IIR Halfband Decimator

Decimate signal using polyphase IIR halfband filter

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Libraries:
DSP System Toolbox / Filtering / Multirate Filters

Description

The IIR Halfband Decimator block performs polyphase decimation of the input signal by a factor of 2. To design the halfband filter, you can specify the block to use an elliptic design or a quasi-linear phase design. The block uses these design methods to compute the filter coefficients. To filter the inputs, the block uses a polyphase structure. The allpass filters in the polyphase structure are in a minimum multiplier form.

Elliptic design introduces nonlinear phase and creates the filter using fewer coefficients than the quasi-linear design. Quasi-linear phase design overcomes phase nonlinearity at the cost of additional coefficients.

Alternatively, instead of designing the halfband filter using a design method, you can specify the filter coefficients directly. When you choose this option, the allpass filters in the two branches of the polyphase implementation can be in a minimum multiplier form or a wave digital form.

You can also use the block to implement the analysis portion of a two-band filter bank to filter a signal into lowpass and highpass subbands. For more details, see Algorithms.

Examples

Design and Implement IIR Halfband Decimator in Simulink

Since R2023b

Design and implement an IIR halfband decimator using the IIR Halfband Decimator block. Pass a noisy input through the decimator. Plot the spectrum of the input and the decimated subband outputs in the spectrum analyzer.

This example also shows how to use the Allow arbitrary frame length for fixed-size input signals parameter.

Open and inspect the DesignAndImplementIIRHalfbandDecimator model. The input signal in the model is a noisy sinusoidal signal with 513 samples per frame and contains two frequencies, one at 1 kHz and the other at 15 kHz. The Random Source block adds white Gaussian noise with a mean of 0 and a variance of 0.05 to this signal. The input signal is a fixed-size signal, that is, the frame length of the signal (513 samples per frame) does not vary during simulation.

To allow an arbitrary frame length for a fixed-size signal, that is, to allow a frame length that is not a multiple of the decimation factor 2, select the Allow arbitrary frame length for fixed-size input signals parameter in the IIR Halfband Decimator block dialog box.

The IIR halfband decimator has a transition width of 4.1 kHz and a stopband attenuation of 80 dB. The decimator outputs the highpass subband signal.

Visualize the magnitude response of the filter by clicking the View Filter Response button in the block dialog box.

Pass the noisy sinusoidal signal through the decimator. Plot the spectrum of the input and the lowpass and highpass subband outputs in two separate spectrum analyzer windows.

The block allows arbitrary frame lengths for variable-size signals irrespective of the setting of the Allow arbitrary frame length for fixed-size input signals parameter. When the input is a variable-size signal, the frame length of the signal can vary during simulation.

Open the DesignAndImplementIIRHalfbandDecimator_Varsize model. The input is a variable-size signal. The input frame length can vary between 257 and 514 samples per frame.

Extract Low Frequency Subband From Speech in Simulink

Since R2023b

Use the IIR Halfband Decimator and IIR Halfband Interpolator blocks to extract and reconstruct the low-frequency subband from a speech signal.

Open and inspect the ExtractLowFrequencySubbandFromSpeechIIR.slx model. The input audio data is a single-channel speech signal with the sample rate of 22050 Hz.

Specify the Sample rate mode parameter of the IIR Halfband Decimator and IIR Halfband Interpolator blocks to Use normalized frequency (0 to 1). This option enables you to specify the transition width of the decimation and interpolation filters in normalized frequency units. Set the transition width to 0.093 in normalized frequency units and the stopband attenuation to 80 dB. The design method is Elliptic by default.

Read the speech signal from the audio file in frames of 1024 samples. The IIR Halfband Decimator block extracts and outputs the lowpass subband of the speech signal. The IIR Halfband Interpolator block reconstructs the lowpass approximation of the speech signal by interpolating the lowpass subband.

The Audio Device Writer block plays the filtered output.

Design and Implement Two-Channel IIR Filter Bank in Simulink

Use the IIR Halfband Decimator and IIR Halfband Interpolator blocks to implement a two-channel filter bank. This example uses an audio file input and shows that the power spectrum of the filter bank output does not differ significantly from the input. Play the output of the filter bank using the Audio Device Writer block.

Open and inspect the TwoChannelIIRFilterBank model. The input audio data is a single-channel speech signal with a sample rate of 22050 Hz.

The IIR Halfband Decimator block acts as an IIR halfband analysis bank as the Output highpass subband parameter is selected in the block dialog box. The IIR Halfband Interpolator block acts as an IIR halfband synthesis bank as the Input highpass subband parameter is selected in the block dialog box.

Set the Sample rate mode parameter in the IIR Halfband Decimator and IIR Halfband Interpolator blocks to Inherit from input port so that the blocks inherit the sample rate from the respective input ports. Set the transition width to 4.1 kHz and the stopband attenuation to 80 dB. The design method is set to Elliptic by default.

Read the speech signal from the audio file in frames of 1024 samples. The IIR halfband analysis filter bank extracts the lowpass and highpass subbands of the speech signal. The IIR halfband synthesis filter bank synthesizes the speech signal from the lowpass and highpass subbands.

Display the power spectrum of the audio input and the output from the synthesis filter bank in the spectrum analyzer. Play the synthesized speech signal using the Audio Device Writer block.

Ports

Input

x — Input signal

column vector | matrix

Specify the input signal as a column vector or a matrix of size P-by-Q.

The block treats each column of the input signal as a separate channel. If the input is a two-dimensional signal, the first dimension represents the channel length (or frame size) and the second dimension represents the number of channels. If the input is a one-dimensional signal, then the block interprets it as a single-channel signal.

This block supports variable-size input signals (frame length changes during simulation).

When you input a variable-size signal, the frame length of the signal can be arbitrary, that is, the input frame length does not have to be a multiple of the decimation factor 2. When you input a fixed-size signal (frame length does not change during simulation), the frame length can be arbitrary only when you select the Allow arbitrary frame length for fixed-size input signals parameter. (since R2023b)

This port is unnamed until you select the Specify coefficients from input port parameter.

Data Types: single | double
Complex Number Support: Yes

coeffs1 — Branch 1 allpass polynomial coefficients

N-by-1 vector | N-by-2 matrix

Specify the allpass polynomial filter coefficients of the first branch as an N-by-1 vector or N-by-2 matrix of N first-order or second-order allpass sections.

Dependencies

To enable this parameter, set:

• Filter specification to Coefficients

• Internal allpass structure to Minimum multiplier

• Clear the Make the first branch a pure delay parameter

• Select the Specify coefficients from input port parameter

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | fixed point

coeffs2 — Branch 2 allpass polynomial coefficients

N-by-1 vector | N-by-2 matrix

Specify the allpass polynomial filter coefficients of the second branch as an N-by-1 vector or N-by-2 matrix of N first-order or second-order allpass sections.

Dependencies

To enable this parameter, set:

• Filter specification to Coefficients

• Internal allpass structure to Minimum multiplier

• Select the Specify coefficients from input port parameter

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | fixed point

Output

LP — Lowpass subband of decimator output

column vector | matrix

Lowpass subband of the decimator output, returned as a column vector or a matrix. As the filter is a halfband filter, the downsampling factor is always 2.

The dimensions of the output signal depend on the dimensions of the input signal and on whether you select the Allow arbitrary frame length for fixed-size input signals parameter. (since R2023b)

This table provides more details on the dimensions of the lowpass subband output signal when you input a fixed-size input signal.

Fixed-Size Input Signal

Input Signal	Output Signal
P-by-Q, where P is a multiple of the decimation factor 2	Fixed-size signal of size (P/2)-by-Q
P-by-Q, where P is not a multiple of the decimation factor 2 (since R2023b)	Variable-size signal with an upper bound of size ceil(P/2)-by-Q when you select Allow arbitrary frame length for fixed-size input signals (since R2023b) If you do not select Allow arbitrary frame length for fixed-size input signals, the block errors. (since R2023b)

This table gives more details on the dimensions of the lowpass subband output signal when you input a variable-size input signal. When you input a variable-size signal (frame length changes during simulation), the Allow arbitrary frame length for fixed-size input signals parameter is visible in the block dialog box but does not have any impact on the input frame length. You can input a variable-size signal of any frame length even if you do not select the Allow arbitrary frame length for fixed-size input signals parameter. (since R2023b)

Variable-Size Input Signal

Input Signal	Output Signal
P-by-Q	Variable-size signal with an upper bound of size ceil(P/2)-by-Q (since R2023b)

This port is unnamed until you select the Output highpass subband parameter.

Data Types: single | double
Complex Number Support: Yes

HP — Highpass subband of decimator output

column vector | matrix

Highpass subband of the decimator output, returned as a column vector or a matrix. As the filter is a halfband filter, the downsampling factor is always 2.

The dimensions of the output signal depend on the dimensions of the input signal and on whether you select the Allow arbitrary frame length for fixed-size input signals parameter. (since R2023b)

This table provides more details on the dimensions of the highpass subband output signal when you input a fixed-size input signal.

Fixed-Size Input Signal

Input Signal	Output Signal
P-by-Q, where P is a multiple of the decimation factor 2	Fixed-size signal of size (P/2)-by-Q
P-by-Q, where P is not a multiple of the decimation factor 2 (since R2023b)	Variable-size signal with an upper bound of size ceil(P/2)-by-Q when you select Allow arbitrary frame length for fixed-size input signals (since R2023b) If you do not select Allow arbitrary frame length for fixed-size input signals, the block errors. (since R2023b)

This table gives more details on the dimensions of the highpass subband output signal when you input a variable-size input signal. When you input a variable-size signal (frame length changes during simulation), the Allow arbitrary frame length for fixed-size input signals parameter is visible in the block dialog box but does not have any impact on the input frame length. You can input a variable-size signal of any frame length even if you do not select the Allow arbitrary frame length for fixed-size input signals parameter. (since R2023b)

Variable-Size Input Signal

Input Signal	Output Signal
P-by-Q	Variable-size signal with an upper bound of size ceil(P/2)-by-Q (since R2023b)

This port is unnamed until you select the Output highpass subband parameter.

Data Types: single | double
Complex Number Support: Yes

Parameters

Filter specification — Filter design parameters

Transition width and stopband attenuation (default) | Filter order and transition width | Filter order and stopband attenuation | Coefficients

Select the parameters that the block uses to design the IIR halfband filter. Because the filter design has only two degrees of freedom, you can specify only two of the three parameters:

• Transition width and stopband attenuation (default) — Design the filter using Transition width (Hz) and Stopband attenuation (dB). This design is the minimum order design.

• Filter order and transition width — Design the filter using Filter order and Transition width (Hz).

• Filter order and stopband attenuation — Design the filter using Filter order and Stopband attenuation (dB).

• Coefficients— Specify the filter coefficients directly using the enabled parameters.

Transition width (Hz) — Transition width in Hz

4.1e3 (default) | positive real scalar

Specify the transition width of the IIR halfband filter as a real positive scalar in Hz or in normalized frequency units (since R2023b).

If you set the Sample rate mode parameter to:

• Specify on dialog or Inherit from input port –– The value of the transition width is in Hz and must be less than half the value of the input sample rate.

• Use normalized frequency (0 to 1) –– The value of the transition width is in normalized frequency units. The value must be a positive scalar less than 1.0.

(since R2023b)

Dependencies

To enable this parameter, set Filter specification to Filter order and transition width or Transition width and stopband attenuation.

Filter order — Order of the IIR halfband filter

9 (default) | positive integer

Specify the filter order as a finite positive integer. If you set Design method to Elliptic, then Filter order must be an odd integer greater than one. If you set Design method to Quasi-linear phase, then Filter order must be a multiple of four.

Dependencies

To enable this parameter, set Filter specification to Filter order and transition width or Filter order and stopband attenuation.

Stopband attenuation (dB) — Minimum attenuation needed in stopband

80 (default) | positive real scalar

Specify the minimum attenuation needed in the stopband of the IIR halfband filter as a real positive scalar in dB.

Dependencies

To enable this parameter, set Filter specification to Filter order and stopband attenuation or Transition width and stopband attenuation.

Design method — Design method

Elliptic (default) | Quasi-linear phase

Specify the design method for the IIR halfband filter.

• Elliptic — The filter has a nonlinear phase and uses few coefficients.

• Quasi-linear phase — The first branch of the polyphase filter structure is a pure delay, which results in an approximately linear phase response.

Dependencies

To enable this parameter, set Filter specification to any option except Coefficients.

Internal allpass structure — Internal allpass filter implementation structure

Minimum multiplier (default) | Wave Digital Filter

Specify the internal allpass filter implementation structure as Minimum multiplier or Wave Digital Filter. Each structure uses a different coefficients set, independently stored in the corresponding coefficients property.

Dependencies

To enable this parameter, set Filter specification to Coefficients.

Make the first branch a pure delay — Make first branch a pure delay

off (default) | on

When you select this check box, the first branch of the polyphase filter structure becomes a pure delay, and the Branch 1 allpass polynomial coefficients and Branch 1 Wave Digital coefficients parameters do not apply.

By default, this check box is not selected.

Dependencies

To enable this parameter, set Filter specification to Coefficients.

Delay length in samples for branch 1 — Length of delay

1 (default) | positive scalar

Specify the length of the first branch delay as a positive scalar.

Dependencies

To enable this parameter, set:

• Filter specification to Coefficients

• Select the Make the first branch a pure delay parameter

Specify coefficients from input port — Specify coefficients from input port

off (default) | on

When you select this check box, you can input the branch 1 allpass polynomial coefficients and branch 2 allpass polynomial coefficients through the input ports coeffs1 and coeffs2. When you clear this check box, you specify the coefficients in the block dialog box through the Branch 1 allpass polynomial coefficients and Branch 2 allpass polynomial coefficients parameters.

Dependencies

To enable this parameter, set:

• Filter specification to Coefficients

• Internal allpass structure to Minimum multiplier

Branch 1 allpass polynomial coefficients — Allpass polynomial filter coefficients of first branch

[0.1284563; 0.7906755] (default) | N-by-1 | N-by-2

Specify the allpass polynomial filter coefficients of the first branch as an N-by-1 vector or N-by-2 matrix of N first-order or second-order allpass sections.

This parameter is tunable, that is, you can change its value during simulation.

Tunable: Yes

Dependencies

To enable this parameter, set:

• Filter specification to Coefficients

• Internal allpass structure to Minimum multiplier

• Clear the Make the first branch a pure delay parameter

• Clear the Specify coefficients from input port parameter

Branch 2 allpass polynomial coefficients — Allpass polynomial filter coefficients of second branch

[0.4295667] (default) | N-by-1 | N-by-2

Specify the allpass polynomial filter coefficients of the second branch as an N-by-1 vector or N-by-2 matrix of N first-order or second-order allpass sections.

This parameter is tunable, that is, you can change its value during simulation.

Tunable: Yes

Dependencies

To enable this parameter, set:

• Filter specification to Coefficients

• Internal allpass structure to Minimum multiplier

• Clear the Specify coefficients from input port parameter

Branch 1 Wave Digital coefficients — Allpass filter coefficients of first branch in wave digital filter form

[0.1284563; 0.7906755] (default) | N-by-1 | N-by-2

Specify the allpass filter coefficients of the first branch in wave digital filter (WDF) form as an N-by-1 vector or N-by-2 matrix of N first-order or second-order allpass sections.

The magnitude of each WDF coefficient must not be greater than 1.

Dependencies

To enable this parameter, set:

• Filter specification to Coefficients

• Internal allpass structure to Wave Digital Filter

• Clear the Make the first branch a pure delay parameter

Branch 2 Wave Digital coefficients — Allpass filter coefficients of second branch in wave digital filter form

[0.4295667] (default) | N-by-1 | N-by-2

Specify the allpass filter coefficients of the second branch in wave digital filter form as an N-by-1 vector or N-by-2 matrix of N first-order or second-order allpass sections.

The magnitude of each WDF coefficient must not be greater than 1.

Dependencies

To enable this parameter, set:

• Filter specification to Coefficients

• Internal allpass structure to Wave Digital Filter

Last section of branch 2 is first order — Last section of branch 2 is first order

off (default) | on

When you select this check box, the block treats the last section of the second branch as a first order section. When the coefficients of the second branch are in an N-by-2 matrix, the block ignores the second element of the last row of the matrix. The last section of the second branch then becomes a first-order section.

When you clear this check box, the block treats the last section of the second branch as a second-order section. When the coefficients of the second branch are in an N-by-1 matrix, the block ignores this parameter.

Dependencies

To enable this parameter, set Filter specification to Coefficients.

Output highpass subband — Output highpass subband

off (default) | on

When you select this check box, the block acts as an analysis filter bank, producing two power-complementary outputs. When you clear this check box, the block acts as an IIR halfband decimator and accepts a single vector or matrix as input.

Sample rate mode — Mode to specify the input sample rate

Use normalized frequency (0 to 1) (default) | Specify on dialog | Inherit from input port

Since R2023b

Specify the input sample rate using one of these options:

• Use normalized frequency (0 to 1) –– Specify the passband-edge and the stopband-edge frequencies in normalized frequency units (0 to 1).

• Specify on dialog –– Specify the input sample rate in the block dialog box using the Input sample rate (Hz) parameter.

• Inherit from input port –– The block inherits the sample rate from the input signal.

Dependencies

To enable this parameter, set Filter specification to any value other than Coefficients.

Input sample rate (Hz) — Sample rate of input signal

44100 (default) | positive real scalar

Specify the sample rate of the input signal as a scalar in Hz.

Dependencies

To enable this parameter, set:

• Filter specification to any value other than Coefficients.

• Sample rate mode to Specify on dialog.

(since R2023b)

Allow arbitrary frame length for fixed-size input signals — Allow arbitrary frame length for fixed-size input signals

off (default) | on

Since R2023b

Specify whether a fixed-size input signal (whose size does not change during simulation) can have an arbitrary frame length, where the frame length does not have to be a multiple of the decimation factor 2. The block uses this parameter only for a fixed-size input signal and ignores it if the input data varies in size during simulation.

When the input signal is a variable-size signal, the signal can have an arbitrary frame length, that is, the frame length does not have to be a multiple of the decimation factor 2.

For fixed-size input signals, if you:

• Select the Allow arbitrary frame length for fixed-size input signals parameter, the frame length of the signal does not have to be a multiple of the decimation factor 2. If the input is not a multiple of the decimation factor, then the output is generally a variable-size signal. Therefore, to support arbitrary input size, the block must also support variable-size operations, which you can enable by selecting the Allow arbitrary frame length for fixed-size input signals parameter.

• Clear the Allow arbitrary frame length for fixed-size input signals parameter, the input frame length must be a multiple of the decimation factor 2.

View Filter Response — View Filter Response

button

Click this button to open the Filter Visualization Tool (FVTool) and display the magnitude and phase response of the IIR Halfband Decimator. The response is based on the values you specify in the block parameters dialog box. Changes made to these parameters update FVTool.

To update the magnitude response while FVTool is running, modify the dialog box parameters and click Apply.

Simulate using — Type of simulation to run

Interpreted execution (default) | Code generation

Specify the type of simulation to run. You can set this parameter to:

• Interpreted execution –– Simulate model using the MATLAB^® interpreter. This option shortens startup time.

• 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.

Block Characteristics

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

Algorithms

Polyphase Implementation with Halfband Filters

When you filter your signal, the IIR halfband decimator uses an efficient polyphase implementation for halfband filters. You can use the polyphase implementation to move the downsample operation before filtering. This change enables you to filter at a lower sampling rate.

IIR halfband filters are generally modeled using two parallel allpass filter branches.

$$H(z)=0.5*[A_1(z^2)+z^{-1}{A}_2(z^2)]$$

Elliptic Design

The allpass filters for elliptic IIR halfband filter are given as

$$A_1(z)={\displaystyle \prod _{k=1}^{K_1}\frac{a_k^{(1)}+z^{-1}}{1+a_k^{(1)}{z}^{-1}}}$$

$$A_2(z)={\displaystyle \prod _{k=1}^{K_2}\frac{a_k^{(2)}+z^{-1}}{1+a_k^{(2)}{z}^{-1}}}$$

Quasi-Linear Phase Design

To achieve a near-linear phase response for IIR halfband filters, make one of the branches a pure delay. In this design, the cost of the filter increases.

The allpass filters for the quasi-linear phase IIR halfband filter are

$$A_1(z)=z^{-k}$$

where k is the length of the delay.

$$A_2(z)={\displaystyle \prod _{K=1}^{K_2^{(1)}}\frac{a_k+z^{-1}}{1+a_k{z}^{-1}}}{\displaystyle \prod _{K=1}^{K_2^{(2)}}\frac{c_k+b_k{z}^{-1}+z^{-2}}{1+b_k{z}^{-1}+c_k{z}^{-2}}}$$

where N is the order of the IIR halfband filter.

You can represent filtering the input signal and then downsampling it by 2 using this figure.

Using the multirate noble identity for downsampling, you can move the downsampling operation before the filtering operation. This change enables you to filter at a lower rate.

To implement the halfband decimator efficiently, this algorithm replaces the delay block and downsampling operator with a commutator switch. When the first input sample is delivered, the commutator switch feeds this input to the first branch and the halfband decimator computes the first output value. As more input samples come in, the switch delivers one sample at a time to each branch alternatively. The decimator generates output every time the first branch generates an output. This halves the sampling rate of the input signal.

Analysis Filter Bank

The transfer function of the complementary highpass filter branch of the analysis filter bank is given by:

$$G(z)=0.5*[A_1(z^2)-z^{-1}{A}_2(z^2)]$$

You can represent the analysis filter bank as in this diagram.

The IIR halfband decimator generates two power-complementary output signals by adding and subtracting the two polyphase branch outputs respectively.

For more information on filter banks, see Overview of Filter Banks.

To summarize, the IIR halfband decimator:

• Decimates the input prior to filtering.

• Acts as an analysis filter bank.

• Has a nonlinear phase response and uses few coefficients with the elliptic design method.

• Has near-linear phase response at the cost of additional coefficients with the quasi-linear phase design method, where one of the branches is a pure delay

Extended Capabilities

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

Version History

Introduced in R2015b

R2024a: Change in the default value of Simulate using parameter

The default value of the Simulate using parameter is now Interpreted execution. With this change, the block uses the MATLAB interpreter for simulation by default.

R2023b: Support for normalized frequencies

When you set the Sample rate mode parameter to Use normalized frequency (0 to 1), you can specify the transition width in normalized frequency units (0 to 1).

R2023b: Support for arbitrary input frame length

This block supports an input signal with an arbitrary frame length when the:

• Input signal is a fixed-size signal (frame length does not change during simulation) and you select the Allow arbitrary frame length for fixed-size input signals parameter.

• Input signal is a variable-size signal (frame length changes during simulation).

When this block supports an input signal with an arbitrary frame length, the input frame length does not have to be a multiple of the decimation factor 2 and the output signal is a variable-size signal.