Dyadic Analysis Filter Bank

Decompose signals into subbands with smaller bandwidths and slower sample rates or compute discrete wavelet transform (DWT)

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Libraries:
DSP System Toolbox / Filtering / Multirate Filters

Description

You can configure this block to decompose a broadband signal into a collection of subbands with smaller bandwidths and slower sample rates. The block uses a series of highpass and lowpass FIR filters to repeatedly divide the input frequency range, as illustrated in Multilevel Filter Banks (the Asymmetric one).

You can specify the filter bank highpass and lowpass filters by providing vectors of filter coefficients. You can do so directly on the block mask. If you have a Wavelet Toolbox™ license, you can specify wavelet-based filters by selecting a wavelet from the Filter parameter. You must set the filter bank structure to asymmetric or symmetric, and specify the number of levels in the filter bank.

For the same input, the DWT configuration of this block does not produce the same results as the Wavelet Toolbox dwt function. Because DSP System Toolbox™ is designed for real-time implementation and Wavelet Toolbox is designed for analysis, the products handle boundary conditions and filter states differently. To make the output of the dwt function match the DWT output of this block, complete the following steps:

1. Set the boundary condition of the dwt function to zero-padding. To do so, type dwtmode('zpd') at the MATLAB^® command line.

2. To match the latency of the block (implemented using FIR filters), add zeros to the input of the dwt function. The number of zeros you add must be equal to the half-length of the filter.

The Dyadic Analysis Filter Bank block is same as the DWT block, but with different default settings.

Examples

Wavelet Reconstruction and Noise Reduction

Uses the Dyadic Analysis Filter Bank and Dyadic Synthesis Filter Bank blocks to show both the perfect reconstruction property of wavelets and an application for noise reduction.

Open Script

Ports

Input

Input 1 — Data input

column vector | matrix

Specify the data input as a column vector or a matrix.

The block always interprets the input signal as frames and operates along the columns. It treats each column of the input as an independent channel, and the number of rows of each channel must be a multiple of 2ⁿ, where n is the number of filter bank levels that you specify in the Number of levels parameter. For example, a frame size of 16 is appropriate for a three-level tree (16 is a multiple of 2³).

For more information on why the input data must meet these requirements, see the figure in Outputs of a 3-Level Asymmetric Dyadic Analysis Filter Bank.

The block decomposes the input signal into either n + 1 or 2ⁿ subbands. To decompose signals with a frame size that is not a multiple of 2ⁿ, use the Two-Channel Analysis Subband Filter block. You can connect multiple copies of the Two-Channel Analysis Subband Filter block to create a multilevel dyadic analysis filter bank.

Data Types: single | double
Complex Number Support: Yes

Output

Output 1 — Dyadic subband decomposition output

column vector | matrix

The block outputs the dyadic subband decomposition output as a column vector or a matrix. The decomposition output contains elements with the highest-frequency subband first followed by subbands in decreasing frequency.

The output characteristics vary depending on the block parameter settings, as summarized in the following list and figure:

• Number of levels parameter set to n

• Tree structure parameter setting:

  • Asymmetric — Block produces n+1 output subbands

  • Symmetric — Block produces 2ⁿ output subbands

• Output parameter setting can be Multiple ports or Single port. When you set the Output parameter to Single port, the block outputs one vector or matrix of concatenated subbands. The following figure illustrates the difference between the two settings for a 3-level asymmetric dyadic analysis filter bank. For an explanation of the illustrated output characteristics, see the table Output Characteristics for an n-Level Dyadic Analysis Filter Bank.

For more information about the filter bank levels and structures, see Dyadic Analysis Filter Banks.

Outputs of a 3-Level Asymmetric Dyadic Analysis Filter Bank

The following table summarizes the different output characteristics of the block when you set it to output from single or multiple ports.

Output Characteristics for an n-Level Dyadic Analysis Filter Bank

	Single Output Port	Multiple Output Ports
Output Description	Block concatenates all the subbands into one vector or matrix, and outputs the concatenated subbands from a single output port. Each output column contains subbands of the corresponding input channel.	Block outputs each subband from a separate output port. The topmost port outputs the subband with the highest frequencies. Each output column contains a subband for the corresponding input channel.
Output Frame Rate	Not applicable	Same as input frame rate (However, the output frame sizes can vary, so the output sample rates can vary.)
Output Dimensions (Frame Size)	Same number of rows and columns as the input.	The output has the same number of columns as the input. The number of output rows is the output frame size. For an input with frame size Mi output yk has frame size Mo,k: Symmetric — All outputs have the frame size Mi / 2n. Asymmetric — The frame size of each output (except the last) is half that of the output from the previous level. The outputs from the last two output ports have the same frame size since they originate from the same level in the filter bank. $$M_{o,k}=\{\begin{array}{cc}{M}_i/2^k& (1\le k\le n)\\ M_i/2^n& (k=n+1)\end{array}$$
Output Sample Rate	Same as input sample rate.	Though the outputs have the same frame rate as the input, they have different frame sizes than the input. Thus, the output sample rates Fso,k are different from the input sample rate Fsi: Symmetric — All outputs have the sample rate Fsi / 2n. Asymmetric — $$F_{so,k}=\{\begin{array}{cc}{F}_{si}/2^k& \left(1\le k\le n\right)\\ F_{si}/2^n& \left(k=n+1\right)\end{array}$$

Data Types: single | double
Complex Number Support: Yes

Parameters

The parameters displayed in the block dialog box 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.

Filter — Type of filter used in subband decomposition

User-defined (default) | Haar | Daubechies | Symlets | Coiflets | Biorthogonal | Reverse Biorthogonal | Discrete Meyer

Specify the highpass and lowpass filters in the filter bank by setting the Filter parameter to one of these 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.

• Wavelet such as Biorthogonal or Daubechies — The block uses the specified wavelet to construct the lowpass and highpass filters using the Wavelet Toolbox wfilters function. Depending on the wavelet, the block enable either the Wavelet order or the Filter order [synthesis / analysis] parameter. The Filter order [synthesis / analysis] parameter allows you to specify different wavelet orders for the analysis and synthesis filter stages. You must have a Wavelet Toolbox license to use wavelets.

See this table for a list of the supported wavelets.

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')

Filter order [synthesis / analysis] — Analysis and synthesis filter orders for biorthogonal filters

[1 / 1] (default) | [1 / 3] | [1 / 5] | [2 / 2] | [2 / 4] | [2 / 6] | [2 / 8] | [3 / 1] | [3 / 3] | [3 / 5] | [3 / 7] | [3 / 9] | [4 / 4] | [5 / 5] | [6 / 8]

Set the order of the wavelet for the synthesis and analysis filter stages. For example, when 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'.

Dependencies

To enable this parameter, set the Filter parameter to Biorthogonal or Reverse Biorthogonal.

Wavelet order — Order for orthogonal wavelets

2 (default) | positive integer

Set the order of the wavelet selected in the Filter parameter.

Dependencies

To enable this parameter, set the Filter parameter to Daubechies, Symlets, or Coiflets.

Lowpass FIR filter coefficients — Lowpass FIR filter coefficients

[0.0352 -0.0854 -0.1350 0.4599 0.8069 0.3327] (default) | row vector

Specifies coefficients used by all the lowpass filters in the filter bank using a vector of filter coefficients (descending powers of z). 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. The default values of this parameter specify a filter based on a Daubechies wavelet with wavelet order 3.

Dependencies

To enable this parameter, set the Filter parameter to User-defined.

Highpass FIR filter coefficients — Highpass FIR filter coefficients

[-0.3327 0.8069 -0.4599 -0.1350 0.0854 0.0352] (default) | row vector

Specifies coefficients used by all the highpass filters in the filter bank using a vector of filter coefficients (descending powers of z). 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. The default values of this parameter specify a filter based on a Daubechies wavelet with wavelet order 3.

Dependencies

To enable this parameter, set the Filter parameter to User-defined.

Number of levels — Number of filter bank levels used in analysis (decomposition)

2 (default) | integer greater than or equal to 1

Select 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 Multilevel Filter Banks. The block icon changes depending on the value of this parameter.

Tree structure — Structure of filter bank

Asymmetric (default) | Symmetric

Select the structure of the filter bank: Asymmetric, or Symmetric. See Multilevel Filter Banks for more information.

The default setting of this parameter is Asymmetric for the Dyadic Analysis Filter Bank block, and Symmetric for the DWT block.

Output — Single port or multiple ports

Multiple ports (default) | Single port

Set to Multiple ports to output each output subband on a separate port. The topmost port outputs the subband with the highest frequency band. Set to Single port to concatenate the subbands into one vector or matrix and output the concatenated subbands on a single port. For more information, see the Output port description.

The default setting of this parameter is Multiple ports for the Dyadic Analysis Filter Bank block and Single port for the DWT block.

Block Characteristics

Data Types	double | single
Direct Feedthrough	no
Multidimensional Signals	no
Variable-Size Signals	no
Zero-Crossing Detection	no

More About

Wavelets

The primary application for dyadic analysis filter banks and dyadic synthesis filter banks is coding for data compression using wavelets.

At the transmitting end, the output of the 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 signal components by a greater number of bits than the less powerful signal components.

At the receiving end, the transmission is decoded and fed to a 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.

See Calculate Channel Latencies Required for Wavelet Reconstruction for an example using the Dyadic Analysis and Dyadic Synthesis Filter Bank blocks.

References

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

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

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

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using Simulink® Coder™.

Generated code relies on the memcpy or memset function (string.h) under certain conditions.

Version History

Introduced before R2006a