01.12.2019 | Research | Ausgabe 1/2019 Open Access

# Better than Rician: modelling millimetre wave channels as two-wave with diffuse power

## 1 Introduction

### 1.1 Related work

### 1.2 Outline and contributions

## 2 Methodology—fading model identification

### 2.1 Mathematical description of TWDP fading

_{1}≥0 and V

_{2}≥0 are the deterministic amplitudes of the non-fluctuating specular components. The phases ϕ

_{1}and ϕ

_{2}are independent and uniformly distributed in (0,2π). The diffuse components are modelled via the law of large numbers as X+jY, where \(X,Y\sim \mathcal {N}\left (0,\sigma ^{2}\right)\). The K-factor is the power ratio of the specular components to the diffuse components:

_{complex}| of TWDP fading is given as:

_{1}and V

_{2}, a clever choice of σ

^{2}normalises Ω, that is Ω≡1. Starting from (4), by using (2) we arrive at:

_{1},…,r

_{n},…r

_{N}) for parameter estimation of the tuple (K,Δ) as described in Section 2.2, and hypothesis testing as described in Section 2.3. The first set is carefully selected to obtain envelopes samples that are as independent as possible. The second set (r

_{1},…,r

_{m},…r

_{M}) is the complement of the first set. We use the elements of the second set to estimate the second moment via:

_{1}(·,·). For Δ→0, Eq. (8) reduces to the well-known Rice CDF:

### 2.2 Parameter estimation and model selection

_{1},…,r

_{n},…r

_{N}) at hand, we are seeking a distribution of which the observed realisations r

_{n}appear most likely. To do so, we estimate the parameter tuple (K,Δ) via the maximum likelihood procedure:

_{R}) or TWDP fading (AIC

_{T}):

_{R}=1, since we estimate the K-factor, only. For TWDP fading U

_{T}=2, as Δ is estimated additionally. The second moment Ω (estimated already with a different data set before the parameter estimation) is not part of the ML estimation (10) and therefore not accounted in the model order U. We choose between Rician fading and TWDP fading based on the lower AIC.

### 2.3 Validation of the chosen model

^{1}. At a significance level α, a null hypothesis is rejected if:

_{i}is the observed bin count in cell i and E

_{i}is the expected bin count in cell i under the null hypothesis \(\mathcal {H}_{0}\). The cell edges are illustrated with vertical lines in Fig. 2. The cell edges are chosen, such that 10 observed bin counts fall into one cell. The estimated parameters of the model are denoted by e. For Rician fading, we estimate e=2 (Ω,K) parameters, and for TWDP fading, we estimate e=3 (Ω,K,Δ) parameters in total. The (1−α) quantile of the chi-square distribution with m−e degrees of freedom is denoted by \(\chi _{(1-\alpha, m-e)}^{2}\). The prescribed confidence level is 1−α=0.01.

## 3 Floor plan and set-ups for MC1 and MC2

## 4 MC1: Scalar-valued wideband measurements

### 4.1 Measurement set-up

_{PLL}. For f

_{LO}≈60 GHz, the scaling factor is s

_{PLL}=120. To avoid crosstalk, we measure the transfer function via the conversion gain (mixer) measurement option of our VNA and operate the transmitter and receiver at different baseband frequencies: 601 to 1100 MHz and 101 to 600 MHz. The set-up is shown in Fig. 6.

### 4.2 Receive power and fading distributions

_{2}in (1) is zero by definition. Whenever the RX power is high, the K-factor is high. Below the K-estimate, the estimate of Δ is shown. Here again, by definition, Δ≡0 whenever we decide for Rician fading. For interacting objects, the parameter Δ tends to be close to one. Note, that decisions based on AIC select TWDP fading mostly when Δ is above 0.3. Smaller Δ values do not change the distribution function sufficiently to justify a higher model order.

## 5 MC2: Vector-valued spatial measurements

### 5.1 Measurement set-up

### 5.2 Receive power

### 5.3 Fading distributions

^{2}. Similar as in the previous section, we partition the measurements into 2 sets. The partitioning is made according to a 3D chequerboard pattern. The first set is used for the estimation of the second moment \(\hat {\Omega }\), and the second set is used for the parameter tuple (K,Δ).

## 6 MC2: Efficient computation of the spatial correlation

^{(z,f)}of one x−y slice at height z at a single frequency f through