguments

synthOut — Merged broadband signal matrix | 3-D array

Object Functions

Specific to dsp.ChannelSynthesizer

Common to All System Objects

Examples

Quadrature Mirror Filter Bank

More About

Synthesis Filter Bank

Prototype Lowpass Filter

Algorithms

Polyphase Implementation

References

Extended Capabilities

C/C++ Code Generation Generate C and C++ code using MATLAB® Coder™.

See Also

Functions

Objects

Blocks

Functions

DSP System Toolbox Documentation

Support

`Specification` — Filter design parameters or coefficients
`'Number of taps per band and stopband attenuation'` (default) | `'Coefficients'`

`NumTapsPerBand` — Number of filter coefficients per frequency band
`12` (default) | positive integer

`StopbandAttenuation` — Stopband attenuation
`80` (default) | positive real scalar

`LowpassCoefficients` — Coefficients of the prototype lowpass filter
`[1×49 double]` (default) | row vector

`input` — Narrowband signals
matrix | 3-D array

`synthOut` — Merged broadband signal
matrix | 3-D array

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.