#
Millimeter Wave Vehicular Channel Emulation: A Framework for Balancing Complexity and Accuracy^{ †}

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## Abstract

**:**

## 1. Introduction

#### 1.1. Channel Emulation

**b**) is reduced as much as possible, ideally making it a completely transparent adaption from (

**a**) to (

**c**). In the case of mmWave, this is only representative for the chosen antenna setup.

**a**) and (

**b**) are equally important and have to be modeled in conjunction. For mmWave, it is crucial to model the antenna position and orientation as part of Stage (

**a**).

**b**), and Stage (

**a**) channel measurements are only taken into account as validation and for tuning purposes, not as a model driving influence. This approach is not as strongly tied to the antenna positions as the other approaches, however very good models are needed to accurately capture the antenna effects.

#### 1.2. Literature Review

#### 1.3. Our Contribution

#### 1.4. Notation

## 2. Reduced Complexity Channel Models and Their Estimation

#### 2.1. Tapped Delay Line

- Only a small number of distinctive MPCs exist.
- If many MPCs exist, they are grouped in a small number of clusters. Within these clusters, all MPCs arrive within the same sampling period, resulting in only one resolvable tap.

#### 2.2. Clustered Delay Line

#### 2.3. Sparse Channel Estimation through c-LASSO

Algorithm 1 c-LASSO algorithm. | ||

1: | procedurec-LASSO($\mathit{h},\mathit{A},M$) | |

2: | $w\leftarrow ]0,1[$ | ▹ Tradeoff between speed and convergence |

3: | $\u03f5\leftarrow {\u03f5}_{0}$ | ▹ Threshold for detecting nonzero magnitudes. |

4: | $\mu \leftarrow {\mu}_{0}$ | |

5: | ${M}_{0}\leftarrow 0$ | |

6: | while ${M}_{i}\ne M$ do | |

7: | $i=i+1$ | |

8: | $\widehat{\mathit{b}}={arg\; min}_{\mathit{b}}\left(\parallel \left(\mathit{h}-\mathit{A}\mathit{b}\right){\parallel}_{2}^{2}+\mu {\parallel \mathit{b}\parallel}_{1}\right)$ | |

9: | if ${M}_{i}<M$ then | ▹ Activate additional taps |

10: | ${\mathit{u}}_{i}=2{\mathit{A}}^{H}\left(\mathit{h}-\mathit{A}\widehat{\mathit{b}}\right)$ | |

11: | $\mathcal{U}=\left\{m\phantom{\rule{0.277778em}{0ex}}\right|\phantom{\rule{0.277778em}{0ex}}1-\frac{{u}_{i}[m]}{\mu}<\u03f5\}$ | ▹ Set of local maxima above threshold. |

12: | $K=\left|\mathcal{U}\right|$ + 1 | |

13: | $\mu =(1-w)\phantom{\rule{4.pt}{0ex}}\mathrm{peak}({\mathit{u}}_{i},K)+w\phantom{\rule{4.pt}{0ex}}\mathrm{peak}({\mathit{u}}_{i},M+1)$ | |

14: | else if ${M}_{i}>M$ then | ▹ Deactivate unnecessary taps |

15: | bisecting between $\mu [i-1]$ and $\mu [i-2]$. | ▹ For details see [34] |

16: | end if | |

17: | end while | |

18: | Refine estimate using Equation (16) | |

19: | return $\widehat{\mathit{b}},\mu $ | |

20: | end procedure |

#### 2.3.1. Configuration for Delay Line Estimation

#### 2.3.2. Configuration for Subband Cluster Estimation

#### 2.3.3. Configuration for Sequential Estimation

#### Propagation of Active Taps

#### Discouraging Spawning New Taps

#### Regularization

## 3. Vehicular Channel Data

#### 3.1. Saleh–Valenzuela Channels

#### 3.2. Vehicular Channel Sounding Campaign

## 4. Results

#### 4.1. Comparison: Peak Search

#### 4.2. Performance Metrics

#### 4.2.1. Absolute Estimation Quality: Mean Squared Error

#### 4.2.2. Balancing Model Order: Akaike Information Criterion

#### 4.3. Performance Analysis for Synthetic Data

#### 4.4. Performance Analysis for Channel Sounding Data

#### 4.5. Discussion: Number of Observed Clusters during the Measurements

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

AIC Akaike Information Criterion |

c-LASSO Complex LASSO |

CDF Cumulative Distribution Function |

DFT Discrete Fourier Transform |

FPGA Field-Programmable Gate Array |

GSCM Geometry-based Stochastic Channel Model |

IDFT Inverse DFT |

LASSO Least Absolute Shrinkage and Selection Operator |

mmWave Millimeter Wave |

MPC Multipath Component |

MSE Mean Squared Error |

OAM Orbital Angular Momentum |

OEW Open-Ended Waveguide |

PDP Power Delay Profile |

RMS Root Mean Squared |

SNR Signal-to-Noise Ratio |

SUV Sports Utility Vehicle |

V2V Vehicle-to-Vehicle |

V2X Vehicle-to-Everything |

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**Figure 1.**Necessary stages for channel emulation: (

**a**) the encountered MPCs of a V2V transmission; (

**b**) a simplified channel impulse response representation of (

**a**); and (

**c**) how a physical device applies the model from (

**b**) to a signal going from Transmitter (Tx) to Receiver (Rx).

**Figure 2.**Example impulse response snapshot from measurements. The strong components clearly arrive in clusters. Potential clusters are indicated with ellipses. The noise floor is estimated via the median method.

**Figure 3.**Sample Saleh–Valenzuela PDPs. The plots show a realization for Conf. (

**a–d**), each produced with the same random seed. The four configurations have progressively dense intra-cluster multipaths.

**Figure 4.**Photograph of the measurement site. The transmitter and receiver are static and urban street traffic is driving by.

**Figure 5.**Zoomed photographs of the measurement site. On the left-hand side, the photo shows the transmit horn antenna mounted on a tripod. As multiple reflection with the transmitter car are below our receiver sensitivity, the TX car is replaced with a tripod. On the right-hand side, the photo shows the open-ended waveguide receive antenna mounted at roof height at the left, rear car side window.

**Figure 6.**Empirical CDFs and Gaussian fits of the estimation error. The error distribution both in real and imaginary part is well approximated as Gaussian, validating the assumptions for the AIC.

**Figure 7.**Relative change of MSE when using the c-LASSO instead of peak search. The results are shown for the four configurations (

**a**–

**d**) of the Saleh–Valenzuela models. The key takeaways are: (1) using $P=4$ shows the overall best performance across the MSE range; and (2) for very densely populated clusters, the c-LASSO noticeably outperforms peak search across a wide range of SNR (Scenarios (

**c**) and especially (

**d**)).

**Figure 8.**Sample impulse responses, estimated cluster impulse responses and cluster locations for $M=16$ and $P\in \{1,2,4\}$ ((

**a**); (

**b**); and (

**c**)).

**Figure 9.**Four-tap delay evolution for an overtaking SUV: memoryless estimations (

**a**,

**c**,

**e**); and sequential estimations (

**b**,

**d**,

**f**). The figure shows that spatial consistency is preserved well using sequential estimation.

**Figure 11.**Optimal number of clusters ${K}_{0}$ for all presented estimation basis and measurements. To the left of the separator, peak search is given as reference. On the right, different c-LASSO configurations are shown. The numbering is consistent with Figure 10.

Paramter unit | Unit | Symbol | Conf. (a) [44] | Conf. (b) | Conf. (c) | Conf. (d) |
---|---|---|---|---|---|---|

Cluster energy decay | ns | $\mathsf{\Gamma}$ | 60 | 60 | 120 | 120 |

Cluster arrival rate | ns ${}^{-1}$ | $\mathsf{\Lambda}$ | 1/300 | 1/90 | 1/90 | 1/30 |

MPC energy decay | ns | $\gamma $ | 20 | 20 | 20 | 20 |

MPC arrival rate | ns ${}^{-1}$ | $\lambda $ | 1/5 | 1/5 | 5 | 5 |

**Table 2.**Channel sounding measurement parameters. For the full parameter set, refer to [28].

Parameter | Value |
---|---|

Center frequency | 60 GHz |

Subcarrier spacing | $4.96$ MHz |

Number of subcarriers | 102 |

Snapshot rate | $129.1$ μs |

Delay resolution | $1.96$ ns |

Recording time | 720 ms |

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## Share and Cite

**MDPI and ACS Style**

Blazek, T.; Zöchmann, E.; Mecklenbräuker, C.F. Millimeter Wave Vehicular Channel Emulation: A Framework for Balancing Complexity and Accuracy. *Sensors* **2018**, *18*, 3997.
https://doi.org/10.3390/s18113997

**AMA Style**

Blazek T, Zöchmann E, Mecklenbräuker CF. Millimeter Wave Vehicular Channel Emulation: A Framework for Balancing Complexity and Accuracy. *Sensors*. 2018; 18(11):3997.
https://doi.org/10.3390/s18113997

**Chicago/Turabian Style**

Blazek, Thomas, Erich Zöchmann, and Christoph F. Mecklenbräuker. 2018. "Millimeter Wave Vehicular Channel Emulation: A Framework for Balancing Complexity and Accuracy" *Sensors* 18, no. 11: 3997.
https://doi.org/10.3390/s18113997