# Least Mean Square (LMS) Filters
[Filtering Functions]

## Functions

void | arm_lms_init_f32 (arm_lms_instance_f32 *S, uint16_t numTaps, float32_t *pCoeffs, float32_t *pState, float32_t mu, uint32_t blockSize)
void | arm_lms_f32 (const arm_lms_instance_f32 *S, float32_t *pSrc, float32_t *pRef, float32_t *pOut, float32_t *pErr, uint32_t blockSize)
void | arm_lms_init_q31 (arm_lms_instance_q31 *S, uint16_t numTaps, q31_t *pCoeffs, q31_t *pState, q31_t mu, uint32_t blockSize, uint32_t postShift)
void | arm_lms_q31 (const arm_lms_instance_q31 *S, q31_t *pSrc, q31_t *pRef, q31_t *pOut, q31_t *pErr, uint32_t blockSize)
void | arm_lms_init_q15 (arm_lms_instance_q15 *S, uint16_t numTaps, q15_t *pCoeffs, q15_t *pState, q15_t mu, uint32_t blockSize, uint32_t postShift)
void | arm_lms_q15 (const arm_lms_instance_q15 *S, q15_t *pSrc, q15_t *pRef, q15_t *pOut, q15_t *pErr, uint32_t blockSize)

## 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 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

void arm_lms_init_f32 | ( | arm_lms_instance_f32 * | S,
|  | uint16_t | numTaps,
|  | float32_t * | pCoeffs,
|  | float32_t * | pState,
|  | float32_t | mu,
|  | uint32_t | blockSize
| ) |  |

Initialization function for floating-point LMS filter.

Parameters:

  [in] | *S | points to an instance of the floating-point LMS filter structure.
  [in] | numTaps | number of filter coefficients.
  [in] | *pCoeffs | points to the coefficient buffer.
  [in] | *pState | points to state buffer.
  [in] | mu | step size that controls filter coefficient updates.
  [in] | blockSize | number of samples to process.

Returns:
  none.

Description:
  `pCoeffs` points to the array of filter coefficients stored in time reversed order:

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

  The initial filter coefficients serve as a starting point for the adaptive filter. `pState` points to an array of length `numTaps+blockSize-1` samples, where `blockSize` is the number of input samples processed by each call to `arm_lms_f32()`.

Definition at line 61 of file arm_lms_init_f32.c.

void arm_lms_f32 | ( | const arm_lms_instance_f32 * | S,
|  | float32_t * | pSrc,
|  | float32_t * | pRef,
|  | float32_t * | pOut,
|  | float32_t * | pErr,
|  | uint32_t | blockSize
| ) |  |

Processing function for floating-point LMS filter.

This function operates on floating-point data types.

Parameters:

  [in] | *S | points to an instance of the floating-point LMS filter structure.
  [in] | *pSrc | points to the block of input data.
  [in] | *pRef | points to the block of reference data.
  [out] | *pOut | points to the block of output data.
  [out] | *pErr | points to the block of error data.
  [in] | blockSize | number of samples to process.

Returns:
  none.

Definition at line 170 of file arm_lms_f32.c.

void arm_lms_init_q31 | ( | arm_lms_instance_q31 * | S,
|  | uint16_t | numTaps,
|  | q31_t * | pCoeffs,
|  | q31_t * | pState,
|  | q31_t | mu,
|  | uint32_t | blockSize,
|  | uint32_t | postShift
| ) |  |

Initialization function for Q31 LMS filter.

Parameters:

  [in] | *S | points to an instance of the Q31 LMS filter structure.
  [in] | numTaps | number of filter coefficients.
  [in] | *pCoeffs | points to coefficient buffer.
  [in] | *pState | points to state buffer.
  [in] | mu | step size that controls filter coefficient updates.
  [in] | blockSize | number of samples to process.
  [in] | postShift | bit shift applied to coefficients.

Returns:
  none.

Description:
  `pCoeffs` points to the array of filter coefficients stored in time reversed order:

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

  The initial filter coefficients serve as a starting point for the adaptive filter. `pState` points to an array of length `numTaps+blockSize-1` samples, where `blockSize` is the number of input samples processed by each call to `arm_lms_q31()`.

Definition at line 66 of file arm_lms_init_q31.c.

void arm_lms_q31 | ( | const arm_lms_instance_q31 * | S,
|  | q31_t * | pSrc,
|  | q31_t * | pRef,
|  | q31_t * | pOut,
|  | q31_t * | pErr,
|  | uint32_t | blockSize
| ) |  |

Processing function for Q31 LMS filter.

Parameters:

  [in] | *S | points to an instance of the Q15 LMS filter structure.
  [in] | *pSrc | points to the block of input data.
  [in] | *pRef | points to the block of reference data.
  [out] | *pOut | points to the block of output data.
  [out] | *pErr | points to the block of error data.
  [in] | blockSize | number of samples to process.

Returns:
  none.

Scaling and Overflow Behavior:
  The function is implemented using an internal 64-bit accumulator. The accumulator has a 2.62 format and maintains full precision of the intermediate multiplication results but provides only a single guard bit. Thus, if the accumulator result overflows it wraps around rather than clips. In order to avoid overflows completely the input signal must be scaled down by log2(numTaps) bits. The reference signal should not be scaled down. After all multiply-accumulates are performed, the 2.62 accumulator is shifted and saturated to 1.31 format to yield the final result. The output signal and error signal are in 1.31 format.

  In this filter, filter coefficients are updated for each sample and the updation of filter cofficients are saturted.

Definition at line 69 of file arm_lms_q31.c.

void arm_lms_init_q15 | ( | arm_lms_instance_q15 * | S,
|  | uint16_t | numTaps,
|  | q15_t * | pCoeffs,
|  | q15_t * | pState,
|  | q15_t | mu,
|  | uint32_t | blockSize,
|  | uint32_t | postShift
| ) |  |

Initialization function for the Q15 LMS filter.

Parameters:

  [in] | *S | points to an instance of the Q15 LMS filter structure.
  [in] | numTaps | number of filter coefficients.
  [in] | *pCoeffs | points to the coefficient buffer.
  [in] | *pState | points to the state buffer.
  [in] | mu | step size that controls filter coefficient updates.
  [in] | blockSize | number of samples to process.
  [in] | postShift | bit shift applied to coefficients.

Returns:
  none.

Description:
  `pCoeffs` points to the array of filter coefficients stored in time reversed order:

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

  The initial filter coefficients serve as a starting point for the adaptive filter. `pState` points to the array of state variables and size of array is `numTaps+blockSize-1` samples, where `blockSize` is the number of input samples processed by each call to `arm_lms_q15()`.

Definition at line 66 of file arm_lms_init_q15.c.

void arm_lms_q15 | ( | const arm_lms_instance_q15 * | S,
|  | q15_t * | pSrc,
|  | q15_t * | pRef,
|  | q15_t * | pOut,
|  | q15_t * | pErr,
|  | uint32_t | blockSize
| ) |  |

Processing function for Q15 LMS filter.

Parameters:

  [in] | *S | points to an instance of the Q15 LMS filter structure.
  [in] | *pSrc | points to the block of input data.
  [in] | *pRef | points to the block of reference data.
  [out] | *pOut | points to the block of output data.
  [out] | *pErr | points to the block of error data.
  [in] | blockSize | number of samples to process.

Returns:
  none.

Scaling and Overflow Behavior:
  The function is implemented using a 64-bit internal accumulator. Both coefficients and state variables are represented in 1.15 format and multiplications yield a 2.30 result. The 2.30 intermediate results are accumulated in a 64-bit accumulator in 34.30 format. There is no risk of internal overflow with this approach and the full precision of intermediate multiplications is preserved. After all additions have been performed, the accumulator is truncated to 34.15 format by discarding low 15 bits. Lastly, the accumulator is saturated to yield a result in 1.15 format.

  In this filter, filter coefficients are updated for each sample and the updation of filter cofficients are saturted.

Definition at line 66 of file arm_lms_q15.c.

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