Quadrature Spatial Scattering Modulation for mmWave Transmission

In this subsection, we employ a geometric Saleh-Valenzuela channel model to charctrize the channel of the proposed QSSM systems. As shown in Fig. 1, we assume that there are $L$ clusters between the receiver and the transmitter. Then, we select the two scatterers with the weakest correlation in each cluster according to the perfect CSI. Therefore, the channel model of the QSSM scheme can be expressed as

where $(\cdot)^{H}$ represents the conjugate transposition and $\beta_{l}$ is the complex gain of the $l$-th scatterer, i.e., $\beta_{l}\sim\mathcal{CN}(0,1),\forall l$, where $\mathcal{CN}(\cdot,\cdot)$ denotes circularly symmetric complex Gaussian distribution. The antenna array response vectors $\boldsymbol{a}_{r}(\theta_{l}^{r})\in\mathbb{C}^{N_{r}\times 1}$ and $\boldsymbol{a}_{t}(\theta_{l}^{t})\in\mathbb{C}^{N_{t}\times 1}$ can be respectively expressed as

where $(\cdot)^{T}$ is the transpose operator, $\phi_{l}^{r}={d_{r}\sin(\theta_{l}^{r})}/{\lambda}$, and $\phi_{l}^{t}={d_{t}\sin(\theta_{l}^{t})}/{\lambda}$. Further, $\theta_{l}^{r}\in[0,2\pi]$ and $\theta_{l}^{t}\in[0,2\pi]$ are the $l$th scatterer’s angle-of-arrival (AoA) and angle-of-departure (AoD) of the receiver and transmitter, respectively. Furthermore, $\lambda$ denotes the signal wavelength, $d_{r}$ and $d_{t}$ represent the distance between adjacent elements in the receive antenna array and the transmit antenna array, respectively.

Note that when $N_{r}$ and $N_{t}$ are large, the resulting beam is very narrow. At this point, we consider that there is no interference between the beams[12, 9, 10, 11]

where $\delta(\cdot)$ denotes the Dirac delta function.

In Fig. 1, the input bits are first modulated by $M$-ary PSK/QAM signal. Then, the modulated signal is decomposed into real and imaginary parts. Next, the transmitter sends quadrature and in-phase beams to carry real and imaginary parts of the signal, respectively. Particularly, two orthogonal beams can activate two candidate scatterers in the scattering cluster. As a result, the spectral efficiency of the QSSM system can be given by

Example 1: In the QSSM system with $M=4$ and $L=4$, we have the data rate $R=6$ b/s/Hz. Table I shows that $b_{1}b_{2}$ denotes the scatterer with the quadrature beam, $b_{3}b_{4}$ denotes the scatterer with the in-phase beam, and $b_{5}$ and $b_{6}$ represent the real part $x_{\Re}$ and imaginary part $x_{\Im}$ of the signal, respectively. We assume that the following input bits

are to be transmitted. Based on Table I, the first 2 bits modulate the active $2^{\#}$ scatterer index $l_{\Re}$ used to carry the real part of $\frac{1}{\sqrt{2}}$ corresponding to 5-th bit. Similarly, the 3-th and 4-th bits represent $3^{\#}$ scatterer activated by the in-phase beam, which carries the imaginary part of $\frac{1}{\sqrt{2}}$ corresponding to 6-th bit. To demonstrate the information in Table I clearly, we draw the QSSM constellation in Fig. 2.

It is worth noting that when the receiver is equipped with a large number of antennas, the 40 MHz bandwidth can achieve centimeter-level positioning accuracy [15]. In the mmWave band, as the communication frequency increases, the positioning accuracy will become more precise. The received signal of the QSSM scheme can be given as

where $P_{s}$ stands for the transmit power, $k_{1},k_{2}=1,2,\ldots,L$, and $\mathbf{n}\in\mathbb{C}^{N_{r}\times 1}$ denotes the additive white Gaussian noise (AWGN) following $\mathcal{CN}(0,N_{0}\mathbf{I}_{N_{r}})$. Note that vector $\mathbf{n}$ contains real and imaginary parts noises, each of part follows $\mathcal{CN}(0,{N_{0}}/{2})$.

To distinguish the direction of the received beam, we equip the receiver with a phase-shifted weight network to monitor the incoming wave direction of all scatterers in real time. The phase-shifted weight network can be described as $\mathbf{R}^{r}=\left[\mathbf{R}^{r}_{1};\mathbf{R}^{r}_{2}\right]^{{H}},$ where $\mathbf{R}^{r}_{1}=[\boldsymbol{a}_{{r}}(\theta_{1}^{r}),\cdots,\boldsymbol{a}_{{r}}(\theta_{k_{1}}^{r}),\cdots,\boldsymbol{a}_{{r}}(\theta_{L}^{r})]$ and $\mathbf{R}^{r}_{2}=[\boldsymbol{a}_{{r}}(\theta_{1}^{r}),\cdots,\boldsymbol{a}_{{r}}(\theta_{k_{2}}^{r}),\cdots,\boldsymbol{a}_{{r}}(\theta_{L}^{r})]$. where every weight vector consists of phase shifters aiming at a specific receive beam. Finally, the index of the scatterer can be obtained by judging the direction of the arrival beam. As such, the received signal after the weights of phase shifters can be expressed as

where

By utilizing Eq. (3), $\mathbf{y}_{r}$ can be calculated as

Finally, the ML detection for the selected scattering path can be characterized as

where $\hat{k}_{1}$ and $\hat{k}_{2}$ denote the indices of detected scatterers in the channel, $\hat{x}_{\Re}$ and $\hat{x}_{\Im}$ stand for real and imaginary parts of the detected PSK/QAM signal.

Based on Eq. (10), the conditional PEP can be derived as follows:

where

Obviously, $G\sim\mathcal{N}(\mu_{G},\sigma_{G}^{2})$, where $\mathcal{N}(\cdot,\cdot)$ denotes the real Gaussian distribution and the mean and variance are expressed as

Since $P_{r}=P(G>0)=P(-\mu_{G}/\sigma_{G})$, Eq. (11) can be rewritten as

where $Q(\cdot)$ denotes the $Q$ function, $\rho={P_{s}}/{N_{0}}$ indicates signal-to-noise ratio (SNR), and $\sqrt{\eta}={{(\beta_{k_{1}}x_{\Re}-\beta_{\hat{k}_{1}}\hat{x}_{\Re})+j(\beta_{{k}_{2}}{x}_{\Im}-\beta_{\hat{k}_{2}}\hat{x}_{\Im})}}$. Since zero mean of $\sqrt{\eta}$ and orthogonal beams of $k_{1}$ and $k_{2}$, the variance of $\sqrt{\eta}$ can be obtained as

Since the signal power is reflected at the receive SNR, we need to normalize $\eta$. Let us define $\gamma=\frac{\eta}{{\bar{\eta}}}$, the probability density function of the central Chi-square distribution with two degrees of freedom can be evaluated as [16]

Combining (14) and (16), the unconditional PEP can be characterized as

Substituting $Q(\gamma)=\frac{1}{\pi}\int_{0}^{\frac{\pi}{2}}\exp\left(-\frac{\gamma^{2}}{2\sin^{2}\theta}\right)d\theta$ into (17), the unconditional PEP can be calculated as

Exchanging the order of integration of $\theta$ and $\gamma$, we have

Addressing the inner integral, Eq. (19) can be recast as

After some algebraic operations, we can obtain

To clarify the trend of the PEP and to gain more valuable insights, we derive the asymptotic PEP of the QSSM in this subsection.

Substituting $Q(\gamma)\approx\frac{1}{12}\exp\left(-\frac{\gamma^{2}}{2}\right)+\frac{1}{4}\exp\left(-\frac{2\gamma^{2}}{3}\right)$ into (17), we can obtain the loose upper bound of $\bar{P}_{r}$ as follows:

In high SNR regions, the constant term added to $\rho\bar{\eta}$ can be ignored. Consequently, Eq. (22) can be evaluated as

With the unconditional PEP and asymptotic PEP in (21) and (23), we provide the corresponding ABEP in (24), shown at the top of next page, where $N([k_{1},k_{2},m]\to[\hat{k}_{1},\hat{k}_{2},\hat{m}])$ denotes the Hamming distance between symbol $[k_{1},k_{2},m]$ and $[\hat{k}_{1},\hat{k}_{2},\hat{m}]$. Besides, $\log_{2}(L^{2}M)$ represents the total number of constellation points in the QSSM constellation diagram.