Least Mean Square (LMS) Filters
[Filtering Functions]

LMS filters are a class of adaptive filters that are able to "learn" an unknown transfer functions. More...

Detailed Description

LMS filters are a class of adaptive filters that are able to "learn" an unknown transfer functions.

LMS filters use a gradient descent method in which the filter coefficients are updated based on the instantaneous error signal. Adaptive filters are often used in communication systems, equalizers, and noise removal. The CMSIS DSP Library contains LMS filter functions that operate on Q15, Q31, and floating-point data types. The library also contains normalized LMS filters in which the filter coefficient adaptation is indepedent of the level of the input signal.

An LMS filter consists of two components as shown below. The first component is a standard transversal or FIR filter. The second component is a coefficient update mechanism. The LMS filter has two input signals. The "input" feeds the FIR filter while the "reference input" corresponds to the desired output of the FIR filter. That is, the FIR filter coefficients are updated so that the output of the FIR filter matches the reference input. The filter coefficient update mechanism is based on the difference between the FIR filter output and the reference input. This "error signal" tends towards zero as the filter adapts. The LMS processing functions accept the input and reference input signals and generate the filter output and error signal.

Internal structure of the Least Mean Square filter

The functions operate on blocks of data and each call to the function processes blockSize samples through the filter. pSrc points to input signal, pRef points to reference signal, pOut points to output signal and pErr points to error signal. All arrays contain blockSize values.

The API functions operate on a block-by-block basis. Internally, the filter coefficients b[n] are updated on a sample-by-sample basis. The convergence of the LMS filter is slower compared to the normalized LMS algorithm.

Algorithm:

The output signal y[n] is computed by a standard FIR filter:

  
     y[n] = b[0] * x[n] + b[1] * x[n-1] + b[2] * x[n-2] + ...+ b[numTaps-1] * x[n-numTaps+1]  
 

The error signal equals the difference between the reference signal d[n] and the filter output:

  
     e[n] = d[n] - y[n].  
 

After each sample of the error signal is computed, the filter coefficients b[k] are updated on a sample-by-sample basis:

  
     b[k] = b[k] + e[n] * mu * x[n-k],  for k=0, 1, ..., numTaps-1  
 

where mu is the step size and controls the rate of coefficient convergence.

In the APIs, pCoeffs points to a coefficient array of size numTaps. Coefficients are stored in time reversed order.

  
    {b[numTaps-1], b[numTaps-2], b[N-2], ..., b[1], b[0]}  
 

pState points to a state array of size numTaps + blockSize - 1. Samples in the state buffer are stored in the order:

  
    {x[n-numTaps+1], x[n-numTaps], x[n-numTaps-1], x[n-numTaps-2]....x[0], x[1], ..., x[blockSize-1]}  
 

Note that the length of the state buffer exceeds the length of the coefficient array by blockSize-1 samples. The increased state buffer length allows circular addressing, which is traditionally used in FIR filters, to be avoided and yields a significant speed improvement. The state variables are updated after each block of data is processed.

Instance Structure

The coefficients and state variables for a filter are stored together in an instance data structure. A separate instance structure must be defined for each filter and coefficient and state arrays cannot be shared among instances. There are separate instance structure declarations for each of the 3 supported data types.

Initialization Functions

There is also an associated initialization function for each data type. The initialization function performs the following operations:

• Sets the values of the internal structure fields.
• Zeros out the values in the state buffer.

Use of the initialization function is optional. However, if the initialization function is used, then the instance structure cannot be placed into a const data section. To place an instance structure into a const data section, the instance structure must be manually initialized. Set the values in the state buffer to zeros before static initialization. The code below statically initializes each of the 3 different data type filter instance structures

  
    arm_lms_instance_f32 S = {numTaps, pState, pCoeffs, mu};  
    arm_lms_instance_q31 S = {numTaps, pState, pCoeffs, mu, postShift};  
    arm_lms_instance_q15 S = {numTaps, pState, pCoeffs, mu, postShift};  
 

where numTaps is the number of filter coefficients in the filter; pState is the address of the state buffer; pCoeffs is the address of the coefficient buffer; mu is the step size parameter; and postShift is the shift applied to coefficients.

Fixed-Point Behavior:

Care must be taken when using the Q15 and Q31 versions of the LMS filter. The following issues must be considered:

• Scaling of coefficients
• Overflow and saturation

Scaling of Coefficients:

Filter coefficients are represented as fractional values and coefficients are restricted to lie in the range [-1 +1). The fixed-point functions have an additional scaling parameter postShift. At the output of the filter's accumulator is a shift register which shifts the result by postShift bits. This essentially scales the filter coefficients by 2^postShift and allows the filter coefficients to exceed the range [+1 -1). The value of postShift is set by the user based on the expected gain through the system being modeled.

Overflow and Saturation:

Overflow and saturation behavior of the fixed-point Q15 and Q31 versions are described separately as part of the function specific documentation below.

Function Documentation