Channel Synthesizer

Polyphase FFT synthesis filter bank

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Libraries:
DSP System Toolbox / Filtering / Multirate Filters

Description

The Channel Synthesizer block 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. When you specify the design parameters, the filter is designed using the designMultirateFIR function.

This block also accepts variable-size inputs. That is, during the simulation, you can change the size of each input channel. The number of channels cannot change.

Examples

Synthesize and Channelize Audio in Simulink

Synthesize and channelize audio signals.

• Open Model

Ports

Input

x — Input narrowband signals

Q-by-M matrix | L-by-M-by-N array

Input narrowband signals, which the channel synthesizer merges to form the broadband signal. Each narrowband signal forms a column in the input signal. The number of columns in the input correspond to the number of frequency bands of the filter bank. If the input is three-dimensional, each matrix corresponds to a separate channel.

This port is unnamed until you set Polyphase filter specification to Coefficients and select the Specify coefficients from input port parameter.

Data Types: single | double
Complex Number Support: Yes

coeffs — Prototype lowpass filter coefficients

row vector

Coefficients of the prototype lowpass filter. 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 block zero-pads the coefficients.

If you specify complex coefficients, the block 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].

Dependencies

This port appears when you set Polyphase filter specification to Coefficients and select the Specify coefficients from input port parameter.

Data Types: single | double
Complex Number Support: Yes

Output

Port_1 — Broadband signal

Q×M-by-1 vector | L×M-by-N matrix

Broadband signal the channel synthesizer forms from the multiple input narrow subbands.

If the input is one of the following:

• Q-by-M matrix — The output is a Q×M-by-1 vector. M is the number of frequency bands.

• Q-by-M-by-N matrix — The output is a Q×M-by-N matrix.

Data Types: single | double
Complex Number Support: Yes

Parameters

If a parameter is listed as tunable, then you can change its value during simulation.

InterpolationFactor — Interpolation factor

Number of frequency bands (default) | positive integer

Since R2024a

Specify the interpolation factor L as one of these:

• Number of frequency bands –– When you change the number of frequency bands, the interpolation factor changes automatically.

• Positive integer greater than or equal to the number of frequency bands M.

The number of frequency bands equals the number of channels in the input signal. If the interpolation factor L equals the number of frequency bands M, then the L/M ratio equals 1, and the synthesizer is known as the critically sampled channel synthesizer.

If L is greater than M, then the L/M ratio must be an integer. For more details, see Algorithm.

Polyphase filter specification — Filter design parameters or coefficients

Number of taps per band and stopband attenuation (default) | Coefficients

• Number of taps per band and stopband attenuation — Specify the filter design parameters through the Number of filter taps per frequency band and Stopband attenuation (dB) parameters. When you specify the design parameters, the filter is designed using the designMultirateFIR function.

• Coefficients — Specify the filter coefficients directly using the Prototype lowpass filter coefficients parameter or input them through the coeffs port.

Number of filter taps per frequency band — Number of filter coefficients per frequency band

12 (default) | positive integer

Number of filter coefficients that each polyphase branch uses. 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 the product of the number of frequency bands and the number of filter taps per frequency band. The number of frequency bands equals the number of columns in the input. 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

To enable this parameter, set Polyphase filter specification to Number of taps per band and stopband attenuation.

Stopband attenuation (dB) — Stopband attenuation

80 (default) | positive real scalar

Stopband attenuation of the lowpass filter, in dB. This value controls the maximum amount of aliasing from one frequency band to the next. As the stopband attenuation increases, the passband ripple decreases.

Dependencies

To enable this parameter, set Polyphase filter specification to Number of taps per band and stopband attenuation.

Specify coefficients from input port — Flag to specify lowpass filter coefficients

off (default) | on

When you select this check box, the lowpass filter coefficients are input through the coeffs port. When you clear this check box, the coefficients are specified on the block dialog through the Prototype lowpass filter coefficients parameter.

Dependencies

To enable this parameter, set Polyspace filter specification to Coefficients.

Prototype lowpass filter coefficients — Coefficients of prototype lowpass filter

rcosdesign(0.25,6,8,'sqrt') (default) | row vector

This parameter specifies the coefficients of the prototype lowpass filter. The default value is the coefficients vector that rcosdesign(0.25,6,8,'sqrt') returns. 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 block zero-pads the coefficients.

If you specify complex coefficients, the block 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

To enable this parameter, set Polyphase filter specification to Coefficients and clear the Specify coefficients from input port parameter.

Complex Number Support: Yes

Simulate using — Type of simulation to run

Interpreted execution (default) | Code generation

Specify the type of simulation to run. You can set this parameter to:

• Interpreted execution –– Simulate model using the MATLAB^® interpreter. This option shortens startup time.

• Code generation –– Simulate model using generated C code. The first time you run a simulation, Simulink^® generates C code for the block. The C code is reused for subsequent simulations as long as the model does not change. This option requires additional startup time but provides faster subsequent simulations.

Block Characteristics

Data Types	double | single
Multidimensional Signals	No
Variable-Size Signals	Yes

More About

Synthesis Filter Bank

The synthesis filter bank consists of a set of parallel bandpass filters that merge multiple input narrowband signals, y₀[m], y₁[m], … , y_M-1[m] 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 $$e^{j{w}_{k}n},w_k=2\pi k/M,k=0,1,\mathrm{...},M-1$$.

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

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.

$$H_0(z)=b_0+b_1{z}^{-1}+\mathrm{...}+b_N{z}^{-N}$$

N + 1 is the length of the prototype filter.

You can rearrange this equation as follows:

$$H_0(z)=\begin{array}{c}\left(b_0+b_L{z}^{-L}+b_{2L}{z}^{-2L}+\mathrm{..}+b_{N-L+1}{z}^{-(N-L+1)}\right)+\\ z^{-1}\left(b_1+b_{L+1}{z}^{-L}+b_{2L+1}{z}^{-2L}+\mathrm{..}+b_{N-L+2}{z}^{-(N-L+1)}\right)+\\ \begin{array}{c}⋮\\ z^{-(L-1)}\left(b_{L-1}+b_{2L-1}{z}^{-L}+b_{3L-1}{z}^{-2L}+\mathrm{..}+b_N{z}^{-(N-L+1)}\right)\end{array}\end{array}$$

L is the number of polyphase components and equals the interpolation factor.

You can write this equation as:

$$H_0(z)=E_0\left(z^L\right)+z^{-1}{E}_1\left(z^L\right)+\mathrm{...}+z^{-(L-1)}{E}_{L-1}\left(z^L\right)$$

E₀(z^L), E₁(z^L), … , E_L−1(z^L) 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 the prototype filter, H₀(z).

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,

$$H_k(z)=h_0+h_1{e}^{jwk}{z}^{-1}+h_2{e}^{j2wk}{z}^{-2}\mathrm{...}+h_N{e}^{jNwk}{z}^{-N}$$

N + 1 is the length of the kth filter.

In polyphase form, the equation is as follows:

$$H_k(z)=\left[1e^{j{w}_k}{e}^{j2w_k}\mathrm{...}{e}^{j(L-1)w_k}\right]\left[\begin{array}{c}{E}_0\left(z^L\right)\\ z^{-1}{E}_1\left(z^L\right)\\ ⋮\\ z^{-(L-1)}{E}_{L-1}\left(z^L\right)\end{array}\right]$$

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

$$H(z)=\left[\begin{array}{ccccc}1& 1& 1& \mathrm{...}& 1\\ 1& e^{j{w}_1}& e^{j2w_1}& \mathrm{...}& e^{j\left(L-1\right)w_1}\\ & & ⋮& & \\ 1& e^{j{w}_{M-1}}& e^{j2w_{M-1}}& \mathrm{...}& e^{j(L-1)w_{M-1}}\end{array}\right]\left[\begin{array}{c}{E}_0\left(z^L\right)\\ z^{-1}{E}_1\left(z^L\right)\\ ⋮\\ z^{-(L-1)}{E}_{L-1}\left(z^L\right)\end{array}\right]$$

Here is the multirate noble identity for interpolation:

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:

$$H(z)=\left[\begin{array}{ccccc}1& 1& 1& \mathrm{...}& 1\\ 1& e^{j{w}_1}& e^{j2w_1}& \mathrm{...}& e^{j(L-1)w_1}\\ & & & & \\ 1& e^{j{w}_{M-1}}& e^{j2w_{M-1}}& \mathrm{...}& e^{j(L-1)w_{M-1}}\end{array}\right]\left[\begin{array}{c}{E}_0(z)\\ E_1(z)\\ ⋮\\ E_{L-1}(z)\end{array}\right]$$

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, and 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.

Extended Capabilities

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

Version History

Introduced in R2017a

R2024a: Support for integer oversampling

Use the new Interpolation factor parameter to specify an interpolation factor L that is greater than or equal to the number of frequency bands M. If L > M, then the ratio L/M must be an integer and the synthesizer is known as an oversampled channel synthesizer.