Channelizer

Polyphase FFT analysis filter bank

Library:
DSP System Toolbox / Filtering / Multirate Filters

Description

The Channelizer block separates a broadband input signal into multiple narrow subbands using an 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. When you specify the design parameters, the filter is designed using the designMultirateFIR function.

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

Ports

Input

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`x` — Broadband signal
L-by-1 column vector | L-by-N matrix

Input broadband signal, which the channelizer splits into multiple narrow bands. 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.

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

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`Port_1` — Multiple narrowband signals
L/M-by-M | L/M-by-M-by-N array

Multiple narrow subbands of the input broadband signal. Each narrow band signal forms a column in the output.

If the input is one of the following:

L-by-1 column vector — The output is a L/M-by-M matrix. M is the number of frequency bands.
L-by-N matrix — The output is a L/M-by-M-by-N matrix.

Data Types: single | double
Complex Number Support: Yes

Parameters

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

`Number of frequency bands` — Number of frequency bands
`8` (default) | positive integer greater than 1

Number of frequency bands into which the block separates the input broadband signal. This parameter indicates the FFT length and the decimation factor used by the 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.

`Oversampling ratio` — Oversampling ratio
`1` (default) | positive integer

Oversampling ratio, specified as a positive scalar divisor of the number of frequency bands.

If the value is greater than 1, the output sample rate is different from the channel spacing. The channelizer is then known as the non-maximally decimated channelizer or oversampled channelizer. For more details, see Algorithm.

`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 Number of frequency bands × Number of filter taps per frequency band. 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 parameter, the lowpass filter coefficients are input through the coeffs port. When you clear this parameter, the coefficients are specified on the block dialog through the Prototype lowpass filter coefficients parameter.

Dependencies

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

`Prototype lowpass filter coefficients` — Coefficients of prototype lowpass filter
`rcosdesign(0.25,6,8,'sqrt')` (default) | row vector

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.

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`

Interpreted execution
Simulate model using the MATLAB^® interpreter. This option shortens startup time and has faster simulation speed compared to Code generation.
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.

Model Examples

High Resolution Filter-Bank-Based Power Spectrum Estimation

Perform high resolution spectral analysis by using an efficient polyphase filter bank sometimes referred to as a channelizer.

Synthesize and Channelize Audio

Synthesize and channelize audio signals.

Block Characteristics

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

More About

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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 H₀(z) in the filter bank. The first branch of the filter bank contains H₀(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:

$e^{- j w_{k} n}, w_{k} = 2 π k / M, k = 0, 1, ..., M - 1$

Using the Prototype Lowpass Filter

The transfer function of the modulated kth bandpass filter is given by:

$H_{k} (z) = H_{0} (z e^{- j w_{k}}), w_{k} = 2 π k / M, k = 1, 2, ..., M - 1$

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, H₀(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, $e^{- j w_{k} n}, w_{k} = 2 π k / M, k = 1, 2, .., M - 1$ , where $w_{k} = 2 π k / M$ , and $k = 1, 2, ..., M - 1$ . The resulting product signals are passed through the lowpass filters, H₀(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.

y₁(n), y₂(n), ..., y_M-1(n) are narrow subband signals translated into baseband.

Algorithms

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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:

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

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:

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

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

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

Maximally decimated channelizer (D = M)

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 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 decimation, you can replace the delays and the decimation factor with a commutator switch. The switch starts on the first branch and moves in the counter clockwise direction as shown in the following diagram.

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

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

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 the following.

The switch starts on the first branch 0, delivers one sample at a time to each branch, and progresses in the counter clockwise direction through the branches 0, M−1, M−2, all the way up to branch 1. When the first set of M input samples are delivered to the M branches of the polyphase structure, the channelizer computes the first set of output values. Hence, the sample rate at the output of the maximally decimated channelizer is fs/M.

When the next set of input data samples are available, the switch starts at branch 0, and delivers these samples one at a time in the counter clock wise direction. Every time the polyphase structure receives a new set of M input samples, the channelizer computes a new set of output values.

Non-maximally decimated channelizer (D < M)

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. In addition to that, the non-maximally decimated channelizers offer increased design freedom, but at the expense of increasing computational cost.

The number of frequency bands, M = RD, where R = 1, 2,…, M, is known as the oversampling ratio, and D is the decimation factor.

When R = 1, M equals D, and the decimator is known as the maximally decimated channelizer. The commutator switch in the filter bank starts at branch 0 and delivers one sample at a time to each branch in the counter clockwise direction. Once all M branches have a data sample, the filter bank computes the output data. For more details, see Maximally decimated channelizer.

When R > 1, the switch starts at the branch (M/R) − 1, loads D samples, delivers one sample at a time to each branch in the counter clockwise direction, and progresses up the stack to branch 0. When a new set of D input samples come in, these samples are delivered to the first set of M/R addresses. The formal contents of these addresses are shifted to the next set of M/R addresses, and this process of data shift continues every time there is a new set of D input samples. All the samples undergo a serpentine shift.

For every D input samples that are fed to the polyphase structure, the channelizer outputs M samples, y₀(m), y₁(m), … , y_M-1(m). This process increases the output sample rate from f_s/M in the case of maximally decimated channelizer, to Rf_s/M in the case of 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 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. 51, no. 4 (2003).

Documentation

Channelizer

Description

Ports

Input

x — Broadband signal L-by-1 column vector | L-by-N matrix

coeffs — Prototype lowpass filter coefficients row vector

Dependencies

Output

Port_1 — Multiple narrowband signals L/M-by-M | L/M-by-M-by-N array

Parameters

Number of frequency bands — Number of frequency bands 8 (default) | positive integer greater than 1

Polyphase filter specification — Filter design parameters or coefficients Number of taps per band and stopband attenuation (default) | Coefficients

Oversampling ratio — Oversampling ratio 1 (default) | positive integer

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

Dependencies

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

Dependencies

Specify coefficients from input port — Flag to specify lowpass filter coefficients off (default) | on

Dependencies

Prototype lowpass filter coefficients — Coefficients of prototype lowpass filter rcosdesign(0.25,6,8,'sqrt') (default) | row vector

Dependencies

Simulate using — Type of simulation to run Interpreted execution (default) | Code generation

Model Examples

High Resolution Filter-Bank-Based Power Spectrum Estimation

Synthesize and Channelize Audio

Block Characteristics

More About

Analysis Filter Bank

Prototype Lowpass Filter

Using the Prototype Lowpass Filter

Shift Narrow Subbands to Baseband

Algorithms

Polyphase Implementation

References

Extended Capabilities

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

See Also

Blocks

Objects

DSP System Toolbox Documentation

Support

`x` — Broadband signal
L-by-1 column vector | L-by-N matrix

`coeffs` — Prototype lowpass filter coefficients
row vector

`Port_1` — Multiple narrowband signals
L/M-by-M | L/M-by-M-by-N array

`Number of frequency bands` — Number of frequency bands
`8` (default) | positive integer greater than 1

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

`Oversampling ratio` — Oversampling ratio
`1` (default) | positive integer

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

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

`Specify coefficients from input port` — Flag to specify lowpass filter coefficients
off (default) | on

`Prototype lowpass filter coefficients` — Coefficients of prototype lowpass filter
`rcosdesign(0.25,6,8,'sqrt')` (default) | row vector

`Simulate using` — Type of simulation to run
`Interpreted execution` (default) | `Code generation`

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