【空中计算】Over-the-air Computing in OFDM Systems

Consider an over-the-air computing system consisting of a receiver，which acts as an fusion center, and $K$ transmitting devices. We assume that we want to calculate a function $f:{\mathbb{R}}^K\to \mathbb{R}$.
We introduce OFDM. Each device can utilize a set of subcarriers L = { 0 , ⋯ , L − 1 } \mathcal{L} = \{0,\cdots,L - 1\} L={0,⋯,L−1} to transmit data for aggregation per subcarrier L=64. The channel of each device is a finite impulse response (FIR) filter $h_k[n],0\le n\le \mu$, where exist interference between $\mu$=4 symbols.

Each device has a set of $L$ symbols to transmit ${\mathbf{X}}_k=[X_k[0],\cdots ,X_k[L-1]]$. Add the cyclic prefix (CP)${\tilde{\mathbf{X}}}_k=[X_k[L-\mu ],\cdots ,X_k[L-1],X_k[0],\cdots ,X_k[L-1]]$.

The available transmit power of the $k$-th user at each subcarrier is denoted as ${\mathbf{b}}_k=[b_k[0],\cdots ,b_k[L-1]]$. The transmit power for each user and subcarrier is denoted by $B=[{\mathbf{b}}_1\cdots {\mathbf{b}}_K]$. The vector transmitted by the $k$-th device for the data symbols ${\mathbf{P}}_k=[P_k[0],\cdots ,P_k[L-1]]=⟨{\mathbf{X}}_k,{\mathbf{b}}_k⟩$.

At the FC, a different receiver gain can be used for each subcarrier, which is denoted as $\mathbf{a}=[a[0],\cdots ,a[L-1]]$.

Each subcarrier can be conceptualized as having a frequency offset $\Delta f$ from its ideal transmit frequency, which is normalized with respect to the bandwidth of two consecutive subcarriers and modeled as a Gaussian random variable $ϵ\sim \mathcal{N}(0,{\sigma }_{ϵ}^2)$, ${\sigma }_{ϵ}^2$=0.01.

With inverse fast Fourier transform (IFFT), the $n$-th symbol transmitted by the $k$-th device is given by $p_k[n]=\frac{1}{\sqrt{L}}{\sum }_{l=0}^{L-1}{P}_k[l]e^{j2\pi n\frac{l}{L}},\forall n\in \mathcal{L}$.

The received signal at the $n$-th sample can be written as $r[n]={\sum }_{k=1}^K{p}_k[n]\ast h_k[n]e^{j2\pi n\frac{ϵ}{L}}+w[n],\forall n\in \mathcal{L}$.

With the fast Fourier transform (FFT) to the received samples, received signal of the $l$-th subcarrier is $\begin{array}{c}R[l]=\frac{1}{L}\sum _{k=1}^K\sum _{n=0}^{L-1}\sum _{m=0}^{L-1}{P}_k[m]H_k[m]e^{-j2\pi n\frac{m-ϵ}{L}}{e}^{j2\pi n\frac{l}{L}}+W[l]\\ =\sum _{k=1}^K\left(P_k[l]H_k[l]S(0)+\sum _{m=0,}^{L-1}{P}_k[m]H_k[m]S(l-m)\right)+W[l],\end{array}$

$W[l]=\frac{1}{\sqrt{L}}{\sum }_{n=0}^{L-1}w[n]e^{-j2\pi n\frac{l}{L}}$ is the FFT of the noise at the $l$-th subcarrier. $H_k[m]=\frac{1}{\sqrt{L}}{\sum }_{n=0}^{L-1}{h}_k[n]e^{-j2\pi n\frac{m}{L}}$ is the channel frequency response of the $k$-th device at the $m$-th subcarrier. Coefficients of the ICI terms which are given as $\begin{array}{rl}S(l-m)& =\frac{1}{L}\sum _{n=0}^{L-1}{e}^{j2\pi (l-m+ϵ)\frac{n}{L}}\\ & =\frac{\sin \left(\pi (l-m+ϵ)\right)}{L\sin \left(\pi \left(\frac{l-m+ϵ}{L}\right)\right)}{e}^{j\pi (l-m-ϵ)\left(1-\frac{1}{L}\right)},\end{array}$

For the $l$-th subcarrier, the OTA calculation is equal to $\hat{Y}[l]=a[l]R[l]$, while the ideally received signal is $Y[l]={\sum }_{k=1}^K{X}_k[l].$

The combined MSE of all subcarriers is given as M S E I C I = ∑ l = 0 L − 1 a 2 [ l ] ∑ k = 1 K ∑ m = 0 L − 1 b k 2 [ m ] H k 2 [ m ] E [ ∣ S ( l − m ) ∣ 2 ] − 2 ∑ l = 0 L − 1 a [ l ] ∑ k = 1 K b k [ l ] H k [ l ] E [ ∣ S ( 0 ) ∣ ] E [ cos ⁡ ( arg ⁡ { S ( 0 ) } ) ] + L K + σ 2 L ∑ l = 0 L − 1 a 2 [ l ] , \begin{aligned} &\mathrm{MSE}^{\mathrm{ICI}}=\sum_{l=0}^{L-1}a^{2}[l]\sum_{k=1}^{K}\sum_{m=0}^{L-1}b_{k}^{2}[m]H_{k}^{2}[m]\mathbb{E}[|S(l-m)|^{2}] \\ &-2\sum_{l=0}^{L-1}a[l]\sum_{k=1}^Kb_k[l]H_k[l]\mathbb{E}[|S(0)|]\mathbb{E}[\cos{(\arg\{S(0)\})}] \\ &+LK+\frac{\sigma^2}L\sum_{l=0}^{L-1}a^2[l], \end{aligned} ​MSEICI=l=0∑L−1​a2[l]k=1∑K​m=0∑L−1​bk2​[m]Hk2​[m]E[∣S(l−m)∣2]−2l=0∑L−1​a[l]k=1∑K​bk​[l]Hk​[l]E[∣S(0)∣]E[cos(arg{S(0)})]+LK+Lσ2​l=0∑L−1​a2[l],​, where arg ⁡ { ⋅ } \arg\{\cdot\} arg{⋅} symbolizes the argument of the specified complex number.

We can formulate the following problem under the total power constraint for each user
$\begin{array}{rlrl}& \underset{\mathbf{a},\mathbf{B}}{\min }& & {\mathrm{MSE}}^{\mathrm{ICI}},\\ & s.t.& & {\mathbb{C}}_1:\Vert b_{\mathbf{k}}{\Vert }_2^2\le P,\forall k\in \mathcal{K}.\end{array}$

${\mathrm{MSE}}^{\mathrm{ICI}}$ is convex for one set of variables when the other is fixed. We use the statistical mean of the ICI coefficients between subcarriers $S(l-m),\forall l,m\in \mathcal{L}$ in $(\mathbf{B}).$ Fixing $\mathbf{a}$ and solving in terms of $b_k[m],\forall k\in \mathcal{K},m\in \mathcal{L}$, the optimal power coefficients can be obtained by considering the Lagrangian function , which is given as

F = ∑ l = 0 L − 1 a 2 [ l ] ∑ k = 1 K ∑ m = 0 L − 1 b k 2 [ m ] H k 2 [ m ] E [ ∣ S ( l − m ) ∣ 2 ] − 2 ∑ l = 0 L − 1 a [ l ] ∑ k = 1 K b k [ l ] H k [ l ] E [ ∣ S ( 0 ) ∣ ] E [ cos ⁡ ( arg ⁡ { S ( 0 ) } ) ] + L K + σ 2 L ∑ l = 0 L − 1 a 2 [ l ] + ∑ k = 1 K λ k ( ∑ m = 0 L − 1 b k 2 [ m ] − P ) . \begin{aligned}\mathcal{F}&=\sum_{l=0}^{L-1}a^2[l]\sum_{k=1}^K\sum_{m=0}^{L-1}b_k^2[m]H_k^2[m]\mathbb{E}[|S(l-m)|^2]\\&-2\sum_{l=0}^{L-1}a[l]\sum_{k=1}^Kb_k[l]H_k[l]\mathbb{E}[|S(0)|]\mathbb{E}[\cos{(\arg\{S(0)\})}]\\&+LK+\frac{\sigma^2}L\sum_{l=0}^{L-1}a^2[l]+\sum_{k=1}^K\lambda_k\left(\sum_{m=0}^{L-1}b_k^2[m]-P\right).\end{aligned} F​=l=0∑L−1​a2[l]k=1∑K​m=0∑L−1​bk2​[m]Hk2​[m]E[∣S(l−m)∣2]−2l=0∑L−1​a[l]k=1∑K​bk​[l]Hk​[l]E[∣S(0)∣]E[cos(arg{S(0)})]+LK+Lσ2​l=0∑L−1​a2[l]+k=1∑K​λk​(m=0∑L−1​bk2​[m]−P).​

After differentiating, b k [ m ] = a [ m ] H k [ m ] E [ ∣ S ( 0 ) ∣ ] E [ cos ⁡ ( arg ⁡ { S ( 0 ) } ) ] ∑ l = 0 L − 1 a 2 [ l ] H k 2 [ l ] E [ ∣ S ( l − m ) ∣ 2 ] + λ k b_k[m]=\frac{a[m]H_k[m]\mathbb{E}[|S(0)|]\mathbb{E}[\cos(\arg\{S(0)\})]}{\sum_{l=0}^{L-1}a^2[l]H_k^2[l]\mathbb{E}[|S(l-m)|^2]+\lambda_k} bk​[m]=∑l=0L−1​a2[l]Hk2​[l]E[∣S(l−m)∣2]+λk​a[m]Hk​[m]E[∣S(0)∣]E[cos(arg{S(0)})]​.

Fixing $\mathbf{B}$ and solving in terms of $a[l],\forall l\in \mathcal{L}$ the optimal MSE is achieved for
a [ l ] = ∑ k = 1 K b k [ l ] H k [ l ] E [ ∣ S ( 0 ) ∣ ] E [ cos ⁡ ( arg ⁡ { S ( 0 ) } ) ] ∑ k = 1 K ∑ m = 0 L − 1 b k 2 [ m ] H k 2 [ m ] E [ ∣ S ( l − m ) ∣ 2 ] + σ 2 / L . a[l]=\frac{\sum_{k=1}^Kb_k[l]H_k[l]\mathbb{E}[|S(0)|]\mathbb{E}[\cos{(\arg{\{S(0)\}})}]}{\sum_{k=1}^K\sum_{m=0}^{L-1}b_k^2[m]H_k^2[m]\mathbb{E}[|S(l-m)|^2]+\sigma^2/L}. a[l]=∑k=1K​∑m=0L−1​bk2​[m]Hk2​[m]E[∣S(l−m)∣2]+σ2/L∑k=1K​bk​[l]Hk​[l]E[∣S(0)∣]E[cos(arg{S(0)})]​.
Iterating between $B$ and $a$ yields a final solution for both transmitting equalization and receiver gain factors.

The channel gains of all $K$ devices and $L$ subcarriers $H_k[n]$, $\forall k\in \mathcal{K}$, $\forall n\in \mathcal{L}$ are modeled as circularly symmetric complex Gaussian random variables with variance equal to 1, i.e., $H_k[n]\sim \mathcal{C}\mathcal{N}(0,1).$ The maximum transmit power for each device is set to $P=10$, so that the average transmit SNR is 10 dB. Unless otherwise specified, the number of devices. We assume that the ISI symbols due to the delay spread of the channel are less than the CP added symbols, which are chosen so that the CP overhead $\mu /L$ is equal to 6.25%.

Help me solve the optimal MSE using MATLAB and draw the optimal MSE curve for K=[10:2:50]