# Formulation and Performance Analysis of Broadband and Narrowband OFDM-Based PLC Systems

^{*}

## Abstract

**:**

## 1. Introduction

## 2. System Model

#### 2.1. Transmitter

#### 2.1.1. Phase Shifting

#### 2.1.2. Tone Mask

- (a)
- The active sub-carriers are allocated in consecutive positions. In this case, $\mathbf{A}$ is defined by$${\mathbf{A}}_{N\times M}=\left(\right)open="["\; close="]">\begin{array}{c}{\mathbf{0}}_{{M}_{1}\times M}\\ {\mathbf{I}}_{M}\\ {\mathbf{0}}_{{M}_{2}\times M}\end{array}$$
- (b)
- The active sub-carriers are in non-consecutive positions, as depicted in Figure 2. In this case, $\mathbf{A}$ is defined by$${\mathbf{A}}_{N\times M}=\left(\right)open="["\; close="]">\begin{array}{cccc}{\mathbf{A}}^{{\ell}_{1}}& \mathbf{0}& \cdots & \mathbf{0}\\ \mathbf{0}& {\mathbf{A}}^{{\ell}_{2}}& \cdots & \mathbf{0}\\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{0}& \mathbf{0}& \cdots & {\mathbf{A}}^{{\ell}_{m}}\end{array}$$$${\mathbf{A}}_{({M}_{{\ell}_{i},1}+{M}_{i}+{M}_{{\ell}_{i},2})\times {M}_{i}}^{{\ell}_{i}}=\left(\right)open="["\; close="]">\begin{array}{c}{\mathbf{0}}_{{M}_{{\ell}_{i},1}\times {M}_{i}}\\ {\mathbf{I}}_{{M}_{i}}\\ {\mathbf{0}}_{{M}_{{\ell}_{i},2}\times {M}_{i}}\end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {\ell}_{act}=\sum _{i=1}^{m}{M}_{i},\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& {\ell}_{inact}=\sum _{i=1}^{m}\left(\right)open="("\; close=")">{M}_{{\ell}_{i},1}+{M}_{{\ell}_{i},2},\hfill \end{array}$$

#### 2.1.3. The Inverse Discrete Fourier Transform (IDFT)

#### 2.1.4. Insertion of a Cyclic Prefix (CP)

#### 2.1.5. Tx Window

#### 2.1.6. Overlapping and Adding

#### 2.2. Channel

#### 2.3. Receiver

#### 2.3.1. RI Removal

#### 2.3.2. Rx Windowing

#### 2.3.3. RI’ Displacement and Addition

#### 2.3.4. Samples Reordering

#### 2.3.5. The Discrete Fourier Transform (DFT)

#### 2.3.6. Frequency Domain Equalizer (FEQ)

#### 2.3.7. Rx Data Arrangement

#### 2.3.8. Phase Shifting

#### 2.4. Recovered Data

## 3. Case Study-I: BB PLC

#### 3.1. Matrix Description

- (a)
- The transceiver only includes a window in the Tx unit. In this case, $R{I}^{\prime}=0$, $\mu -RI-R{I}^{\prime}=756$, and$${\mathbf{R}}_{4096\times 4852}=\left[\begin{array}{cc}{\mathbf{0}}_{4096\times 756}& {\mathbf{I}}_{4096}\end{array}\right],$$$${\mathbf{V}}_{4096\times 4096}^{rx}={\mathbf{I}}_{4096},$$$${\mathbf{P}}_{4096}={\mathbf{I}}_{4096},$$$${\mathbf{K}}_{4096\times 4096}=\left(\right)open="["\; close="]">\begin{array}{cc}{\mathbf{0}}_{3600\times 496}& {\mathbf{I}}_{3600}\\ {\mathbf{I}}_{496}& {\mathbf{0}}_{496\times 3600}\end{array}$$
- (b)
- The transceiver only incorporates a window at the Rx unit. In this scheme, assuming $R{I}^{\prime}=496$, we have $\mu -RI-R{I}^{\prime}=756$, and$${\mathbf{R}}_{4592\times 5348}=\left[\begin{array}{cc}{\mathbf{0}}_{4592\times 756}& {\mathbf{I}}_{4592}\end{array}\right],$$$${\mathbf{V}}_{4592}^{rx}=\mathrm{diag}\left(\begin{array}{ccc}{\mathbf{v}}_{1\times 496}^{rr}\phantom{\rule{1.em}{0ex}}& {\mathbf{1}}_{1\times 3600}\phantom{\rule{1.em}{0ex}}& {\mathbf{v}}_{1\times 496}^{rf}\end{array}\right),$$$$\begin{array}{c}\hfill {\left(\right)}_{{\mathbf{v}}_{1\times 496}^{rr}}1,n=\left(\right)open="\{"\; close>\begin{array}{cc}\frac{0.2}{71}n,\hfill & 0\le n\le 70,\\ \phantom{\rule{1.em}{0ex}}\hfill & \\ 0.2+\frac{0.6}{355}(n-70),\hfill & 70\le n\le 222,\\ \phantom{\rule{1.em}{0ex}}\hfill & \\ 0.8+\frac{0.2}{71}(n-431),\hfill & 431\le n\le 495,\end{array}\end{array}$$$${\mathbf{v}}_{1\times 496}^{rf}\left[n\right]={\mathbf{v}}_{1\times 496}^{rr}[-n+4592],\phantom{\rule{1.em}{0ex}}4096\le n\le 4591,$$$${\mathbf{P}}_{4096\times 4592}=\left(\right)open="["\; close="]">\begin{array}{c}{\mathbf{0}}_{3600\times 496}\\ {\mathbf{I}}_{496}\end{array}\begin{array}{c}{\mathbf{I}}_{4096}\end{array}$$$${\mathbf{K}}_{4096\times 4096}={\mathbf{I}}_{4096}.$$
- (c)
- The transceiver incorporates double window (Tx and Rx units). In this second scheme, we assume $R{I}^{\prime}=130$; thus $\mu -RI-R{I}^{\prime}=626$, and$${\mathbf{R}}_{4226\times 4852}=\left[\begin{array}{cc}{\mathbf{0}}_{4226\times 626}& {\mathbf{I}}_{4226}\end{array}\right].$$If the following rising slope is assumed:$${\left(\right)}_{{\mathbf{v}}_{1\times R{I}^{\prime}}^{rr}}1,n$$$${\mathbf{v}}_{1\times R{I}^{\prime}}^{rf}\left[n\right]={\mathbf{v}}_{1\times R{I}^{\prime}}^{rr}[-n+N+R{I}^{\prime}],\phantom{\rule{0.166667em}{0ex}}N\le n\le N+R{I}^{\prime}-1,$$$${\mathbf{V}}_{4226}^{rx}=\mathrm{diag}\left(\begin{array}{ccc}{\mathbf{v}}_{1\times 130}^{rr}\phantom{\rule{1.em}{0ex}}& {\mathbf{1}}_{1\times 3966}\phantom{\rule{1.em}{0ex}}& {\mathbf{v}}_{1\times 130}^{rf}\end{array}\right),$$$$\begin{array}{c}\hfill {\left(\right)}_{{\mathbf{v}}_{1\times 130}^{rr}}1,n=\left(\right)open="\{"\; close>\begin{array}{cc}\frac{0.2}{19}n,\hfill & 0\le n\le 18,\\ \phantom{\rule{1.em}{0ex}}\hfill & \\ 0.2+\frac{0.6}{94}(n-19),\hfill & 19\le n\le 111,\\ \phantom{\rule{1.em}{0ex}}\hfill & \\ 0.8+\frac{0.2}{19}(n-112),\hfill & 112\le n\le 129,\end{array}\end{array}$$$${\mathbf{v}}_{1\times 130}^{rf}\left[n\right]={\mathbf{v}}_{1\times 130}^{rr}[-n+4226],\phantom{\rule{1.em}{0ex}}4096\le n\le 4227.$$Finally,$${\mathbf{P}}_{4096\times 4226}=\left(\right)open="["\; close="]">\begin{array}{cc}\frac{{\mathbf{0}}_{3966\times 130}}{{\mathbf{I}}_{260}}& {\mathbf{I}}_{4226}\end{array}$$$${\mathbf{K}}_{4096\times 4096}=\left(\right)open="["\; close="]">\begin{array}{cc}{\mathbf{0}}_{3600\times 496}& {\mathbf{I}}_{3600}\\ {\mathbf{I}}_{496}& {\mathbf{0}}_{496\times 3600}\end{array}$$
- (d)
- The transceiver includes double window (Tx and Rx units) with $R{I}^{\prime}=260$, which is the maximum value possible for this scheme. Then, $\mu -RI-R{I}^{\prime}=496$, and$${\mathbf{R}}_{4356\times 4852}=\left[\begin{array}{cc}{\mathbf{0}}_{4356\times 496}& {\mathbf{I}}_{4356}\end{array}\right],$$$${\mathbf{V}}_{4356}^{rx}=\mathrm{diag}\left(\begin{array}{ccc}{\mathbf{v}}_{1\times 260}^{rr}\phantom{\rule{1.em}{0ex}}& {\mathbf{1}}_{1\times 3836}\phantom{\rule{1.em}{0ex}}& {\mathbf{v}}_{1\times 260}^{rf}\end{array}\right),$$$$\begin{array}{c}\hfill {\left(\right)}_{{\mathbf{v}}_{1\times 260}^{rr}}1,n=\left(\right)open="\{"\; close>\begin{array}{cc}\frac{0.2}{37}n,\hfill & 0\le n\le 36,\\ \phantom{\rule{1.em}{0ex}}\hfill & \\ 0.2+\frac{0.6}{186}(n-37),\hfill & 37\le n\le 222,\\ \phantom{\rule{1.em}{0ex}}\hfill & \\ 0.8+\frac{0.2}{37}(n-223),\hfill & 223\le n\le 259,\end{array}\end{array}$$$${\mathbf{v}}_{1\times 260}^{rf}\left[n\right]={\mathbf{v}}_{1\times 260}^{rr}[-n+4356],\phantom{\rule{1.em}{0ex}}4096\le n\le 4355,$$$${\mathbf{P}}_{4096\times 4356}=\left(\right)open="["\; close="]">\begin{array}{cc}\frac{{\mathbf{0}}_{3836\times 260}}{{\mathbf{I}}_{260}}& {\mathbf{I}}_{4096}\end{array}$$$${\mathbf{K}}_{4096\times 4096}=\left(\right)open="["\; close="]">\begin{array}{cc}{\mathbf{0}}_{3836\times 260}& {\mathbf{I}}_{3836}\\ {\mathbf{I}}_{260}& {\mathbf{0}}_{260\times 3836}\end{array}$$

#### 3.2. Channel and Noise Models

#### 3.3. Simulations

^{3}iterations. The BER obtained for the four different schemes are shown in Figure 5. As can be seen, there are practically no differences in the results for the different window schemes within each noise scenario (GBN, ALL). In addition, for almost the low SNR values, the sum of all PLC noises is a better scenario in terms of BER, than that in which GBN is present. There is a crossing point around 30 dB SNR that reflects a change in performance.

## 4. Case Study-II: NB PLC

#### 4.1. Matrix Description

- (a)
- The transceiver only includes a window in the Tx unit. In this case, $R{I}^{\prime}=0$, $\mu -RI-R{I}^{\prime}=22$, and$${\mathbf{R}}_{256\times 278}=\left[\begin{array}{cc}{\mathbf{0}}_{256\times 22}& {\mathbf{I}}_{256}\end{array}\right],$$$${\mathbf{V}}_{256\times 256}^{rx}={\mathbf{I}}_{256},$$$${\mathbf{P}}_{256}={\mathbf{I}}_{256},$$$${\mathbf{K}}_{256\times 256}=\left(\right)open="["\; close="]">\begin{array}{cc}{\mathbf{0}}_{248\times 8}& {\mathbf{I}}_{248}\\ {\mathbf{I}}_{8}& {\mathbf{0}}_{8\times 248}\end{array}$$
- (b)
- The transceiver only incorporates a window at the Rx unit. In this scheme, assuming $R{I}^{\prime}=8$, we have $\mu -RI-R{I}^{\prime}=22$, and$${\mathbf{R}}_{264\times 286}=\left[\begin{array}{cc}{\mathbf{0}}_{264\times 22}& {\mathbf{I}}_{264}\end{array}\right],$$If the following rising slope is assumed:$$\begin{array}{c}\hfill {\left(\right)}_{{\mathbf{v}}_{1\times R{I}^{\prime}}^{rr}}1,n=\left(\right)open="\{"\; close>\begin{array}{cc}\frac{0.2}{{k}_{1}}n,\hfill & 0\le n\le {k}_{1}-1,\\ \phantom{\rule{1.em}{0ex}}\hfill & \\ 0.2+\frac{0.6}{{k}_{2}}(n-{k}_{1}),\hfill & {k}_{1}\le n\le {k}_{3}-1,\\ \phantom{\rule{1.em}{0ex}}\hfill & \\ 0.8+\frac{0.2}{{k}_{1}}(n-{k}_{3}),\hfill & {k}_{3}\le n\le R{I}^{\prime}-1,\end{array}\end{array}$$$${\mathbf{v}}_{1\times R{I}^{\prime}}^{rf}\left[n\right]={\mathbf{v}}_{1\times R{I}^{\prime}}^{rr}[-n+N+R{I}^{\prime}],\phantom{\rule{1.em}{0ex}}N\le n\le N+R{I}^{\prime}-1,$$$${\mathbf{V}}_{264}^{rx}=\mathrm{diag}\left(\begin{array}{ccc}{\mathbf{v}}_{1\times 8}^{rr}\phantom{\rule{1.em}{0ex}}& {\mathbf{1}}_{1\times 248}\phantom{\rule{1.em}{0ex}}& {\mathbf{v}}_{1\times 8}^{rf}\end{array}\right),$$$$\begin{array}{c}\hfill {\left(\right)}_{{\mathbf{v}}_{1\times 8}^{rr}}1,n=\left(\right)open="\{"\; close>\begin{array}{cc}\frac{0.2}{2}n,\hfill & 0\le n\le 1,\\ \phantom{\rule{1.em}{0ex}}\hfill & \\ 0.2+\frac{0.6}{5}(n-2),\hfill & 2\le n\le 5,\\ \phantom{\rule{1.em}{0ex}}\hfill & \\ 0.8+\frac{0.2}{2}(n-7),\hfill & 6\le n\le 7,\end{array}\end{array}$$$${\mathbf{v}}_{1\times 8}^{rf}\left[n\right]={\mathbf{v}}_{1\times 8}^{rr}[-n+264],\phantom{\rule{1.em}{0ex}}256\le n\le 263,$$Finally,$${\mathbf{P}}_{256\times 264}=\left(\right)open="["\; close="]">\begin{array}{c}{\mathbf{0}}_{248\times 8}\\ {\mathbf{I}}_{8}\end{array}\begin{array}{c}{\mathbf{I}}_{256}\end{array}$$$${\mathbf{K}}_{256\times 256}={\mathbf{I}}_{256}.$$
- (c)
- The transceiver incorporates double window (Tx and Rx units). In this second scheme, we assume $R{I}^{\prime}=7$; thus $\mu -RI-R{I}^{\prime}=15$, and$${\mathbf{R}}_{263\times 278}=\left[\begin{array}{cc}{\mathbf{0}}_{263\times 15}& {\mathbf{I}}_{263}\end{array}\right],$$If the following rising slope is assumed:$${\left(\right)}_{{\mathbf{v}}_{1\times R{I}^{\prime}}^{rr}}1,n$$$${\mathbf{v}}_{1\times R{I}^{\prime}}^{rf}\left[n\right]={\mathbf{v}}_{1\times R{I}^{\prime}}^{rr}[-n+N+R{I}^{\prime}],\phantom{\rule{1.em}{0ex}}N\le n\le N+R{I}^{\prime}-1,$$$${\mathbf{V}}_{263}^{rx}=\mathrm{diag}\left(\begin{array}{ccc}{\mathbf{v}}_{1\times 7}^{rr}\phantom{\rule{1.em}{0ex}}& {\mathbf{1}}_{1\times 248}\phantom{\rule{1.em}{0ex}}& {\mathbf{v}}_{1\times 7}^{rf}\end{array}\right),$$$$\begin{array}{c}\hfill {\left(\right)}_{{\mathbf{v}}_{1\times 7}^{rr}}1,n=\left(\right)open="\{"\; close>\begin{array}{cc}\frac{0.2}{2}n,\hfill & 0\le n\le 1,\\ \phantom{\rule{1.em}{0ex}}\hfill & \\ 0.2+\frac{0.6}{10}(n-2),\hfill & 2\le n\le 11,\\ \phantom{\rule{1.em}{0ex}}\hfill & \\ 0.8+\frac{0.2}{2}(n-12),\hfill & 12\le n\le 13,\end{array}\end{array}$$$${\mathbf{v}}_{1\times 7}^{rf}\left[n\right]={\mathbf{v}}_{1\times 7}^{rr}[-n+270],\phantom{\rule{1.em}{0ex}}256\le n\le 269,$$Finally,$${\mathbf{P}}_{256\times 263}=\left(\right)open="["\; close="]">\begin{array}{cc}\frac{{\mathbf{0}}_{249\times 7}}{{\mathbf{I}}_{7}}& {\mathbf{I}}_{256}\end{array}$$$${\mathbf{K}}_{256\times 256}=\left(\right)open="["\; close="]">\begin{array}{cc}{\mathbf{0}}_{248\times 8}& {\mathbf{I}}_{248}\\ {\mathbf{I}}_{8}& {\mathbf{0}}_{8\times 248}\end{array}$$
- (d)
- The transceiver incorporates double window (Tx and Rx units) but assuming $R{I}^{\prime}=14$, which is its maximum value possible for this scheme; thus $\mu -RI-R{I}^{\prime}=8$, and$${\mathbf{R}}_{270\times 278}=\left[\begin{array}{cc}{\mathbf{0}}_{270\times 8}& {\mathbf{I}}_{270}\end{array}\right],$$$${\mathbf{V}}_{270}^{rx}=\mathrm{diag}\left(\begin{array}{ccc}{\mathbf{v}}_{1\times 14}^{rr}\phantom{\rule{1.em}{0ex}}& {\mathbf{1}}_{1\times 242}\phantom{\rule{1.em}{0ex}}& {\mathbf{v}}_{1\times 14}^{rf}\end{array}\right),$$$$\begin{array}{c}\hfill {\left(\right)}_{{\mathbf{v}}_{1\times 14}^{rr}}1,n=\left(\right)open="\{"\; close>\begin{array}{cc}\frac{0.2}{2}n,\hfill & 0\le n\le 1,\\ \phantom{\rule{1.em}{0ex}}\hfill & \\ 0.2+\frac{0.6}{10}(n-2),\hfill & 2\le n\le 11,\\ \phantom{\rule{1.em}{0ex}}\hfill & \\ 0.8+\frac{0.2}{2}(n-12),\hfill & 12\le n\le 13,\end{array}\end{array}$$$${\mathbf{v}}_{1\times 14}^{rf}\left[n\right]={\mathbf{v}}_{1\times 14}^{rr}[-n+270],\phantom{\rule{1.em}{0ex}}256\le n\le 269,$$$${\mathbf{P}}_{256\times 270}=\left(\right)open="["\; close="]">\begin{array}{cc}\frac{{\mathbf{0}}_{242\times 14}}{{\mathbf{I}}_{14}}& {\mathbf{I}}_{256}\end{array}$$$${\mathbf{K}}_{256\times 256}=\left(\right)open="["\; close="]">\begin{array}{cc}{\mathbf{0}}_{248\times 8}& {\mathbf{I}}_{248}\\ {\mathbf{I}}_{8}& {\mathbf{0}}_{8\times 248}\end{array}$$

#### 4.2. Channel and Noise Models

#### 4.3. Simulations

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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# | Abbrev. | Comment |
---|---|---|

1 | TxWin | 1901 PLC system with windowing at transmission side. |

2 | RxWin | 1901 PLC system with windowing at reception side. |

3 | dbWin | 1901 PLC system with double windowing: at transmission and reception side. |

4 | dbWin-max | 1901 PLC system double windowing: at transmission and reception side but with maximun admissible value of roll-off for the reception window. |

Matrix | Description |
---|---|

$\mathsf{\Phi}$ | Phase shifting |

$\mathbf{A}$ | Tone mask |

${\mathbf{W}}^{-1}$ | Inverse discrete Fourier transform |

$\mathsf{\Gamma}$ | CP insertion |

${\mathbf{V}}^{tx}$ | Transmitter (Tx) window |

$\mathbf{h}$ | Channel impulse response |

$\mathbf{H}$ | Channel matrix |

$\mathbf{R}$ | Discards the samples of the Tx rolloff interval |

${\mathbf{V}}^{rx}$ | Receiver (Rx) window |

$\mathbf{P}$ | Displaces and adds the samples of the Rx rolloff interval |

$\mathbf{K}$ | Samples reordering |

$\mathbf{W}$ | Discrete Fourier transform |

$\mathbf{E}$ | Frequency domain equalizer |

$\mathbf{B}$ | Inverse of the masking function A |

${\mathsf{\Phi}}^{-1}$ | Inverse phase shifting |

**Table 3.**Tx parameters defined in the BB FFT PHY [16].

Parameter | Description | Value (Samples) |
---|---|---|

N | FFT size | 4096 |

M | Active sub-carriers ^{1} | 917 |

$\mu $ | CP length (samples) | 1252 |

$RI$ | Samples of the Roll-off interval | 496 |

^{1}Distributed in nine blocks according to the suggested tone mask given in ([16] sub clause 13.9.7).

# | Abbrev. | Comment. |
---|---|---|

1 | BB-TxWin | Broadband PLC system with windowing at transmission side only. |

2 | BB-RxWin | Broadband PLC system with windowing at reception side only. |

3 | BB-dbWin | Broadband PLC system with double windowing: at transmission and reception side. |

4 | BB-dbWin-max | Broadband PLC system double windowing: at transmission and reception side but with maximun RI’ value. |

# | Abbrev. | Comment. |
---|---|---|

1 | NB-TxWin | Narrowband PLC system with windowing at transmission side only. |

2 | NB-RxWin | Narrowband PLC system with windowing at reception side only. |

3 | NB-dbWin | Narrowband PLC system with double windowing: at transmission and reception side. |

4 | NB-dbWin-max | Narrowband PLC system double windowing: at transmission and reception side but with maximun admissible RI’ value. |

Parameter | Description | Value (Samples) |
---|---|---|

N | Size of FFT | 256 |

M | Active carriers † | 36 |

$\mu $ | Length of CP | 30 |

$RI$ | Samples of roll-off interval in Tx. | 8 |

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## Share and Cite

**MDPI and ACS Style**

García-Gangoso, F.; Blanco-Velasco, M.; Cruz-Roldán, F.
Formulation and Performance Analysis of Broadband and Narrowband OFDM-Based PLC Systems. *Sensors* **2021**, *21*, 290.
https://doi.org/10.3390/s21010290

**AMA Style**

García-Gangoso F, Blanco-Velasco M, Cruz-Roldán F.
Formulation and Performance Analysis of Broadband and Narrowband OFDM-Based PLC Systems. *Sensors*. 2021; 21(1):290.
https://doi.org/10.3390/s21010290

**Chicago/Turabian Style**

García-Gangoso, Fausto, Manuel Blanco-Velasco, and Fernando Cruz-Roldán.
2021. "Formulation and Performance Analysis of Broadband and Narrowband OFDM-Based PLC Systems" *Sensors* 21, no. 1: 290.
https://doi.org/10.3390/s21010290