# PHY, MAC, and RLC Layer Based Estimation of Optimal Cyclic Prefix Length

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. PHY-Only BER-Based Indication of Oversized CP Length

^{−3}and 10

^{−6}which are commonly referred to as the upper limits for degraded and acceptable bit-oriented digital transmission systems performance, respectively [21]. By applying these BER thresholds (in the absence of equivalent ones for the state-of-the-art access-level wireless networks of interest here), with long enough power delay profile, the corresponding CP length values were found to be around 2.25 and 4.60 μs, respectively, with very little BER reduction for CP length above 3 μs, Figure 1.

#### 1.2. Motivation for PHY/MAC/RLC-Based CP Length Model

## 2. Optimal CP Length Model

_{0}to RV

_{3}) of the codeblock can be sent until the codeblock CRC at the receiver indicates error-free transmission. Eventual residual post-HARQ erroneous codeblock is handed over to the RLC-layer ARQ process, which in that case makes the final retransmission [22].

#### 2.1. Effective Average Codeblock Length

_{CB}= L bits, and the BLER for the i-th redundancy version as: $BLE{R}_{\mathrm{RV}i};i=0,1,2,3$. We can justifiably consider the latter as a monotonically decreasing function of CP length ${\tau}_{\mathrm{CP}}$; therefore, the larger ${\tau}_{\mathrm{CP}}$, the smaller $BLE{R}_{\mathrm{RV}i}$.

_{i}only if the just-finished transmission of RV

_{i}

_{−1}results with CRC indication of an erroneous codeblock, where the probability of an error-free RV

_{i}is approximated by the 1-complement of the related BLER

_{RVi}. Thus, the overall used length equals L just for the error-free RV

_{0}, whereas it increases to 2L if the RV

_{0}is erroneous, but RV

_{1}is error-free. This rise continues to 3L if RV

_{0}and RV

_{1}are erroneous, but RV

_{2}is error-free, and to 4L with erroneous RV

_{0}, RV

_{1}and RV

_{2}, and error-free RV

_{3}being the last IR-HARQ transmission. Finally, the overall length 5L is accumulated after erroneous RV

_{3}, as the post-HARQ remaining errors are dealt by RLC’s ARQ, which sends the last retransmission. Whether is it error-free (which is much more likely) or not is irrelevant for our CP length model, as no more retransmission is sent except, eventually, at the transmission layer, all the way up the stack, which is not in our scope here.

_{i}with respect to the first transmission (RV

_{0}) at the same SNR value. (This is inverse to the definition of the more common coding gain ${G}_{\mathrm{RV}i/0}$, which allows RV

_{i}to have that much reduced SNR with respect to RV

_{0}but still retain the same BLER value [22].)

_{i}with respect to RV

_{0}, pertain to the target value of BLER = 0.1 [27,28].

_{0}are represented in Figure 2 by the lengths of the vertical lines drawn from the points where the plots reach the target value $BLE{R}_{\mathrm{RV}i}={10}^{-1};i=1,2,3$ up to the intersections with the RV

_{0}curve. Thus, in this example, we can see that $\Delta BLE{R}_{\mathrm{RV}1/0}$, $\Delta BLE{R}_{\mathrm{RV}2/0}$ and $\Delta BLE{R}_{\mathrm{RV}3/0}$ are approximately equal to 7.5, 9.5 and 10 times, respectively.

#### 2.2. Optimal CP Length for Minimal Codeblock Average Gross Length

_{UNC}[25]. By considering the time-dispersion (targeted by CP) dominant impairment causing errors, BER

_{UNC}is a function of CP length: $BE{R}_{\mathrm{UNC}}=BER\left({\tau}_{\mathrm{CP}}^{}\right)$ [2].

_{0}regarding the uncoded block transmission, then $BLE{R}_{\mathrm{RV}0}\left({\tau}_{\mathrm{CP}}^{}\right)$ can be expressed as:

^{−1}[22] and the maximal block length of L = 6144 bits in (10) results in a small value of BER ≈ 1.63∙10

^{−5}.

#### 2.3. Time-Dispersion-Only Related Residual BER for Optimal CP Length

## 3. Numerical Results

#### 3.1. Setup of Coding and Channel Parameters

#### 3.1.1. Power-Delay Profile

#### 3.1.2. BLER Reductions

^{−1}level. Applying the same graphical means to measure BLER reduction as in Figure 2, by drawing vertical lines from the BLER = 10

^{−1}points up to the RV

_{0}curve, it is obvious that all BLER reductions are almost equal to 1/0.1, i.e., $\Delta BLE{R}_{\mathrm{RV}i/0}=10,i=1,2,3$.

_{i}to preserve the BLER of RV

_{0}with ${G}_{\mathrm{RV}i/0}$ times lower energy per bit to noise power spectral density ratio E

_{b}/N

_{0}but as the increase in the RV

_{0′s}E

_{b}/N

_{0}that makes its $BLE{R}_{\mathrm{RV}0}$ reduced to $BLE{R}_{\mathrm{RVi}}$.

_{0}, with and without the coding gain ${G}_{\mathrm{RV}i/0}$.

^{−1}is achieved with $BER\approx 0.1/L$, which, for the maximal block length in LTE (L = 6144 bits), amounts BER ≈ 1.63·10

^{−5}and determines the near-optimal “operating point” of (31) to be at ${E}_{\mathrm{b}}/{N}_{0}\approx 9.3$ dB, whereas the absolute upper-bound BLER = 1 is reached with BER ≈ 1.63·10

^{−4}at ${E}_{\mathrm{b}}/{N}_{0}=8.1$ dB already, i.e., with just as little as 1.2 dB SNR degradation between the target BLER value and the outage-related one.

_{i}, which has reached the target performance $BLE{R}_{\mathrm{RV}i}\approx {10}^{-1}$ with the coding gain ${G}_{\mathrm{RV}i/0}>1.2$ dB, the RV

_{0}was likely with the outage-state performance: $BLE{R}_{\mathrm{RV}0}\approx 1$, as various physical channel impairments (expressed as the AWGN-equivalent abstracts) easily overcome the 1.2 dB margin and produce many erroneous (especially large) blocks.

_{0}and the uncoded block transmission, to adopt in (29) and (30).

^{−1}to the “saturating” value 1 occurs with just a fraction of dB of E

_{b}/N

_{0}degradation, as it is shown for 16 QAM in the illustrative Figure 4.

#### 3.2. Analysis of Numerical Results

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**$BLE{R}_{\mathrm{RV}m}=0.1;\begin{array}{c}\end{array}m=0,1,2,3$ vs. SNR for CQI = 6 [26]; added here: $\Delta BLE{R}_{\mathrm{RV}m/0};m=1,2,3$.

Parameter/Numerology | Subcarrier Spacing (kHz) | OFDM Symbol Length (μs) | CP Length (μs) |
---|---|---|---|

0 | 15 | 66.67 | 4.69 |

1 | 30 | 33.33 | 2.34 |

2 | 60 | 16.67 | 1.17 |

3 | 120 | 8.33 | 0.57 |

4 | 140 | 4.17 | 0.29 |

Rms Delay Spread | Codeblock Length | ||
---|---|---|---|

$\sqrt{\stackrel{-}{{\tau}_{i}^{2}}}$ | L = 1536 | L = 3072 | L = 6144 |

100 ns | 79.3% | 77.7% | 74.9% |

200 ns | 56.1% | 51.1% | 46.7% |

300 ns | 29.6% | 22.5% | 17.4% |

400 ns | 3.4% | −17.0% | −19.2% |

Rms Delay Spread | Codeblock Length | ||
---|---|---|---|

$\sqrt{\stackrel{-}{{\tau}_{i}^{2}}}$ | L = 1536 | L = 3072 | L = 6144 |

100 ns | 80.6% | 77.9% | 76.3% |

200 ns | 57.0% | 53.2% | 47.7% |

300 ns | 31.3% | 26.0% | 19.1% |

400 ns | 7.6% | −16.1% | −18.5% |

Rms Delay Spread | Codeblock Length | ||
---|---|---|---|

$\sqrt{\stackrel{-}{{\tau}_{i}^{2}}}$ | L = 1536 | L = 3072 | L = 6144 |

100 ns | 81.4% | 79.2% | 77.0% |

200 ns | 59.1% | 53.5% | 49.2% |

300 ns | 34.7% | 27.1% | 21.8% |

400 ns | 9.6% | −14.9% | −17.6% |

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Lipovac, A.; Lipovac, V.; Modlic, B.
PHY, MAC, and RLC Layer Based Estimation of Optimal Cyclic Prefix Length. *Sensors* **2021**, *21*, 4796.
https://doi.org/10.3390/s21144796

**AMA Style**

Lipovac A, Lipovac V, Modlic B.
PHY, MAC, and RLC Layer Based Estimation of Optimal Cyclic Prefix Length. *Sensors*. 2021; 21(14):4796.
https://doi.org/10.3390/s21144796

**Chicago/Turabian Style**

Lipovac, Adriana, Vlatko Lipovac, and Borivoj Modlic.
2021. "PHY, MAC, and RLC Layer Based Estimation of Optimal Cyclic Prefix Length" *Sensors* 21, no. 14: 4796.
https://doi.org/10.3390/s21144796