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
TimeDelay Channel Estimation Through Adaptive Filtering
Adaptively estimate the time delay for a noisy input signal using the LMS adaptive FIR algorithm.
Adaptive Filter Convergence
Compares the rate of convergence for adaptive filters using different LMS algorithms.
Ports
Input
Input — Input signal
scalar  column vector
Input signal, specified as a scalar or column vector.
When the input is fixedpoint, 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 fixedpoint, the desired signal must be a signed fixedpoint.
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
Stepsize — Stepsize
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 fixedpoint, the stepsize must be a signed fixedpoint.
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 Nonzero
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 floatingpoint signals. Obeys the
Weights
parameter for fixedpoint 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
 SignError LMS
 SignData LMS
 SignSign 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 theStepsize
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
 Nonzero 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 aRising edge
orFalling edge
Nonzero 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 fixedpoint 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
fixedpoint operation. When you clear this parameter, the block wraps
the result of its fixedpoint operation. For details on
saturate
and wrap
, see overflow
mode for fixedpoint 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 binarypoint 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 binarypoint 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 Stepsize
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 FixedPoint Data Types and Multiplication Data Types.
Lock data type settings against changes by the fixedpoint tools — Prevent fixedpoint tools from overriding data types
off
(default)  on
Select this parameter to prevent the fixedpoint tools from overriding the data types you specify in the block dialog box.
Block Characteristics
Data Types 

Direct Feedthrough 

Multidimensional Signals 

VariableSize Signals 

ZeroCrossing Detection 

More About
LMS Filter Algorithms
When you select LMS
for the Algorithm
parameter, the block calculates the filter weights by
using the least meansquare (LMS) algorithm. This algorithm is defined by these equations.
$$\begin{array}{c}y(n)={w}^{T}(n1)u(n)\\ e(n)=d(n)y(n)\\ w(n)=\alpha w(n1)+f(u(n),e(n),\mu )\end{array}$$
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 doubleprecision floatingpoint input, ε is the output of the
eps
function. For singleprecision floatingpoint input, ε is the output ofeps("single")
. For fixedpoint input, ε is 0.SignError LMS ––
$$f(u(n),e(n),\mu )=\mu \text{sign}(e(n)){u}^{*}(n)$$
SignData LMS ––
$$f(u(n),e(n),\mu )=\mu e(n)\text{sign}(u(n))$$
where u(n) is real.
SignSign 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. 
FixedPoint Data Types
The following diagrams show the data types used within the LMS Filter block for fixedpoint 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} 
^{1}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.
You can set the data type of the parameters, weights, products, quotient, and accumulators in the block mask. Fixedpoint 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
orNormalized LMS
.Input signal is realvalued.
Input signal has a data type of
single
ordouble
.
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
SignError LMS
,SignData LMS
,SignSign LMS
,LMS
orNormalized LMS
.Input signal is realvalued with real filter coefficients
Input signal is complexvalued with real or complex filter coefficients.
Data type of the input signal is
single
(ARM CortexA processors)Data type of the input signal is
single
ordouble
(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 toMaximize execution speed
.
In the Interface pane, under Software environment, clear nonfinite 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 CortexA 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.
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.
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

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

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

This block supports code generation for complex signals.
HDL Coder does not support the
Normalized LMS
algorithm of the LMS Filter.The
Reset
port supports onlyBoolean
andunsigned
inputs.The
Adapt
port supports onlyBoolean
inputs.Filter length must be greater than or equal to 2.
Version History
Introduced before R2006aR2024b: 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 complexvalued by using the model configuration parameter Leverage target hardware instruction set extensions.
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