Dyadic Synthesis Filter Bank (DSP Blockset)

Reconstruct a signal from its multirate bandlimited components.

Library

Multirate Filters, in Filtering

Description

The Dyadic Synthesis Filter Bank block typically reconstructs a signal that was decomposed by the Dyadic Analysis Filter Bank block. The reconstruction or synthesis process is the inverse of the analysis process, and restores the original signal by upsampling, filtering, and summing the bandlimited inputs in stages corresponding to the analysis process. The typical (asymmetric) n-level filter bank structure is shown below:

At each level, the two bandlimited inputs (one low-frequency, one high-frequency, both with the same sample rate) are upsampled by a factor of 2 to match the sample rate of the input to the next stage. They are then filtered by a highpass (HP) and lowpass (LP) filter pair with coefficients calculated to cancel (in the subsequent summation) the aliasing introduced in the corresponding dyadic analysis filter stage. The output from each (upsample-filter-sum) level has twice the bandwidth and twice the sample rate of the input to that level (hence "dyadic").

The Lowpass FIR filter coefficients and Highpass FIR filter coefficients parameters specify the filter coefficients to be used for every highpass and lowpass (respectively) filter in the structure. The values of these coefficients are typically computed together with the dyadic analysis coefficients using the wavelet family functions in the Wavelet Toolbox (see the Wavelet Toolbox User's Guide for more information).

The Tree structure parameter specifies an asymmetric (or wavelet) tree, as shown above, or a symmetric structure, as shown below. Note that the symmetric structure reconstructs a signal that was symmetrically decomposed by the Dyadic Analysis Filter Bank block (i.e., both the high- and low-frequency subbands were divided at each level). The asymmetric structure reconstructs a signal that was asymmetrically decomposed by the Dyadic Analysis Filter Bank block (i.e., only the low-frequency subbands were divided at each level).

The asymmetric structure in the first figure (Tree structure set to Asymmetric) has n+1 inputs, where n is the number of levels. The sample rate and bandwidth of the output are twice the sample rate and bandwidth of the top input. The sample rate and bandwidth of each additional input (except the last) are half that of the input to the previous level.

The bottom two inputs share the same sample rate and bandwidth since they are processed by the same level.

and

Note that in sample-based mode, the different sample rates described above are represented by different port rates at the block inputs. In frame-based mode, the different input sample rates are reflected in the input frame sizes rather than the actual input port periods.

The symmetric structure (Tree structure set to Symmetric) has 2ⁿ inputs, where n is the number of levels. The sample rate and bandwidth of each input are equal.

The Frame-based inputs parameter allows you to choose between sample-based and frame-based operation.

Sample-Based Operation

When the check box is not selected (default), the block assumes that the input is a 1-by-N sample vector or M-by-N sample matrix. Each of the N vector elements (or M*N matrix elements) is treated as an independent channel, and the block filters each channel independently over time. The output is the same size as the input at each port, with one channel for each input channel. As described earlier, for the asymmetric tree structure, each input port has a different period.

Example:

Frame-Based Operation

When the Frame-based inputs check box is selected, the block assumes that the input at each port is an M-by-N frame matrix. Each of the N frames in the matrix contains M sequential time samples from an independent signal. The illustration below shows a 8-by-4 matrix input.

The Number of channels parameter specifies the number of independent channels (columns), N, in the matrix, and the block filters each channel independently over time. The number of columns in each output is therefore the same as the number of columns in the input.

Note that each input port has the same sample period as the output port. The increase in the output sample rate results from the larger output frame size, as shown in the example below.

Frame-based operation provides substantial increases in throughput rates, at the expense of greater model latency.

Applications

The primary application for asymmetric dyadic synthesis filter banks is coding for compression using wavelets.

At the transmitting end, the output of a dyadic analysis filter bank is fed to a lossy compression scheme, which typically assigns the number of bits for each filter bank output in proportion to the relative energy in that frequency band. This represents the more powerful components of the signal by a greater number of bits than the less powerful signal components.

At the receiving end, the transmission is decoded and fed to the dyadic synthesis filter bank to reconstruct the original signal. The filter coefficients of the complementary analysis and synthesis stages are designed to cancel aliasing introduced by the filtering and resampling.

Note
If you expect to generate code for the Dyadic Synthesis Filter Bank block using the Real-Time Workshop, you should ensure that inputs are contiguous in memory. See the Contiguous Copy block for more information.

Dialog Box

Lowpass FIR filter coefficients: A vector of filter coefficients (descending powers of z) to be shared by all the lowpass filters in the filter bank.
Highpass FIR filter coefficients: A vector of filter coefficients (descending powers of z) to be shared by all the highpass filters in the filter bank.
Number of levels: The number of filter bank levels. An n-level asymmetric structure has n+1 inputs; an n-level symmetric structure has 2ⁿ inputs.
Tree structure: The structure of the filter bank, Asymmetric (wavelet) or Symmetric.
Frame-based inputs: Selects frame-based operation.
Number of channels: For frame-based operation, the number of columns (channels) in the input matrix.

References

Fliege, N. J. Multirate Digital Signal Processing: Multirate Systems, Filter Banks, Wavelets. West Sussex, England: John Wiley & Sons, 1994.

Strang, G. and T. Nguyen. Wavelets and Filter Banks. Wellesley, MA: Wellesley-Cambridge Press, 1996.

Vaidyanathan, P. P. Multirate Systems and Filter Banks. Englewood Cliffs, NJ: Prentice Hall, 1993.