LMS Filter

Compute output, error, and weights using least mean squares (LMS) adaptive algorithm

• 

Libraries:
DSP System Toolbox / Filtering / Adaptive Filters
DSP System Toolbox HDL Support / Filtering

Description

The LMS Filter block can implement an adaptive FIR filter by using five different algorithms. The block estimates the filter weights or coefficients needed to minimize the error, e(n), between the output signal y(n) and the desired signal, d(n). The output is the filtered input signal, which is the estimate of the desired signal. The Error port outputs the result of subtracting the output signal from the desired signal.

Under specific conditions, this block also supports SIMD code generation. For details, see Code Generation.

Examples

Time-Delay Channel Estimation Through Adaptive Filtering

Adaptively estimate the time delay for a noisy input signal using the LMS adaptive FIR algorithm.

• Open Model

Adaptive Filter Convergence

Compares the rate of convergence for adaptive filters using different LMS algorithms.

• Open Script

Ports

Input

Input — Input signal

scalar | column vector

Input signal, specified as a scalar or column vector.

When the input is fixed-point, it must be signed.

When you set the Algorithm parameter to Sign—Error LMS, Sign—Data LMS, or Sign—Sign LMS, the data input through the Input port must be real.

Data Types: single | double | fixed point

Desired — Desired signal

scalar | column vector

Desired signal, specified as a scalar or column vector. The desired signal must have the same data type, complexity, and dimensions as the Input signal.

When Input is fixed-point, the desired signal must be a signed fixed-point.

When you set the Algorithm parameter to Sign—Error LMS, Sign—Data LMS, or Sign—Sign LMS, the data input through the Desired port must be real.

Data Types: single | double | fixed point

Step-size — Step-size

scalar

Enter the step size μ. For convergence of the normalized LMS equations, 0<µ<2. Input type must match the type of the Input port.

When Input is fixed-point, the step-size must be a signed fixed-point.

Dependencies

This port appears only when you set the Specify step size via parameter to Input port.

Data Types: single | double | fixed point

Adapt — Update filter weights

scalar

When the input to this port is greater than zero, the block continuously updates the filter weights. When the input to this port is less than or equal to zero, the filter weights remain at their current values.

Dependencies

This port appears only when you set the Adapt port parameter to on.

Data Types: single | double | int8 | int16 | int32 | Boolean

Reset — Reset filter weights

scalar

Signal to reset the value of the filter weights to their initial values, specified as a scalar. The block resets the filter weights whenever a reset event is detected at the Reset port. The reset signal rate must be the same rate as the data signal input.

For reset event types, see the Reset parameter.

Dependencies

This port appears only when you set the Reset port parameter to Rising edge, Falling edge, Either edge, or Non-zero sample.

Data Types: single | double | int8 | int16 | int32 | Boolean

Output

Output — Estimate of desired signal

scalar | column vector

Estimate of the desired signal, returned as a scalar or a column vector. It is the same size and complexity as the input signal.

The output signal has the same data type as the desired signal.

Data Types: single | double | fixed point

Error — Error between output and desired signals

scalar | column vector

Error between the output and desired signals, returned as a scalar or a column vector. This error is the result of subtracting the output signal from the desired signal.

The error signal has the same data type as the desired signal.

Data Types: single | double | fixed point

Wts — Filter weights

scalar | column vector

Filter weights, returned as a scalar or a column vector. For each iteration, the block outputs the current updated filter weights from this port.

The weights data type must match the type of the Input port for floating-point signals. Obeys the Weights parameter for fixed-point signals.

Dependencies

This port appears only when you set the Output filter weights parameter to On.

Data Types: single | double | fixed point

Parameters

Main Tab

Algorithm — Select algorithm

LMS (default) | Normalized LMS | Sign-Error LMS | Sign-Data LMS | Sign-Sign LMS

Choose the algorithm used to calculate the filter weights.

Filter length — Filter length

32 (default) | scalar

Enter the length of the FIR filter weights vector.

Specify step size via — Specify step size via

Dialog (default) | Input port

• Dialog –– Specify step size by using the Step size (mu) parameter.

• Input port –– Specify step size by using the Step-size port.

Step size (mu) — Step size

0.1 (default) | positive scalar

Enter the step size μ. For convergence of the normalized LMS equations, 0<µ<2.

Tunable: Yes

Dependencies

This parameter appears only when you set the Specify step size via parameter to Dialog.

Leakage factor (0 to 1) — Leakage factor

1.0 (default) | scalar

Enter the leakage factor, 0 < 1 – μα ≤ 1.

Tunable: Yes

Initial value of filter weights — Initial value of filter weights

0 (default) | vector | scalar

Enter the initial filter weights w(0) as a vector or a scalar. When you enter a scalar, the block uses the scalar value to create a vector of filter weights. This vector length is equal to the filter length and all of its values are equal to the scalar value.

Adapt port — Enable Adapt port

off (default) | on

Select this check box to enable the Adapt input port.

Reset port — Reset port

None (default) | Rising edge | Falling edge | Either edge | Non-zero sample

When you want to reset the value of filter weights to their initial values, use the Reset port parameter. The reset signal must be the same rate as the data signal input.

Select None to disable the Reset port. To enable the Reset port, select one of the following from the list:

• Rising edge — Triggers a reset operation when the Reset input does one of the following:

  • Rises from a negative value to a positive value or zero

  • Rises from zero to a positive value, where the rise is not a continuation of a rise from a negative value to zero (see the following figure)

    

• Falling edge — Triggers a reset operation when the Reset input does one of the following:

  • Falls from a positive value to a negative value or zero

  • Falls from zero to a negative value, where the fall is not a continuation of a fall from a positive value to zero (see the following figure)

    

• Either edge — Triggers a reset operation when the Reset input is a Rising edge or Falling edge

• Non-zero sample — Triggers a reset operation at each sample time that the Reset input is not zero

Output filter weights — Output filter weights

on (default) | off

Select the Output filter weights parameter to export the filter weights from the Wts port. For each iteration, the block outputs the current updated filter weights from this port.

Data Type Tab

Rounding mode — Method of rounding operation

Floor (default) | Ceiling | Convergent | Nearest | Round | Simplest | Zero

Specify the rounding mode for fixed-point operations as one of the following:

• Floor

• Ceiling

• Convergent

• Nearest

• Round

• Simplest

• Zero

For more details, see rounding mode.

Saturate on integer overflow — Method of overflow action

off (default) | on

When you select this parameter, the block saturates the result of its fixed-point operation. When you clear this parameter, the block wraps the result of its fixed-point operation. For details on saturate and wrap, see overflow mode for fixed-point operations.

Parameters — Parameters

Same word length as first input (default) | Specify word length | Binary point scaling

Choose how you specify the word length and the fraction length of the leakage factor and step size:

• Same word length as first input –– The word length of the leakage factor and step size match that of the first input to the block. In this mode, the fraction length of the leakage factor and step size is automatically set to the binary-point only scaling that provides you with the best precision possible given the value and word length of the coefficients.

• Specify word length –– You can enter the word length of the leakage factor and step size, in bits. In this mode, the fraction length of the leakage factor and step size is automatically set to the binary-point only scaling that provides you with the best precision possible given the value and word length of the coefficients.

• Binary point scaling –– You can enter the word length and the fraction length of the leakage factor and step size, in bits. The leakage factor and the step size must have the same word length, but the fraction lengths can differ.

For the Specify step size via parameter, if you choose Input port, the word length of the leakage factor is the same as the word length of the step size input at the Step-size port. The fraction length of the leakage factor is automatically set to the best precision possible based on the word length of the leakage factor.

Dependencies

This parameter is visible only if you set the Specify step size via parameter to Dialog

Weights — Word and fraction length of filter weights

Same as first input (default) | Binary point scaling

Choose how you specify the word length and fraction length of the filter weights of the block:

• Same as first input –– The word length and fraction length of the filter weights match those of the first input to the block.

• Binary point scaling –– You can enter the word length and the fraction length of the filter weights, in bits.

Product and quotient — Word and fraction length of product and quotient

Same as first input (default) | Binary point scaling

Choose how you specify the word length and fraction length of u'u, W'u, $$\mu \cdot e$$, $$Q\cdot u$$, and the quotient, Q. Here, u is the input vector, W is the vector of filter weights, μ is the step size, e is the error, and Q is the quotient, which is defined as $$Q=\frac{\mu \cdot e}{u\text{'}u}$$

• Same as first input –– The word length and fraction length of these quantities match those of the first input to the block.

• Binary point scaling –– You can enter the word length and the fraction length of these quantities, in bits. The word length of the quantities must be the same, but the fraction lengths can differ.

Accumulator — Word and fraction lengths of accumulators

Same as first input (default) | Binary point scaling

Use this parameter to specify how you want to designate the word and fraction lengths of the accumulators for the u'u and W'u operations.

Note

Do not use this parameter to designate the word and fraction lengths of the accumulator for the $$Q\cdot u$$ operation. The accumulator data type for this quantity is automatically set to be the same as the product data type. The minimum, maximum, and overflow information for this accumulator is logged as part of the product information. Autoscaling treats this product and accumulator as one data type.

• Same as first input –– These characteristics match those of the input to the block.

• Binary point scaling –– You can enter the word length and the fraction length of the accumulators, in bits. The word length of both the accumulators must be the same, but the fraction lengths can differ.

For illustrations depicting the use of the accumulator data type in this block, see Fixed-Point Data Types and Multiplication Data Types.

Lock data type settings against changes by the fixed-point tools — Prevent fixed-point tools from overriding data types

off (default) | on

Select this parameter to prevent the fixed-point tools from overriding the data types you specify in the block dialog box.

Block Characteristics

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

More About

LMS Filter Algorithms

When you select LMS for the Algorithm parameter, the block calculates the filter weights by using the least mean-square (LMS) algorithm. This algorithm is defined by these equations.

$$\begin{array}{c}y(n)=w^T(n-1)u(n)\\ e(n)=d(n)-y(n)\\ w(n)=\alpha w(n-1)+f(u(n),e(n),\mu )\end{array}$$

The various LMS adaptive filter algorithms available in this block are defined as:

• LMS ––

  $$f(u(n),e(n),\mu )=\mu e(n)u^{*}(n)$$

• Normalized LMS ––

  $$f(u(n),e(n),\mu )=\mu e(n)\frac{u^{\ast }(n)}{\epsilon +u^H(n)u(n)}$$

  In Normalized LMS, to overcome potential numerical instability in the update of the weights, a small positive constant, ε, has been added in the denominator. For double-precision floating-point input, ε is the output of the eps function. For single-precision floating-point input, ε is the output of eps("single"). For fixed-point input, ε is 0.

• Sign-Error LMS ––

  $$f(u(n),e(n),\mu )=\mu \text{sign}(e(n))u^{*}(n)$$

• Sign-Data LMS ––

  $$f(u(n),e(n),\mu )=\mu e(n)\text{sign}(u(n))$$

  where u(n) is real.

• Sign-Sign LMS ––

  $$f(u(n),e(n),\mu )=\mu \text{sign}(e(n))\text{sign}(u(n))$$

  where u(n) is real.

Variable	Description
n	The current time index
u(n)	The vector of buffered input samples at step n
u*(n)	The complex conjugate of the vector of buffered input samples at step n
w(n)	The vector of filter weight estimates at step n
y(n)	The filtered output at step n
e(n)	The estimation error at step n
d(n)	The desired response at step n
µ	The adaptation step size
α	The leakage factor (0 < α ≤ 1)
ε	A constant that corrects any potential numerical instability that occurs during the update of weights.

Fixed-Point Data Types

The following diagrams show the data types used within the LMS Filter block for fixed-point signals. The table summarizes the definitions of variables used in the diagrams.

Variable	Definition
u	Input vector
W	Vector of filter weights
µ	Step size
e	Error
Q	Quotient, $$Q=\frac{\mu \cdot e}{u\text{'}u}$$
Product u'u	Product data type in Energy calculation diagram
Accumulator u'u	Accumulator data type in Energy calculation diagram
Product W'u	Product data type in Convolution diagram
Accumulator W'u	Accumulator data type in Convolution diagram
Product $$\mu \cdot e$$	Product data type in Product of step size and error diagram
Product $$Q\cdot u$$	Product and accumulator data type in Weight update diagram. 1

¹The accumulator data type for this quantity is automatically set to be the same as the product data type. The minimum, maximum, and overflow information for this accumulator is logged as part of the product information. Autoscaling treats this product and accumulator as one data type.

You can set the data type of the parameters, weights, products, quotient, and accumulators in the block mask. Fixed-point inputs, outputs, and mask parameters of this block must have these characteristics:

• The input signal and the desired signal must have the same word length, but their fraction lengths can differ.

• The step size and leakage factor must have the same word length, but their fraction lengths can differ.

• The output signal and the error signal have the same word length and the same fraction length as the desired signal.

• The quotient and the product output of the u'u, W'u, $$\mu \cdot e$$, and $$Q\cdot u$$ operations must have the same word length, but their fraction lengths can differ.

• The accumulator data type of the u'u and W'u operations must have the same word length, but their fraction lengths can differ.

The output of the multiplier is in the product output data type if at least one of the inputs to the multiplier is real. If both of the inputs to the multiplier are complex, the result of the multiplication is in the accumulator data type. For details, see Multiplication Data Types.

References

[1] Hayes, M.H. Statistical Digital Signal Processing and Modeling. New York: John Wiley & Sons, 1996.

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.

Generate SIMD code using Intel^® AVX2 code replacement library

Note

Requires Embedded Coder^® license

The LMS Filter block supports SIMD code generation using Intel AVX2 code replacement library under these conditions:

• You set Algorithm to LMS or Normalized LMS.

• Input signal is real-valued.

• Input signal has a data type of single or double.

To generate SIMD code from this block using this workflow, see Use Intel AVX2 Code Replacement Library to Generate SIMD Code from Simulink Blocks.

Generate SIMD code by leveraging target hardware instruction set extensions (since R2023b)

Note

Requires Simulink^® Coder™ or Embedded Coder license

You can generate SIMD code for the LMS Filter block on all Intel platforms, ARM^® Cortex^®-A processors, and Apple silicon processors by using the model configuration parameter Leverage target hardware instruction set extensions under these conditions:

• You set Algorithm to Sign-Error LMS, Sign-Data LMS, Sign-Sign LMS, LMS or Normalized LMS.

• Input signal is real-valued with real filter coefficients

• Input signal is complex-valued with real or complex filter coefficients.

• Data type of the input signal is single (ARM Cortex-A processors)

• Data type of the input signal is single or double (Intel platforms)

In addition, configure your model appropriately. In the Modeling tab of the Simulink model window, click Model Settings and configure these parameters under Code Generation.

• In the Optimization pane:

  • Provide a specific instruction set in the Leverage target hardware instruction set extensions parameter.

  • Select the Optimize reductions parameter.

  • Under Optimization levels, set Level to Maximum and Priority to Maximize execution speed.

• In the Interface pane, under Software environment, clear non-finite numbers.

To generate SIMD code from this block using this workflow, see Use Target Hardware Instruction Set Extensions to Generate SIMD Code from Simulink Blocks for Intel Platforms, Use Target Hardware Instruction Set Extensions to Generate SIMD Code from Simulink Blocks for Apple silicon, and Use Target Hardware Instruction Set Extensions to Generate SIMD Code from Simulink Blocks for ARM Cortex-A Processors.

For computationally intensive operations on supported blocks, SIMD intrinsics can significantly improve the performance of the generated code on Intel platforms. For more details, see Optimize Code for Reduction Operations by Using SIMD (Simulink Coder).

For more information on SIMD code generation in DSP System Toolbox™, see SIMD Code Generation.

HDL Code Generation

Generate VHDL, Verilog and SystemVerilog code for FPGA and ASIC designs using HDL Coder™.

HDL Coder™ provides additional configuration options that affect HDL implementation and synthesized logic.

HDL Architecture

By default, the LMS Filter implementation uses a linear sum for the FIR section of the filter.

The LMS Filter implements a tree summation (which has a shorter critical path) under the following conditions:

• The LMS Filter is used with real data.

• The word length of the Accumulator W'u data type is at least ceil(log2(filter length)) bits wider than the word length of the Product W'u data type.

• The Accumulator W'u data type has the same fraction length as the Product W'u data type.

HDL Block Properties

ConstrainedOutputPipeline	Number of registers to place at the outputs by moving existing delays within your design. Distributed pipelining does not redistribute these registers. The default is 0. For more details, see ConstrainedOutputPipeline (HDL Coder).
InputPipeline	Number of input pipeline stages to insert in the generated code. Distributed pipelining and constrained output pipelining can move these registers. The default is 0. For more details, see InputPipeline (HDL Coder).
OutputPipeline	Number of output pipeline stages to insert in the generated code. Distributed pipelining and constrained output pipelining can move these registers. The default is 0. For more details, see OutputPipeline (HDL Coder).

Complex Data Support

This block supports code generation for complex signals.

Restrictions

• HDL Coder does not support the Normalized LMS algorithm of the LMS Filter.

• The Reset port supports only Boolean and unsigned inputs.

• The Adapt port supports only Boolean inputs.

• Filter length must be greater than or equal to 2.

Version History

Introduced before R2006a

R2024b: Generate SIMD code for Apple silicon processors

In R2024b, if you have Embedded Coder, you can generate SIMD code for the LMS Filter block for Apple silicon processors by using the model configuration parameter Leverage target hardware instruction set extensions. For more information, see Use Target Hardware Instruction Set Extensions to Generate SIMD Code from Simulink Blocks for Apple silicon.

R2023b: Generate SIMD Code for Complex Signals

In R2023b, if you have Embedded Coder, you can generate SIMD code for the LMS Filter block when the input signal is complex-valued by using the model configuration parameter Leverage target hardware instruction set extensions.