EngeeDSP.Channelizer

A standard set of filters for analysis consists of a series of parallel bandpass filters that separate the input broadband signal. on a number of narrow sub-bands. Each bandpass filter stores a different part of the input signal. After reducing the bandwidth by one of the bandpass filters, the signal is sampled at a lower sampling rate corresponding to the new bandwidth.

To effectively implement a set of filters for analysis, the channel converter uses a prototype low-pass filter.

The prototype low-pass filter has a pulse response , normalized two-way bandwidth and the cutoff frequency . — the number of frequency bands, that is, the branches of the filter set for analysis. This value corresponds to the length of the FFT used by the filter set. it can be large, on the order of 2048 or more. The attenuation in the delay band determines the minimum level of interference (contour distortion) from one frequency band to another. The ripples in the bandwidth must be small so that the input signal is not distorted in the bandwidth.

The prototype of the low-pass filter corresponds to in the filter set. The first branch of the filter set contains , followed by a decimator. The others the branches contain filters that are modulated versions of the prototype filter. The modulation coefficient is determined by the following equation:

The transfer function of the modulated The th bandpass filter is given by the formula

This figure shows the frequency response filters.

To obtain the frequency response of the filter , where , it is necessary to evenly shift the frequency response of the prototype filter by a multiple of . Each subband filter , where , is derived from the filter prototype.

Below is an equivalent representation of the frequency response diagram with , varying in the range of .

Frequency components of the input signal are converted to a frequency band by multiplication for complex exponents:

The resulting signals pass through low-pass filters . The output signal of the low-pass filter has a relatively narrow bandwidth. The frequency of the signal is lowered in accordance with the new bandwidth. The decimation coefficient is selected , where — the number of branches of the filter set for analysis. If A channel converter is called a channel converter with oversampling or without maximum decimation.

The figure shows a set of filters for analysis using a prototype low-pass filter.

— signals of narrow sub-bands converted to the main frequency band.

A set of filters for analysis can be effectively implemented using a polyphase structure. For more information about the set of filters for analysis, see A set of filters for analysis.

To obtain a polyphase structure, let’s start with the transfer function of the prototype low-pass filter.:

where — the length of the filter prototype.

This equation can be reformulated as follows:

where — the number of polyphase components.

This equation can be written as follows:

where — polyphase components of the low-pass filter prototype .

The rest of the filters in the filter set , where , are modulated versions of this prototype filter.

The transfer function The th modulated bandpass filter can be written as .

Replacing on and ,

In the polyphase form, the equation looks like this:

For everyone channels in the filter set transfer function It is set as follows:

When A channel converter is called a channel converter with maximum decimation or a channel converter with critical sampling.

Here is a multi-rate remarkable identity for frequency reduction (decimation), suggesting that .

For example, consider the first branch of a set of filters containing a low-pass filter.

Replace its polyphase representation.

After applying the remarkable decimation identity, it is possible to replace the delays and decimation coefficient with a switching device. The switching device starts moving from the first branch 0 and moves counterclockwise, as shown in the following diagram. The output battery receives processed input samples from each branch of the polyphase structure and accumulates these processed samples until the switching device moves to branch 0. When the switching device moves to branch 0, the battery outputs the accumulated value.

For everyone channels in the filter set transfer function It is set as follows:

The matrix on the left is the inverse discrete Fourier transform (ODFT) matrix. Using the ODPF matrix, the effective implementation of a set of filters based on a prototype low-pass filter is as follows.

When the channel converter receives the first input sample, the switching device feeds this input to branch 0, and the channel converter calculates the first set of output values. As new input samples arrive, the switching device moves counterclockwise through the branches , and up to branch 0, applying one count to each branch. When the switching device reaches branch 0, the channel converter outputs the following set of output values. This process continues as the data becomes available. Each time the switching device reaches the first branch 0, the channel converter outputs . Each branch of the channel converter actually outputs one sample for each received counts. Therefore, the sampling frequency at the output of the channel converter is .

If you use -point FFT, the implementation will look like this.

When the switching device transmits the first sample to branch 0, the channel converter calculates the first set of output values. As new data arrives, the switching device moves clockwise to the branches , and so on to branch 1, passing one count to each branch. When the switching device reaches branch 0, the channel converter outputs the following set of output values. This process continues as new data becomes available.

When A channel converter is called a channel converter without maximizing the sampling rate or a channel converter with oversampling. In this configuration, the sampling frequency of the output signal differs from the step between the channels. Channel converters provide greater design freedom without maximizing the sampling rate, but at the cost of increased computational costs.

If the relation is is equal to an integer greater than 1 and less or equal , a channel converter is called a channel converter with integer oversampling. If the relation is If it is not an integer, then the channel converter is called a channel converter with rational oversampling.

In this configuration, when the first input sample arrives, the switching device feeds this input to branch 0, and the channel converter calculates the first set of output values. As new input samples arrive, the switching device moves counterclockwise through the branches , and up to branch 0, feeding one count at a time to each branch. When the switching device switches to branch 0, the channel converter outputs the following set of output values. This process continues as the data becomes available. Each time the switching device switches to the first branch 0, the channel converter outputs .

As new data arrives and these samples are sent by the switching device to the first addresses, the formal contents of these addresses are shifted to the following set addresses, and this data shifting process continues each time a new set of input counts.

For each The channel converter outputs input samples supplied to the polyphase structure. counts, . This process increases the sampling rate of the output data with in the case of a channel converter with maximum decimation up to in the case of a channel converter without maximum decimation.

For more information, see [2].

After each sequence of data -points are received on a divided by -stepwise polyphase filter, output signals -steps are calculated and processed for submission to -point FFT. Shifting the data through the filter introduces a frequency-dependent phase shift. To correct this phase shift and superimpose all bands on a constant component, after the polyphase filters and before -a cyclic shift buffer is inserted with a point FFT.

With a switchboard followed by -a stepwise polyphase filter, a cyclic shift buffer and a DFT matrix, an effective implementation of a set of filters based on a prototype low-pass filter is as follows.