Dyadic Synthesis Filter Bank (DSP Blockset)

Reconstruct a signal from subbands with smaller bandwidths and slower sample rates

Library

Description

Note This block always outputs frame-based signals, and its inputs must be of certain sizes. To get sample-based outputs or to use input subbands that do not fit the criteria of this block, use the Two-Channel Synthesis Subband Filter block. (You can connect multiple copies of the Two-Channel Synthesis Subband Filter block to create a multilevel dyadic synthesis filter bank.)

The Dyadic Synthesis Filter Bank block reconstructs a signal decomposed by the Dyadic Analysis Filter Bank block. The block takes in subbands of a signal, and uses them to reconstruct the signal by using a series of highpass and lowpass FIR filters as illustrated in Figure 7-9, n-Level Asymmetric Dyadic Synthesis Filter Bank,. The reconstructed signal has a wider bandwidth and faster sample rate than the input subbands.

You can specify the filter bank's highpass and lowpass filters by providing vectors of filter coefficients. If you install the Wavelet Toolbox, you can also specify wavelet-based filters by selecting a wavelet from the Filter parameter.

Note To use a dyadic synthesis filter bank to perfectly reconstruct the output of a dyadic analysis filter bank, the number of levels and tree structures of both filter banks must be the same. In addition, the filters in the synthesis filter bank must be designed to perfectly reconstruct the outputs of the analysis filter bank. Otherwise, the reconstruction will not be perfect.

This block automatically computes wavelet-based perfect reconstruction filters if the wavelet selection in the Filter parameter of this block is the same as the Filter parameter setting of the corresponding Dyadic Analysis Filter Bank block. The use of wavelets requires the Wavelet Toolbox. To learn how to design your own perfect reconstruction filters, see References.

For more information about filter banks and the block, see the other sections of this reference page.

Sections of This Reference Page

Review of Dyadic Synthesis Filter Banks

Dyadic synthesis filter banks are constructed from the following basic unit. The unit can be cascaded to construct dyadic synthesis filter banks with either a asymmetric or symmetric tree structure as illustrated in Figure 7-9, n-Level Asymmetric Dyadic Synthesis Filter Bank, and Figure 7-10, n-Level Symmetric Dyadic Synthesis Filter Bank,.

Each unit consists of a lowpass (LP) and highpass (HP) FIR filter pair, preceded by an interpolation by a factor of 2. The filters are halfband filters with a cutoff frequency of F_s/ 4, a quarter of the input sampling frequency. Each filter passes the frequency band that the other filter stops.

The unit takes in adjacent high-frequency and low-frequency subbands, and reconstructs them into a wide-band signal. Compared to each subband input, the output has twice the bandwidth and twice the sample rate.

Note The following figures illustrate the concept of a filter bank, but not how the block implements a filter bank; the block uses a more efficient polyphase implementation.

Figure 7-9: n-Level Asymmetric Dyadic Synthesis Filter Bank

Use the above figure and the following figure to compare the two tree structures of the dyadic synthesis filter bank. Note that in the asymmetric structure, the low-frequency subband input to each level is the output of the previous level, while the high-frequency subband input to each level is an input to the filter bank. In the symmetric structure, both the low- and high-frequency subband inputs to each level are outputs from the previous level.

Figure 7-10: n-Level Symmetric Dyadic Synthesis Filter Bank

The following table summarizes the key characteristics of symmetric and asymmetric dyadic synthesis filter banks.

Notable Characteristics of Asymmetric and Symmetric Dyadic Synthesis Filter Banks

n-level Symmetric
n-level Asymmetric

Input Paths Through the Filter Bank
The low-frequency subband input to each level (except the first) is the output of the previous level. The low-frequency subband input to the first level, and the high-frequency subband input to each level, are inputs to the filter bank.
Both the high-frequency and low-frequency input subbands to each level (except the first) are the outputs of the previous level. The inputs to the first level are the inputs to the filter bank.

Number of Input Subbands
2ⁿ
n+1

Bandwidth and
Number of Samples in Input Subbands
All inputs subbands have bandwidth BW / 2ⁿ and N / 2ⁿ samples, where the output has bandwidth BW and N samples.
For an output with bandwidth BW and N samples, the kth input subband has the following bandwidth and number of samples.

Input Sample Periods
All input subbands have a sample period of 2ⁿ(T_so), where the output sample period is T_so.

Sample period of kth input subband

where the output sample period is T_so.

Total Number of
Input Samples
The number of samples in the output is always equal to the total number of samples in all of the input subbands.

Wavelet Applications
In wavelet applications, the highpass and lowpass wavelet-based filters are carefully selected so that the aliasing introduced by the decimation in the dyadic analysis filter bank is exactly canceled in the reconstruction of the signal in the dyadic synthesis filter bank.

**Notable Characteristics of Asymmetric and Symmetric Dyadic Synthesis Filter Banks**
	n-level Symmetric	n-level Asymmetric
Input Paths Through the Filter Bank	The low-frequency subband input to each level (except the first) is the output of the previous level. The low-frequency subband input to the first level, and the high-frequency subband input to each level, are inputs to the filter bank.	Both the high-frequency and low-frequency input subbands to each level (except the first) are the outputs of the previous level. The inputs to the first level are the inputs to the filter bank.
Number of Input Subbands	2ⁿ	n+1
Bandwidth and Number of Samples in Input Subbands	All inputs subbands have bandwidth BW / 2ⁿ and N / 2ⁿ samples, where the output has bandwidth BW and N samples.	For an output with bandwidth BW and N samples, the kth input subband has the following bandwidth and number of samples.
Input Sample Periods	All input subbands have a sample period of 2ⁿ(T_so), where the output sample period is T_so.	Sample period of kth input subband where the output sample period is T_so.
Total Number of Input Samples	The number of samples in the output is always equal to the total number of samples in all of the input subbands.
Wavelet Applications	In wavelet applications, the highpass and lowpass wavelet-based filters are carefully selected so that the aliasing introduced by the decimation in the dyadic analysis filter bank is exactly canceled in the reconstruction of the signal in the dyadic synthesis filter bank.

Input Requirements (Setting the Input Parameter)

The inputs to this block are usually the outputs of a Dyadic Analysis Filter Bank block. Since the Dyadic Analysis Filter Bank block can output from either a single port or multiple ports, the Dyadic Synthesis Filter Bank block accepts inputs to either a single port or multiple ports.

The Input parameter sets whether the block accepts inputs from a single port or multiple ports, and thus determines the input requirements, as summarized in the following lists and figure.

Note Any output of a Dyadic Analysis Filter Bank block whose parameter settings match the corresponding settings of this block is a valid input to this block. For example, the setting of the Dyadic Analysis Filter Bank block parameter, Output, must be the same as this block's Input parameter (Single port or Multiple ports).

Valid Inputs for Input = Single port.

Inputs must be sample-based vectors or sample-based matrices of concatenated subbands.
Each input column contains the subbands for an independent signal
Upper input rows contain the high-frequency subbands, and the lower rows contain the low-frequency subbands.

Valid Inputs for Input = Multiple ports.

Inputs must be a frame-based vector or frame-based matrix for each subband, each of which is input to a separate input port.
The columns of each input contains a subband for an independent signal
The input to the topmost input port is the subband containing the highest frequencies, and the input to the bottommost port is the subband containing the lowest frequencies.

Figure 7-11: Valid Inputs to a 3-Level Asymmetric Dyadic Synthesis Filter Bank

For general information about the filter banks, see Review of Dyadic Synthesis Filter Banks.

Output Characteristics

The following table summarizes the output characteristics for both frame-based inputs, and concatenated subband inputs. For an illustration of why the output characteristics exist, see Figure 7-11, Valid Inputs to a 3-Level Asymmetric Dyadic Synthesis Filter Bank,.

Frame-Based Inputs
(Input = Multiple ports)
Concatenated Subband Inputs (Input = Single port)

Output Frame Status
Outputs are always frame based regardless of the input frame status. Each output column is an independent channel, reconstructed from the corresponding channel in the inputs.

Output Frame Rate
Same as the input frame rate.
Same as the input rate (the rate of the concatenated subband inputs).

Output Frame Dimensions

The output has the same number of columns as the inputs.

The number of output rows depends on the tree structure of the filter bank:

Asymmetric -- The number of output rows is twice the number of rows in the input to the topmost input port.

Symmetric -- The number of output rows is the product of the number of input ports and the number of rows in an input to any input port.

The output has the same number of rows and columns as the input.

	Frame-Based Inputs (Input = Multiple ports)	Concatenated Subband Inputs (Input = Single port)
Output Frame Status	Outputs are always frame based regardless of the input frame status. Each output column is an independent channel, reconstructed from the corresponding channel in the inputs.
Output Frame Rate	Same as the input frame rate.	Same as the input rate (the rate of the concatenated subband inputs).
Output Frame Dimensions	The output has the same number of columns as the inputs. The number of output rows depends on the tree structure of the filter bank: Asymmetric -- The number of output rows is twice the number of rows in the input to the topmost input port. Symmetric -- The number of output rows is the product of the number of input ports and the number of rows in an input to any input port.	The output has the same number of rows and columns as the input.

For general information about the filter banks, see Review of Dyadic Synthesis Filter Banks.

Specifying Filter Bank Filters

You must specify the highpass and lowpass filters in the filter bank by setting the Filter parameter to one of the following options:

User defined -- Allows you to explicitly specify the filters with two vectors of filter coefficients in the Lowpass FIR filter coefficients and Highpass FIR filter coefficients parameters. The block uses the same lowpass and highpass filters throughout the filter bank. The two filters should be halfband filters, where each filter passes the frequency band that the other filter stops. To use this block to perfectly reconstruct a signal decomposed by a Dyadic Analysis Filter Bank block, the filters in this block must be designed to perfectly reconstruct the outputs of the analysis filter bank. To learn how to design your own perfect reconstruction filters, see References.

Wavelet such as Biorthogonal or Daubechies -- The block uses the specified wavelet to construct the lowpass and highpass filters using the Wavelet Toolbox function, wfilters. Depending on the wavelet, the block may enable either the Wavelet order or Filter order [synthesis / analysis] parameter. (The latter parameter allows you to specify different wavelet orders for the analysis and synthesis filter stages.) To use this block to reconstruct a signal decomposed by a Dyadic Analysis Filter Bank block, you must set both blocks to use the same wavelets with the same order. You must install the Wavelet Toolbox to use wavelets.

**Table 7-11: Specifying Filters with the Filter Parameter and Related Parameters**
Filter	Sample Setting for Related Filter Specification Parameters	Corresponding Wavelet Function Syntax
User-defined	Filters based on Daubechies wavelets with wavelet order `3`: Lowpass FIR filter coefficients = [0.0352 -0.0854 -0.1350 0.4599 0.8069 0.3327] Highpass FIR filter coefficients = [-0.3327 0.8069 -0.4599 -0.1350 0.0854 0.0352]	None
Haar	None	`wfilters('haar')`
Daubechies	Wavelet order = `4`	`wfilters('db4')`
Symlets	Wavelet order = `3`	`wfilters('sym3')`
Coiflets	Wavelet order = `1`	`wfilters('coif1')`
Biorthogonal	Filter order [synthesis / analysis] = `[3/1]`	`wfilters('bior3.1')`
Reverse Biorthogonal	Filter order [synthesis / analysis] = `[3/1]`	`wfilters('rbio3.1')`
Discrete Meyer	None	`wfilters('dmey')`

Examples

See Examples in the Dyadic Analysis Filter Bank block reference.

Dialog Box

The parameters displayed in the block dialog vary depending on the setting of the Filter parameter. Only some of the parameters described below are visible in the dialog box at any one time.

Note To use this block to reconstruct a signal decomposed by a Dyadic Analysis Filter Bank block, all the parameters in this block must be the same as the corresponding parameters in the Dyadic Analysis Filter Bank block (except the Lowpass FIR filter coefficients and Highpass FIR filter coefficients; see the descriptions of these parameters).

Filter: The type of filter used to determine the high- and low-pass FIR filters in the dyadic synthesis filter bank:

Select User defined to explicitly specify the filter coefficients in the Lowpass FIR filter coefficients and Highpass FIR filter coefficients parameters.
Select a wavelet such as Biorthogonal or Daubechies to specify a wavelet-based filter. The block uses the Wavelet Toolbox function, wfilters, to construct the filters. Extra parameters such as Wavelet order or Filter order [synthesis / analysis] may become enabled. For a list of the supported wavelets, see Table 7-11, Specifying Filters with the Filter Parameter and Related Parameters,.

Lowpass FIR filter coefficients: A vector of filter coefficients (descending powers of z) that specifies coefficients used by all the lowpass filters in the filter bank. This parameter is enabled when you set Filter to User defined. The lowpass filter should be a half-band filter that passes the frequency band stopped by the filter specified in the Highpass FIR filter coefficients parameter. To perfectly reconstruct a signal decomposed by the Dyadic Analysis Filter Bank, the filters in this block must be designed to perfectly reconstruct the outputs of the analysis filter bank. Otherwise, the reconstruction will not be perfect. The default values of this parameter specify a perfect reconstruction filter for the default settings of the Dyadic Analysis Filter Bank (based on a Daubechies wavelet with wavelet order 3).
Highpass FIR filter coefficients: A vector of filter coefficients (descending powers of z) that specifies coefficients used by all the highpass filters in the filter bank. This parameter is enabled when you set Filter to User defined. The highpass filter should be a half-band filter that passes the frequency band stopped by the filter specified in the Lowpass FIR filter coefficients parameter. To perfectly reconstruct a signal decomposed by the Dyadic Analysis Filter Bank, the filters in this block must be designed to perfectly reconstruct the outputs of the analysis filter bank. Otherwise, the reconstruction will not be perfect. The default values of this parameter specify a perfect reconstruction filter for the default settings of the Dyadic Analysis Filter Bank (based on a Daubechies wavelet with wavelet order 3).
Wavelet order: The order of the wavelet selected in the Filter parameter. This parameter is enabled only when you set Filter to certain types of wavelets, as shown in Table 7-11, Specifying Filters with the Filter Parameter and Related Parameters,.
Filter order [synthesis / analysis]: The order of the wavelet for the synthesis and analysis filter stages. For example, if you set the Filter parameter to Biorthogonal and set the Filter order [synthesis / analysis] parameter to [2 / 6], the block calls the wfilters function with input argument 'bior2.6'. This parameter is enabled only when you set Filter to certain types of wavelets, as shown in Table 7-11, Specifying Filters with the Filter Parameter and Related Parameters,.
Number of levels: The number of filter bank levels. An n-level asymmetric structure has n+1 outputs, and an n-level symmetric structure has 2ⁿ outputs, as shown in Figure 7-9, n-Level Asymmetric Dyadic Synthesis Filter Bank, and Figure 7-10, n-Level Symmetric Dyadic Synthesis Filter Bank,. The block's icon displays the value of this parameter in the lower-left corner.
Tree structure: The structure of the filter bank: Asymmetric, or Symmetric. See Figure 7-9, n-Level Asymmetric Dyadic Synthesis Filter Bank, and Figure 7-10, n-Level Symmetric Dyadic Synthesis Filter Bank,.
Input: Set to Multiple ports to accept each input subband at a separate port (the topmost port accepts the subband with the highest frequency band). Set to Single port to accept one vector or matrix of concatenated subbands at a single port. For more information, see Input Requirements (Setting the Input Parameter).

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.

Supported Data Types

Double-precision floating point
Single-precision floating point

To learn how to convert to the above data types in MATLAB and Simulink, see Supported Data Types and How to Convert to Them.

See Also

Dyadic Analysis Filter Bank
DSP Blockset

Two-Channel Synthesis Subband Filter
DSP Blockset

Dyadic Analysis Filter Bank	DSP Blockset
Two-Channel Synthesis Subband Filter	DSP Blockset

See Multirate Filters for related information. Also see Filtering for a list of all DSP Blockset filtering blocks.

Dyadic Analysis Filter Bank Edge Detector