# Validation of Parallel Distributed Adaptive Signal Processing (PDASP) Framework through Processing-Inefficient Low-Cost Platforms

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## Abstract

**:**

## 1. Introduction

## 2. Parallel Distributed Adaptive Signal Processing (PDASP) Architecture

Algorithm 1: Working procedure of PDASP with Diffused components for $N\times N$ MIMO communication system |

Initialize: ${\mathbf{W}}_{k-1},{\mathbf{a}}_{k-2},{\widehat{\mathbf{r}}}_{k-1},{\mathrm{\Psi}}_{k-1},{\mathrm{\Psi}}_{k-2},{\mathbf{e}}_{k-2},{\mathbf{g}}_{k-2}$ |

parallel procedure for ${M}_{1}$, ${M}_{2}$, ${M}_{3}$ and ${M}_{4}$ |

for k=0:N |

Process node ${M}_{3}$ |

at time ${t}_{1}$: ${\widehat{\mathbf{r}}}_{k}^{T}={\mathbf{x}}_{k}^{T}{\mathbf{W}}_{k-1}$ |

at time ${t}_{1}$: ${\mathbf{e}}_{k-1}={\mathbf{r}}_{k-1}-{\widehat{\mathbf{r}}}_{k-1}$ |

at time ${t}_{1}$: wait $\mathrm{\Delta}{T}_{\mathbf{e}}$ |

Process node ${M}_{2}$ |

at time ${t}_{1}$: ${\mathbf{a}}_{k-1}={\mathbf{x}}_{k-1}^{T}{\mathrm{\Psi}}_{k-1}$ |

at time ${t}_{1}$: ${\mathbf{g}}_{k-1}=\frac{{\displaystyle {\mathbf{a}}_{k-1}}}{{\displaystyle {\mathbf{a}}_{k-1}^{T}{\mathbf{x}}_{k-1}+{\sigma}_{v,k}}}$ |

at time ${t}_{1}$: wait $\mathrm{\Delta}{T}_{\mathbf{g}}$ |

Process node ${M}_{1}$ |

at time ${t}_{1}$: ${\mathrm{\Psi}}_{k}={\mathrm{\Psi}}_{k-2}-{\mathbf{g}}_{k-2}{\mathbf{a}}_{k-2}^{T}$ |

Process node ${M}_{4}$ |

at time ${t}_{1}$: ${\mathbf{W}}_{k}={\mathbf{W}}_{k-1}+{\mathbf{e}}_{k-2}{\mathbf{g}}_{k-2}^{T}$ |

at time ${t}_{1}$: wait $\mathrm{\Delta}{T}_{\mathbf{W}}$ |

at time ${t}_{2}$: Transmit ${\mathbf{g}}_{k-1}$ from ${M}_{2}$ to ${M}_{1}$ and ${M}_{4}$ |

at time ${t}_{3}$: Transmit ${\mathrm{\Psi}}_{k-1}$ from ${M}_{1}$ to ${M}_{2}$, ${\mathbf{W}}_{k-1}$ from ${M}_{4}$ to ${M}_{3}$ |

at time ${t}_{4}$: Transmit ${\mathbf{e}}_{k-2}$ from ${M}_{3}$ to ${M}_{4}$, ${\mathbf{a}}_{k-2}$ from ${M}_{2}$ to ${M}_{1}$ |

end for |

## 3. Communication Load-Balancing Procedure

## 4. Complexity Analysis

## 5. Measurement Results

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**PDASP architecture for low-complexity MIMO channel estimator with non-aligned time indexes [17].

**Figure 3.**Working procedure of low complexity MIMO Algorithm (

**a**). Sequential procedure (

**b**). Parallel working procedure.

**Figure 5.**Sequential and Distributed Multiplication Complexity for Various MIMO Systems with LoS and Diffused Components.

**Figure 6.**Sequential and Distributed Addition Complexity for Various MIMO Systems with LoS and Diffused Components.

**Figure 7.**Sequential and Distributed Processing Time in μsec for Various MIMO Systems with LoS and Diffused Components.

Multipath Components | $2\times 2$ MIMO | $3\times 3$ MIMO | $4\times 4$ MIMO |
---|---|---|---|

L = 0 | $2\times 2$ | $3\times 3$ | $4\times 4$ |

L = 1 | $2\times 4$ | $3\times 6$ | $4\times 8$ |

L = 2 | $2\times 6$ | $3\times 9$ | $4\times 12$ |

L = 3 | $2\times 8$ | $3\times 12$ | $4\times 16$ |

Algorithm Part | Multiplication Complexity | Addition Complexity |
---|---|---|

${\mathrm{\Psi}}_{k}$ | $2{(N+NL)}^{2}$ | ${(N+NL)}^{2}$ |

${\mathbf{g}}_{k}$ | ${(N+NL)}^{2}+NL+1$ | ${(N+NL)}^{2}-N$ |

${\mathbf{e}}_{k}$ | ${N}^{2}(L+1)$ | ${N}^{2}(L+1)$ |

${\mathbf{W}}_{k}$ | ${N}^{2}(L+1)$ | ${N}^{2}(L+1)$ |

Complexity | Diffused Components | $2\times 2$ MIMO | $3\times 3$ MIMO | $4\times 4$ MIMO |
---|---|---|---|---|

Multiplication Complexity | $L=0$ | 57.14% | 62.79% | 65.75% |

Multiplication Complexity | $L=1$ | 56.14% | 59.50% | 61.24% |

Multiplication Complexity | $L=2$ | 55.04% | 57.44% | 58.70% |

Addition Complexity | $L=0$ | 75.00% | 75.00% | 75.00% |

Addition Complexity | $L=1$ | 66.66% | 66.66% | 66.66% |

Addition Complexity | $L=2$ | 62.50% | 62.50% | 62.50% |

**Table 4.**Percentage Improvement in Processing Time for Various MIMO communication Systems Using PDASP architecture.

Diffused Components | $2\times 2$ MIMO | $3\times 3$ MIMO | $4\times 4$ MIMO |
---|---|---|---|

$L=0$ | 71.84% | 75.49% | 78.63% |

$L=1$ | 70.56% | 71.46% | 71.02% |

$L=2$ | 68.82% | 66.90% | 61.54% |

Aurdino Platform | Multipath Components | $2\times 2$ MIMO | $3\times 3$ MIMO | $4\times 4$ MIMO |
---|---|---|---|---|

NANO | $L=0$ | Working | Working | Working |

UNO | $L=0$ | Working | Working | Working |

MEGA | $L=0$ | Working | Working | Working |

NANO | $L=1$ | Working | Memory Error | Memory Error |

UNO | $L=1$ | Working | Working | Memory Error |

MEGA | $L=1$ | Working | Working | Working |

NANO | $L=2$ | Memory Error | Memory Error | Memory Error |

UNO | $L=2$ | Working | Memory Error | Memory Error |

MEGA | $L=2$ | Working | Working | Working |

**Table 6.**Distributed Memory Limitation Comparison for Various MIMO Systems with Multipath Components.

Aurdino Platform | Multipath Components | $2\times 2$ MIMO | $3\times 3$ MIMO | $4\times 4$ MIMO |
---|---|---|---|---|

NANO | $L=0$ | Working | Working | Working |

UNO | $L=0$ | Working | Working | Working |

MEGA | $L=0$ | working | Working | Working |

NANO | $L=1$ | Working | Working | Working |

UNO | $L=1$ | Working | Working | Working |

MEGA | $L=1$ | Working | Working | Working |

NANO | $L=2$ | Working | Memory Error | Memory Error |

UNO | $L=2$ | Working | Working | Working |

MEGA | $L=2$ | Working | Working | Working |

Processing Time | Diffused Components | $2\times 2$ MIMO | $3\times 3$ MIMO | $4\times 4$ MIMO |
---|---|---|---|---|

Sequential Algorithm | $L=0$ | 696 | 1420 | 2396 |

Combined Time ($M1,S1,S2,S3$) | $L=0$ | 500 | 1068 | 1832 |

Percentage Improvement | $L=0$ | 28.16% | 24.78% | 23.53% |

Sequential Algorithm | $L=1$ | 1780 | 3532 | 6116 |

Combined Time ($M1,S1,S2,S3$) | $L=1$ | 1408 | 2848 | 4984 |

Percentage Improvement | $L=1$ | 20.89% | 19.36% | 18.50% |

Sequential Algorithm | $L=2$ | 3272 | 6852 | 11816 |

Combined Time ($M1,S1,S2,S3$) | $L=2$ | 2604 | 5536 | 9740 |

Percentage Improvement | $L=2$ | 20.41% | 19.20% | 17.56% |

**Table 8.**Maximum Time Taken for One Complete Iteration Using PDASP Architecture for Various MIMO Systems.

Diffused Components | $2\times 2$ MIMO | $3\times 3$ MIMO | $4\times 4$ MIMO |
---|---|---|---|

$L=0$ | 9520 μs | 11,449 μs | 1533 μs |

$L=1$ | 16,934 μs | 26,320 μs | 38,986 μs |

$L=2$ | 27,143 μs | 43,380 μs | 66,898 μs |

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**MDPI and ACS Style**

Raza, H.; Ahmad, I.; Khan, N.M.; Abbasi, W.; Anwar, M.S.; Ahmad, S.; El-Affendi, M.A. Validation of Parallel Distributed Adaptive Signal Processing (PDASP) Framework through Processing-Inefficient Low-Cost Platforms. *Mathematics* **2022**, *10*, 4600.
https://doi.org/10.3390/math10234600

**AMA Style**

Raza H, Ahmad I, Khan NM, Abbasi W, Anwar MS, Ahmad S, El-Affendi MA. Validation of Parallel Distributed Adaptive Signal Processing (PDASP) Framework through Processing-Inefficient Low-Cost Platforms. *Mathematics*. 2022; 10(23):4600.
https://doi.org/10.3390/math10234600

**Chicago/Turabian Style**

Raza, Hasan, Ishtiaq Ahmad, Noor M. Khan, Waseem Abbasi, Muhammad Shahid Anwar, Sadique Ahmad, and Mohammed A. El-Affendi. 2022. "Validation of Parallel Distributed Adaptive Signal Processing (PDASP) Framework through Processing-Inefficient Low-Cost Platforms" *Mathematics* 10, no. 23: 4600.
https://doi.org/10.3390/math10234600