LMS Filter

Compute output, error, and weights using LMS adaptive algorithm

Library:
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.

Ports

Input

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`Input` — Input signal
scalar | vector

Connect the signal that you want to filter, specified as a scalar or column vector.

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

Data Types: single | double | fixed point

`Desired` — Desired signal
vector | matrix

Connect the desired signal, specified as a vector or matrix. 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.

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 | vector

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 | vector

Signal to reset the value of the filter weights to their initial values, specified as a vector. 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

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`Output` — Output signal
scalar | vector

Outputs the filtered input signal, which is the estimate of the desired signal. 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 | vector

Outputs 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` — Updated filter weights
scalar | 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

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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
`on` (default) | `off`

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, $μ \cdot e$ , $Q \cdot u$ , 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 $Q = \frac{μ \cdot e}{u' u}$

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 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 on the block dialog.

Model Examples

Time-Delay Estimation

Adaptively estimate the time delay for a noisy input signal using the LMS adaptive FIR algorithm. The peak in the filter taps vector indicates the time-delay estimate.

Adaptive Filter Convergence

The convergence path taken by different adaptive filtering algorithms. The plot is a sequence of points of the form (w1,w2) where w1 and w2 are the weights of the adaptive filter. The blue dots in the figure indicate the contour lines of the error surface. Zoom into the graph to see properties of convergence path by selecting Zoom In from the Tools menu.

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

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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.

$\begin{matrix} y (n) = w^{T} (n - 1) u (n) \\ e (n) = d (n) - y (n) \\ w (n) = α w (n - 1) + f (u (n), e (n), μ) \end{matrix}$

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

LMS ––

$f (u (n), e (n), μ) = μ e (n) u^{*} (n)$
Normalized LMS ––

$f (u (n), e (n), μ) = μ e (n) \frac{u^{*} (n)}{ε + u^{H} (n) u (n)}$
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 2.2204460492503131e-016. For single-precision floating-point input, ε is 1.192092896e-07. For fixed-point input, ε is 0.
Sign-Error LMS ––

$f (u (n), e (n), μ) = μ sign (e (n)) u^{*} (n)$
Sign-Data LMS ––

$f (u (n), e (n), μ) = μ e (n) sign (u (n))$
where u(n) is real.
Sign-Sign LMS ––

$f (u (n), e (n), μ) = μ sign (e (n)) sign (u (n))$
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.

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 memcpy or memset functions (string.h) under certain conditions.

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

Algorithm is set to LMS or Normalized LMS.
Input signal is real-valued.
Input signal has a data type of single or double.

The SIMD technology significantly improves the performance of the generated code.

HDL Code Generation
Generate Verilog and VHDL 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.

Fixed-Point Conversion
Design and simulate fixed-point systems using Fixed-Point Designer™.

If the input is fixed point, it must be signed integer or signed fixed point with power-of-two slope and zero bias.

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, $Q = \frac{μ \cdot e}{u' u}$
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 $μ \cdot e$	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, $μ \cdot e$ , 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.

Documentation

LMS Filter

Description

Ports

Input

Input — Input signal scalar | vector

Desired — Desired signal vector | matrix

Step-size — Step-size scalar

Dependencies

Adapt — Update filter weights scalar | vector

Dependencies

Reset — Reset filter weights scalar | vector

Dependencies

Output

Output — Output signal scalar | vector

Error — Error between Output and Desired signals scalar | vector

Wts — Updated filter weights scalar | vector

Dependencies

Parameters

Main Tab

Algorithm — Select algorithm LMS (default) | Normalized LMS | Sign-Error LMS | Sign-Data LMS | Sign-Sign LMS

Filter length — Filter length 32 (default) | scalar

Specify step size via — Specify step size via Dialog (default) | Input port

Step size (mu) — Step size 0.1 (default) | positive scalar

Dependencies

Leakage factor (0 to 1) — Leakage factor 1.0 (default) | scalar

Initial value of filter weights — Initial value of filter weights 0 (default) | vector | scalar

Adapt port — Enable Adapt port on (default) | off

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

Output filter weights — Output filter weights on (default) | off

Data Type Tab

Rounding mode — Method of rounding operation Floor (default) | Ceiling | Convergent | Nearest | Round | Simplest | Zero

Saturate on integer overflow — Method of overflow action off (default) | on

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

Dependencies

Weights — Word and fraction length of filter weights Same as first input (default) | Binary point scaling

Product and quotient — Word and fraction length of product and quotient Same as first input (default) | Binary point scaling

Accumulator — Word and fraction lengths of accumulators Same as first input (default) | Binary point scaling

Lock data type settings against changes by the fixed-point tools — Prevent fixed-point tools from overriding data types off (default) | on

Model Examples

Time-Delay Estimation

Adaptive Filter Convergence

Block Characteristics

More About

LMS Filter Algorithms

References

Extended Capabilities

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

HDL Code Generation Generate Verilog and VHDL code for FPGA and ASIC designs using HDL Coder™.

Fixed-Point Conversion Design and simulate fixed-point systems using Fixed-Point Designer™.

See Also

Blocks

Topics

DSP System Toolbox Documentation

Support

`Input` — Input signal
scalar | vector

`Desired` — Desired signal
vector | matrix

`Step-size` — Step-size
scalar

`Adapt` — Update filter weights
scalar | vector

`Reset` — Reset filter weights
scalar | vector

`Output` — Output signal
scalar | vector

`Error` — Error between Output and Desired signals
scalar | vector

`Wts` — Updated filter weights
scalar | vector

`Algorithm` — Select algorithm
`LMS` (default) | `Normalized LMS` | `Sign-Error LMS` | `Sign-Data LMS` | `Sign-Sign LMS`

`Filter length` — Filter length
`32` (default) | scalar

`Specify step size via` — Specify step size via
`Dialog` (default) | `Input port`

`Step size (mu)` — Step size
`0.1` (default) | positive scalar

`Leakage factor (0 to 1)` — Leakage factor
`1.0` (default) | scalar

`Initial value of filter weights` — Initial value of filter weights
`0` (default) | vector | scalar

`Adapt port` — Enable Adapt port
`on` (default) | `off`

`Reset port` — Reset port
`None` (default) | `Rising edge` | `Falling edge` | `Either edge` | `Non-zero sample`

`Output filter weights` — Output filter weights
`on` (default) | `off`

`Rounding mode` — Method of rounding operation
`Floor` (default) | `Ceiling` | `Convergent` | `Nearest` | `Round` | `Simplest` | `Zero`

`Saturate on integer overflow` — Method of overflow action
off (default) | on

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

`Weights` — Word and fraction length of filter weights
`Same as first input` (default) | `Binary point scaling`

`Product and quotient` — Word and fraction length of product and quotient
`Same as first input` (default) | `Binary point scaling`

`Accumulator` — Word and fraction lengths of accumulators
`Same as first input` (default) | `Binary point scaling`

`Lock data type settings against changes by the fixed-point tools` — Prevent fixed-point tools from overriding data types
`off` (default) | `on`

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

HDL Code Generation
Generate Verilog and VHDL code for FPGA and ASIC designs using HDL Coder™.

Fixed-Point Conversion
Design and simulate fixed-point systems using Fixed-Point Designer™.