CMSIS-DSP: Correlation

| CMSIS-DSP Version 1.4.7 CMSIS DSP Software Library

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## Functions
void | arm_correlate_f32 (float32_t *pSrcA, uint32_t srcALen, float32_t *pSrcB, uint32_t srcBLen, float32_t *pDst)
| Correlation of floating-point sequences.
void | arm_correlate_fast_opt_q15 (q15_t *pSrcA, uint32_t srcALen, q15_t *pSrcB, uint32_t srcBLen, q15_t *pDst, q15_t *pScratch)
| Correlation of Q15 sequences (fast version) for Cortex-M3 and Cortex-M4.
void | arm_correlate_fast_q15 (q15_t *pSrcA, uint32_t srcALen, q15_t *pSrcB, uint32_t srcBLen, q15_t *pDst)
| Correlation of Q15 sequences (fast version) for Cortex-M3 and Cortex-M4.
void | arm_correlate_fast_q31 (q31_t *pSrcA, uint32_t srcALen, q31_t *pSrcB, uint32_t srcBLen, q31_t *pDst)
| Correlation of Q31 sequences (fast version) for Cortex-M3 and Cortex-M4.
void | arm_correlate_opt_q15 (q15_t *pSrcA, uint32_t srcALen, q15_t *pSrcB, uint32_t srcBLen, q15_t *pDst, q15_t *pScratch)
| Correlation of Q15 sequences.
void | arm_correlate_opt_q7 (q7_t *pSrcA, uint32_t srcALen, q7_t *pSrcB, uint32_t srcBLen, q7_t *pDst, q15_t *pScratch1, q15_t *pScratch2)
| Correlation of Q7 sequences.
void | arm_correlate_q15 (q15_t *pSrcA, uint32_t srcALen, q15_t *pSrcB, uint32_t srcBLen, q15_t *pDst)
| Correlation of Q15 sequences.
void | arm_correlate_q31 (q31_t *pSrcA, uint32_t srcALen, q31_t *pSrcB, uint32_t srcBLen, q31_t *pDst)
| Correlation of Q31 sequences.
void | arm_correlate_q7 (q7_t *pSrcA, uint32_t srcALen, q7_t *pSrcB, uint32_t srcBLen, q7_t *pDst)
| Correlation of Q7 sequences.

## Description

Correlation is a mathematical operation that is similar to convolution. As with convolution, correlation uses two signals to produce a third signal. The underlying algorithms in correlation and convolution are identical except that one of the inputs is flipped in convolution. Correlation is commonly used to measure the similarity between two signals. It has applications in pattern recognition, cryptanalysis, and searching. The CMSIS library provides correlation functions for Q7, Q15, Q31 and floating-point data types. Fast versions of the Q15 and Q31 functions are also provided.

Algorithm
  Let `a[n]` and `b[n]` be sequences of length `srcALen` and `srcBLen` samples respectively. The convolution of the two signals is denoted by

                        c[n] = a[n] * b[n]

  In correlation, one of the signals is flipped in time

                        c[n] = a[n] * b[-n]

  and this is mathematically defined as

  The `pSrcA` points to the first input vector of length `srcALen` and `pSrcB` points to the second input vector of length `srcBLen`. The result `c[n]` is of length `2 * max(srcALen, srcBLen) - 1` and is defined over the interval `n=0, 1, 2, ..., (2 * max(srcALen, srcBLen) - 2)`. The output result is written to `pDst` and the calling function must allocate `2 * max(srcALen, srcBLen) - 1` words for the result.

Note

  The `pDst` should be initialized to all zeros before being used.

Fixed-Point Behavior

  Correlation requires summing up a large number of intermediate products. As such, the Q7, Q15, and Q31 functions run a risk of overflow and saturation. Refer to the function specific documentation below for further details of the particular algorithm used.

Fast Versions

  Fast versions are supported for Q31 and Q15. Cycles for Fast versions are less compared to Q31 and Q15 of correlate and the design requires the input signals should be scaled down to avoid intermediate overflows.

Opt Versions

  Opt versions are supported for Q15 and Q7. Design uses internal scratch buffer for getting good optimisation. These versions are optimised in cycles and consumes more memory(Scratch memory) compared to Q15 and Q7 versions of correlate

## Function Documentation

void arm_correlate_f32 | ( | float32_t * | pSrcA,
|  | uint32_t | srcALen,
|  | float32_t * | pSrcB,
|  | uint32_t | srcBLen,
|  | float32_t * | pDst
| ) |  |

Parameters

  [in] | *pSrcA | points to the first input sequence.
  [in] | srcALen | length of the first input sequence.
  [in] | *pSrcB | points to the second input sequence.
  [in] | srcBLen | length of the second input sequence.
  [out] | *pDst | points to the location where the output result is written. Length 2 * max(srcALen, srcBLen) - 1.

Returns
  none.

References srcALen, and srcBLen.

void arm_correlate_fast_opt_q15 | ( | q15_t * | pSrcA,
|  | uint32_t | srcALen,
|  | q15_t * | pSrcB,
|  | uint32_t | srcBLen,
|  | q15_t * | pDst,
|  | q15_t * | pScratch
| ) |  |

Parameters

  [in] | *pSrcA | points to the first input sequence.
  [in] | srcALen | length of the first input sequence.
  [in] | *pSrcB | points to the second input sequence.
  [in] | srcBLen | length of the second input sequence.
  [out] | *pDst | points to the location where the output result is written. Length 2 * max(srcALen, srcBLen) - 1.
  [in] | *pScratch | points to scratch buffer of size max(srcALen, srcBLen) + 2*min(srcALen, srcBLen) - 2.

Returns
  none.

Restrictions
  If the silicon does not support unaligned memory access enable the macro UNALIGNED_SUPPORT_DISABLE In this case input, output, scratch buffers should be aligned by 32-bit

Scaling and Overflow Behavior:

  This fast version uses a 32-bit accumulator with 2.30 format. The accumulator maintains full precision of the intermediate multiplication results but provides only a single guard bit. There is no saturation on intermediate additions. Thus, if the accumulator overflows it wraps around and distorts the result. The input signals should be scaled down to avoid intermediate overflows. Scale down one of the inputs by 1/min(srcALen, srcBLen) to avoid overflow since a maximum of min(srcALen, srcBLen) number of additions is carried internally. The 2.30 accumulator is right shifted by 15 bits and then saturated to 1.15 format to yield the final result.

  See `arm_correlate_q15()` for a slower implementation of this function which uses a 64-bit accumulator to avoid wrap around distortion.

References __SIMD32, _SIMD32_OFFSET, arm_copy_q15(), arm_fill_q15(), srcALen, and srcBLen.

void arm_correlate_fast_q15 | ( | q15_t * | pSrcA,
|  | uint32_t | srcALen,
|  | q15_t * | pSrcB,
|  | uint32_t | srcBLen,
|  | q15_t * | pDst
| ) |  |

Parameters

  [in] | *pSrcA | points to the first input sequence.
  [in] | srcALen | length of the first input sequence.
  [in] | *pSrcB | points to the second input sequence.
  [in] | srcBLen | length of the second input sequence.
  [out] | *pDst | points to the location where the output result is written. Length 2 * max(srcALen, srcBLen) - 1.

Returns
  none.

Scaling and Overflow Behavior:

  This fast version uses a 32-bit accumulator with 2.30 format. The accumulator maintains full precision of the intermediate multiplication results but provides only a single guard bit. There is no saturation on intermediate additions. Thus, if the accumulator overflows it wraps around and distorts the result. The input signals should be scaled down to avoid intermediate overflows. Scale down one of the inputs by 1/min(srcALen, srcBLen) to avoid overflow since a maximum of min(srcALen, srcBLen) number of additions is carried internally. The 2.30 accumulator is right shifted by 15 bits and then saturated to 1.15 format to yield the final result.

  See `arm_correlate_q15()` for a slower implementation of this function which uses a 64-bit accumulator to avoid wrap around distortion.

References __SIMD32, _SIMD32_OFFSET, srcALen, and srcBLen.

void arm_correlate_fast_q31 | ( | q31_t * | pSrcA,
|  | uint32_t | srcALen,
|  | q31_t * | pSrcB,
|  | uint32_t | srcBLen,
|  | q31_t * | pDst
| ) |  |

Parameters

  [in] | *pSrcA | points to the first input sequence.
  [in] | srcALen | length of the first input sequence.
  [in] | *pSrcB | points to the second input sequence.
  [in] | srcBLen | length of the second input sequence.
  [out] | *pDst | points to the location where the output result is written. Length 2 * max(srcALen, srcBLen) - 1.

Returns
  none.

Scaling and Overflow Behavior:

  This function is optimized for speed at the expense of fixed-point precision and overflow protection. The result of each 1.31 x 1.31 multiplication is truncated to 2.30 format. These intermediate results are accumulated in a 32-bit register in 2.30 format. Finally, the accumulator is saturated and converted to a 1.31 result.

  The fast version has the same overflow behavior as the standard version but provides less precision since it discards the low 32 bits of each multiplication result. In order to avoid overflows completely the input signals must be scaled down. The input signals should be scaled down to avoid intermediate overflows. Scale down one of the inputs by 1/min(srcALen, srcBLen)to avoid overflows since a maximum of min(srcALen, srcBLen) number of additions is carried internally.

  See `arm_correlate_q31()` for a slower implementation of this function which uses 64-bit accumulation to provide higher precision.

References srcALen, and srcBLen.

void arm_correlate_opt_q15 | ( | q15_t * | pSrcA,
|  | uint32_t | srcALen,
|  | q15_t * | pSrcB,
|  | uint32_t | srcBLen,
|  | q15_t * | pDst,
|  | q15_t * | pScratch
| ) |  |

Parameters

  [in] | *pSrcA | points to the first input sequence.
  [in] | srcALen | length of the first input sequence.
  [in] | *pSrcB | points to the second input sequence.
  [in] | srcBLen | length of the second input sequence.
  [out] | *pDst | points to the location where the output result is written. Length 2 * max(srcALen, srcBLen) - 1.
  [in] | *pScratch | points to scratch buffer of size max(srcALen, srcBLen) + 2*min(srcALen, srcBLen) - 2.

Returns
  none.

Restrictions
  If the silicon does not support unaligned memory access enable the macro UNALIGNED_SUPPORT_DISABLE In this case input, output, scratch buffers should be aligned by 32-bit

Scaling and Overflow Behavior:

  The function is implemented using a 64-bit internal accumulator. Both inputs are 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. This approach provides 33 guard bits and there is no risk of overflow. The 34.30 result is then truncated to 34.15 format by discarding the low 15 bits and then saturated to 1.15 format.

  Refer to `arm_correlate_fast_q15()` for a faster but less precise version of this function for Cortex-M3 and Cortex-M4.

References __SIMD32, _SIMD32_OFFSET, arm_copy_q15(), arm_fill_q15(), srcALen, and srcBLen.

void arm_correlate_opt_q7 | ( | q7_t * | pSrcA,
|  | uint32_t | srcALen,
|  | q7_t * | pSrcB,
|  | uint32_t | srcBLen,
|  | q7_t * | pDst,
|  | q15_t * | pScratch1,
|  | q15_t * | pScratch2
| ) |  |

Parameters

  [in] | *pSrcA | points to the first input sequence.
  [in] | srcALen | length of the first input sequence.
  [in] | *pSrcB | points to the second input sequence.
  [in] | srcBLen | length of the second input sequence.
  [out] | *pDst | points to the location where the output result is written. Length 2 * max(srcALen, srcBLen) - 1.
  [in] | *pScratch1 | points to scratch buffer(of type q15_t) of size max(srcALen, srcBLen) + 2*min(srcALen, srcBLen) - 2.
  [in] | *pScratch2 | points to scratch buffer (of type q15_t) of size min(srcALen, srcBLen).

Returns
  none.

Restrictions
  If the silicon does not support unaligned memory access enable the macro UNALIGNED_SUPPORT_DISABLE In this case input, output, scratch1 and scratch2 buffers should be aligned by 32-bit

Scaling and Overflow Behavior:

  The function is implemented using a 32-bit internal accumulator. Both the inputs are represented in 1.7 format and multiplications yield a 2.14 result. The 2.14 intermediate results are accumulated in a 32-bit accumulator in 18.14 format. This approach provides 17 guard bits and there is no risk of overflow as long as `max(srcALen, srcBLen)<131072`. The 18.14 result is then truncated to 18.7 format by discarding the low 7 bits and saturated to 1.7 format.

References __SIMD32, _SIMD32_OFFSET, arm_fill_q15(), srcALen, and srcBLen.

void arm_correlate_q15 | ( | q15_t * | pSrcA,
|  | uint32_t | srcALen,
|  | q15_t * | pSrcB,
|  | uint32_t | srcBLen,
|  | q15_t * | pDst
| ) |  |

Parameters

  [in] | *pSrcA | points to the first input sequence.
  [in] | srcALen | length of the first input sequence.
  [in] | *pSrcB | points to the second input sequence.
  [in] | srcBLen | length of the second input sequence.
  [out] | *pDst | points to the location where the output result is written. Length 2 * max(srcALen, srcBLen) - 1.

Returns
  none.

Scaling and Overflow Behavior:

  The function is implemented using a 64-bit internal accumulator. Both inputs are 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. This approach provides 33 guard bits and there is no risk of overflow. The 34.30 result is then truncated to 34.15 format by discarding the low 15 bits and then saturated to 1.15 format.

  Refer to `arm_correlate_fast_q15()` for a faster but less precise version of this function for Cortex-M3 and Cortex-M4.

  Refer the function `arm_correlate_opt_q15()` for a faster implementation of this function using scratch buffers.

References __SIMD32, _SIMD32_OFFSET, srcALen, and srcBLen.

void arm_correlate_q31 | ( | q31_t * | pSrcA,
|  | uint32_t | srcALen,
|  | q31_t * | pSrcB,
|  | uint32_t | srcBLen,
|  | q31_t * | pDst
| ) |  |

Parameters

  [in] | *pSrcA | points to the first input sequence.
  [in] | srcALen | length of the first input sequence.
  [in] | *pSrcB | points to the second input sequence.
  [in] | srcBLen | length of the second input sequence.
  [out] | *pDst | points to the location where the output result is written. Length 2 * max(srcALen, srcBLen) - 1.

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. There is no saturation on intermediate additions. Thus, if the accumulator overflows it wraps around and distorts the result. The input signals should be scaled down to avoid intermediate overflows. Scale down one of the inputs by 1/min(srcALen, srcBLen)to avoid overflows since a maximum of min(srcALen, srcBLen) number of additions is carried internally. The 2.62 accumulator is right shifted by 31 bits and saturated to 1.31 format to yield the final result.

  See `arm_correlate_fast_q31()` for a faster but less precise implementation of this function for Cortex-M3 and Cortex-M4.

References srcALen, and srcBLen.

void arm_correlate_q7 | ( | q7_t * | pSrcA,
|  | uint32_t | srcALen,
|  | q7_t * | pSrcB,
|  | uint32_t | srcBLen,
|  | q7_t * | pDst
| ) |  |

Parameters

  [in] | *pSrcA | points to the first input sequence.
  [in] | srcALen | length of the first input sequence.
  [in] | *pSrcB | points to the second input sequence.
  [in] | srcBLen | length of the second input sequence.
  [out] | *pDst | points to the location where the output result is written. Length 2 * max(srcALen, srcBLen) - 1.

Returns
  none.

Scaling and Overflow Behavior:

  The function is implemented using a 32-bit internal accumulator. Both the inputs are represented in 1.7 format and multiplications yield a 2.14 result. The 2.14 intermediate results are accumulated in a 32-bit accumulator in 18.14 format. This approach provides 17 guard bits and there is no risk of overflow as long as `max(srcALen, srcBLen)<131072`. The 18.14 result is then truncated to 18.7 format by discarding the low 7 bits and saturated to 1.7 format.

  Refer the function `arm_correlate_opt_q7()` for a faster implementation of this function.

References srcALen, and srcBLen.