LMS Update
Estimate weights of least mean squares (LMS) adaptive filter
Libraries:
DSP System Toolbox /
Filtering /
Adaptive Filters
Description
The LMS Update block estimates the weights of an least mean squares (LMS) adaptive filter. Specify data and error as inputs and the block computes the filter weights based on the algorithm specified in the Algorithm parameter. For more details on the algorithms, see Algorithms.
You can configure the block to operate as a tappeddelay line FIR filter or as an adaptive linear combiner using the Adaptive filter mode parameter. The block can also adapt multiple filters independently when you specify the Number of adaptive filters parameter to a value greater than 1.
You can use this block to compute the adaptive filter weights in applications such as system identification, inverse modeling, and filteredx LMS algorithms, which are used in acoustic noise cancellation. For more details, see References.
Examples
Active Noise Control with Simulink RealTime
Design a realtime active noise control system using a Speedgoat® Simulink® RealTime™ target.
(Audio Toolbox)
Adapt Multiple Filters Using LMS Update Block
Adapt multiple filters independently using the same LMS Update block. Use the Adapt port of the LMS Update block to selectively enable or disable the filters from being adapted.
Model Adaptive Linear Combiner using LMS Update Block
Use LMS Update block as an adaptive linear combiner.
Ports
Input
Input — Data input
scalar  column vector
Data input to the adaptive filter, specified as one of these:
Scalar –– When you set the Adaptive filter mode parameter to
Tapped delayline FIR filter
, the input to the block should be a scalar.The block algorithm buffers the input samples to generate the vector $$u(n)=\left[\text{u}(n),\text{u}(n1),\mathrm{...},\text{u}(nL+1)\right]$$, where L is the length of the adaptive filter and n is the time index. The block algorithm then passes the vector through the filter.
Column vector of length L –– When you set the Adaptive filter mode parameter to
Adaptive linear combiner
, the block does not buffer the input samples, and the input to the block should be in the form of a column vector. The block multiplies the vector with the filter weights to generate the filter output. In this case, there is no relationship between the input samples.
When you set the Number of adaptive filters parameter to a value greater than 1, the block assumes that all filters run on the same input.
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
Complex Number Support: Yes
Error — Error
scalar  column vector
Specify the error between the output signal and the desired signal as a:
Scalar when the number of adaptive filters N = 1.
Column vector of length N when N > 1.
When you set the Algorithm parameter to
Sign—Error LMS
,
Sign—Data LMS
, or
Sign—Sign LMS
, the error input
through the Error port must be real.
Data Types: single
 double
Complex Number Support: Yes
Mu — Filter adaptation step size
0.1 (default)  scalar in the range [0,1]
Specify the filter adaptation step size as a scalar in the range [0,1].
Dependency
To enable this port, set Step size source to
Input port
.
Data Types: single
 double
Adapt — Update filter weights
real scalar  vector
The input to this port can be one of these:
Scalar –– If the number of adaptive filters (N) is greater than 1, then the block updates the filter weights for all filters if the value at the port is nonzero and does not update if the value at the port is 0.
Vector of length N –– If the number of adaptive filters (N) is greater than 1, then the block determines whether to update each filter during run time depending on the value of the corresponding element in the Adapt vector input. When the Adapt port input element is not 0, the block updates the filter weights. When the input is 0, the filter weights do not change.
For an example, see Adapt Multiple Filters Using LMS Update Block.
Dependency
To enable this port, select the Enable adapt input parameter on the block dialog.
Data Types: single
 double
 Boolean
 int16
 int32
 int64
 int8
 uint16
 uint32
 uint64
 uint8
Reset — Reset filter weights
real scalar
When the input to this port is not 0, the block resets the filter weights to their initial values. When the input to this port is 0, the filter weights do not change.
Dependency
To enable this port, select the Enable reset input parameter on the block dialog.
Data Types: single
 double
 Boolean
 int16
 int32
 int64
 int8
 uint16
 uint32
 uint64
 uint8
Output
Wts — Filter weights
row vector  matrix
The block outputs the filter weights as a 1byL row vector or a NbyL matrix, where N is the number of adaptive filters, and L is the length of each filter.
Data Types: single
 double
Parameters
Algorithm — LMS adaptive algorithm
LMS
(default)  Normalized LMS
 Sign—Error LMS
 Sign—Data LMS
 Sign—Sign LMS
The block uses one of the listed algorithms to compute the filter weights. For more details on the algorithms, see Algorithms.
Filter length — Length of the filter
32 (default)  positive integer
Specify the length of each adaptive filter L as a positive integer.
When there are N adaptive filters, the weights vector the block generates through the Wts output port is an NbyL matrix.
Number of adaptive filters — Number of adaptive filters
1 (default)  positive integer
Specify the number of adaptive filters N the block supports as a positive integer.
When you specify this value to be an integer greater than 1, each filter can be chosen to adapt independently during run time by providing the Adapt input as a logical vector of length N. All filters are assumed to run on the same input and have a common step size, leakage factor, and reset input.
The weights the block generates through the Wts output port is an NbyL matrix, where L is the length of each filter.
For an example that shows how to adapt multiple filters, see Adapt Multiple Filters Using LMS Update Block.
Step size source — Method to specify the step size
Property
(default)  Input port
Property
— Specify the filter adaptation size using the Step size (mu) parameter.Input port
— Pass filter adaptation size using the Mu input port.
Step size (mu) — Size of the adaptation step
0.1 (default)  nonnegative scalar
Step size (mu) indicates the amount by which the filter weights are updated in each iteration. Choose an optimal step size so that the filter is stable and the convergence speed is optimal.
This parameter is tunable. You can change its value even during the simulation.
Tunable: Yes
Dependency
To enable this parameter, set Step size source to
Property
.
Leakage factor (0 to 1) — Leakage factor
1.0 (default)  real scalar in the range [0,1]
Leakage factor (0 to 1) prevents unbounded growth of
the filter coefficients by reducing the drift of the coefficients from their
optimum values. A leakage factor of 1.0
indicates no
leakage. If you encounter coefficient drift, that is, large fluctuation
about the optimum solution, decrease the leakage factor until the
coefficient fluctuation becomes small.
This parameter is tunable. You can change its value even during the simulation.
Tunable: Yes
Initial value of filter weights — Initial value of filter weights
0 (default)  real scalar  vector  matrix
Specify the initial value of filter weights as one of the following:
Scalar
Vector of size 1byL –– N is set to 1
Matrix of size NbyL –– N > 1 and L > 1
where N is the number of adaptive filters specified in Number of adaptive filters and L is the length of each filter specified in Filter length.
This parameter specifies the initial value of the filter weights w(n−1). The block uses this value to compute the weights w(n), when n = 1. For more information, see Algorithms.
Adaptive filter mode — Adaptive filter mode
Tapped delayline FIR
filter
(default)  Adaptive linear combiner
Specify the adaptive filter mode as one of the following:
Tapped delayline FIR filter
–– The block assumes the variable portion of the adaptive filter to be a tapped delayline FIR filter. In this mode, the input samples are related using this equation:$$u(n)=\left[\text{u}(n),\text{u}(n1),\mathrm{...},\text{u}(nL+1)\right]$$
where n is the current time index and L is the length of the adaptive filter specified in the Filter length parameter.
The block accepts scalar input samples in this mode. The block algorithm buffers the samples to form the u(n) vector.
Adaptive linear combiner
–– The block does not buffer the input samples, and the input you provide to the block must be a column vector equal in length to the Filter length parameter. In this case, there is no relationship between the input samples.
For more details, see Algorithms.
Enable adapt input — Update the filter weights
off (default)  on
When you select this check box, the Adapt input port appears on the block. When the input to this port is greater than 0, the block updates the filter weights. When the input to this port is less than or equal to 0, the filter weights do not change.
Enable reset input — Reset the filter weights
off (default)  on
When you select this check box, the Reset input port appears on the block. When the input to this port is greater than 0, the block resets the filter weights to their initial values. When the input to this port is less than or equal to 0, the filter weights do not change.
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 

Direct Feedthrough 

Multidimensional Signals 

VariableSize Signals 

ZeroCrossing Detection 

Algorithms
In a closed loop adaptive filter, the weights or coefficients of the filter are adjusted until the error is minimized. The error is the difference between the filter output and the desired signal. In this block, the error signal is specified through the Error input port.
The block computes filter weight estimates using $$w(n)=\alpha w(n1)+f(u(n),e(n),\mu )$$.
The function $$f(u(n),e(n),\mu )$$ is defined according to the LMS algorithm you specify through the Algorithm parameter:
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 the
Normalized LMS
algorithm, ε is a small positive constant that overcomes the potential numerical instability in the update of weights.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 realSignSign LMS
— $$f(u(n),e(n),\mu )=\mu \text{sign}(e(n))\text{sign}(u(n))$$, where u(n) is real
In the equations:
n — The current time index.
u(n) — The vector of input samples at step n.
When you set the Adaptive filter mode parameter to
Tapped delayline FIR filter
, the input to the block is a scalar value and the block buffers the input samples to generate the u(n) vector.When you set the Adaptive filter mode parameter to
Adaptive linear combiner
, the block does not buffer the samples. The input to the block should be in the form of a column vector. In this case, there is no relationship between the input samples.u*(n) — The complex conjugate of the vector of input samples at step n.
w(n) — The vector of filter weight estimates at step n.
e(n) — The estimation error at step n.
µ — The adaptation step size.
α — The leakage factor (0 ≤ α ≤ 1).
Multifilter Support
You can update the weights of multiple adaptive filters using the same LMS Update block. To enable this, specify the Number of adaptive filters parameter to be a value greater than 1. You can then adapt each filter independently during run time by providing the input at the Adapt port as a logical vector of length N. The block algorithm assumes that all filters run on the same input, adapt using the same algorithm, and have the same filter length, step size, leakage factor, and reset input.
References
[1] Madisetti, Vijay, and Douglas Williams. "Introduction to Adaptive Filters." The Digital Signal Processing Handbook. Boca Raton, FL: CRC Press, 1999.
[2] Akhtar, M. T., M. Abe, M. Kawamata. "Modifiedfilteredx LMS algorithm based active noise control systems with improved online secondarypath modeling." IEEE Symposium on Circuits and Systems, 2004.
[3] Orfanidis, Sophocles J. Optimum Signal Processing. An Introduction. 2nd Ed. Englewood Cliffs, NJ: PrenticeHall, 1996.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.
Version History
Introduced in R2016bR2024a: 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.
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