Signal Recovery Performance Analysis in Wireless Sensing with Rectangular-Type Analog Joint Source-Channel Coding

The 3:1 mapping of rectangular type is depicted in Fig. 1. The mapped signal is the accumulated length of the lines from the origin to the mapped point. The mapping from source point $(s_{1},s_{2},s_{3})$ to the point $(x_{s},y_{s},z_{s})$ on the the AJSCC curve can be written by $x_{s}=\frac{d}{{R_{1}}}s_{1}$, $y_{s}=s_{2}+\lambda_{y}$ and $z_{s}=s_{3}+\lambda_{z}$, where the $\lambda_{y}$ and $\lambda_{z}$ are error terms due to the discrete nature of AJSCC on y and z axes. The high SNR is defined by the SNR range such that, the probability that received point (“Signal at the receiver” in Fig. 1) is located on the adjacent line (as opposed to mapped line) can be neglected in the derivation and evaluation. In other words, the received point is assumed to locate on the same line mapped at the transmitter albeit with variation. The medium/low SNR scenario is defined by the SNR range where such probability cannot be neglected in the derivation and evaluation. Hence it is possible that the received point is located on the adjacent lines (either up or down) (discussed later in Sect. III-B).

Based on the mapping method, $y_{s}=round\left({\frac{{s_{2}}}{{\Delta_{y}}}}\right)\cdot\Delta_{y}$, where $\Delta_{y}$ is the spacing of the lines in the y-axis and has the value of $R_{2}/(L_{1}-1)$. Hence, the error term $\left|{\lambda_{y}}\right|$ can be written as,

If we assume now a uniform source signal distribution of $s_{2}$, the $\left|{\lambda_{y}}\right|$ is a random variable uniformly distributed in $[0,{\Delta_{y}}/2]$. The error term $\left|{\lambda_{z}}\right|$ can also be written by $\left|{\lambda_{z}}\right|=\left|{s_{3}-\tilde{s}_{3}}\right|=\left|{s_{3}-round\left({\frac{{s_{3}}}{{\Delta_{z}}}}\right)\cdot\Delta_{z}}\right|$, and $\Delta_{z}$=$R_{3}/(L_{2}-1)$. The error term $\left|{\lambda_{z}}\right|$ is also a random variable uniformly distributed in $[0,{\Delta_{z}}/2]$.

As mentioned above, in case of high SNR, the probability that the received point moves to another parallel line is so small that it can be neglected in the analysis. For Gaussian noise and in such SNR regime, the received signal can be expressed by $x_{r}=x_{s}+n$, $y_{r}=y_{s}$, and $z_{r}=z_{s}$. The random variable $n$ is the Gaussian noise with the variance of $\sigma_{n}^{2}$. This Gaussian noise is an assumption in our analytical work. The recovered source signal is denoted by $(\tilde{s}_{1},\tilde{s}_{2},\tilde{s}_{3})$, where $\tilde{s}_{1}$ can be written as $\tilde{s}_{1}=\frac{{R_{1}}}{d}x_{r}=\frac{{R_{1}L_{1}L_{2}}}{{D_{\max}}}(x_{s}+n)=s_{1}+\frac{{R_{1}L_{1}L_{2}}}{{D_{\max}}}n$. We can observe that greater the values of $L_{1}$ and $L_{2}$, the larger the error. The received signal $\tilde{s}_{2}$ can be written as $\tilde{s}_{2}=y_{s}=s_{2}+\lambda_{y}$, where $\lambda_{y}$ is the error term. Similarly, we have $\tilde{s}_{3}=z_{s}=s_{3}+\lambda_{z}$. The sum MSE of the three signals is $MSE_{3,H}=E\{{\left|{s_{1}-\tilde{s}_{1}}\right|^{2}}\}+E\{{\left|{s_{2}-\tilde{s}_{2}}\right|^{2}}\}+E\{{\left|{s_{3}-\tilde{s}_{3}}\right|^{2}}\}$, which can be further written as,

The Probability Density Function (PDF) of $\left|{\lambda_{y}}\right|$ is $p_{\left|{\lambda_{y}}\right|}=\{\begin{array}[]{l}2/\Delta_{y},{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}\left|{\lambda_{y}}\right|\in[0,\Delta_{y}/2]{\kern 1.0pt}{\kern 1.0pt}\\ 0{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt},{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}otherwise.{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}{\kern 1.0pt}\\ \end{array}$. The expectation of $\left|{\lambda_{y}}\right|^{2}$ in (2) is thus written as $E\{{\kern 1.0pt}\left|{\lambda_{y}}\right|^{2}\}=\int\limits{{\kern 1.0pt}\left|{\lambda_{y}}\right|^{2}}p_{\left|{\lambda_{y}}\right|}d{\left|{\lambda_{y}}\right|}=\frac{1}{{12}}\Delta_{y}^{2}=\frac{1}{{12}}\frac{{R_{2}^{2}}}{{(L_{1}-1)^{2}}}$. Similarly, the expectation of $\left|{\lambda_{z}}\right|^{2}$ is $E\{{\kern 1.0pt}\left|{\lambda_{z}}\right|^{2}\}=\frac{1}{{12}}\frac{{R_{3}^{2}}}{{(L_{2}-1)^{2}}}$. The noise $n$ is assumed to follow a Normal distribution $\mathcal{N}(0,\sigma_{n}^{2})$.

The MSE for 3:1 mapping can then be expressed in the following closed form,

For the $N$:1 mapping, the source signals, denoted as $(s_{1},s_{2},...,s_{N})$, are mapped to points on the $N$-dimensional space, $(x_{1},x_{2},...,x_{N})$. The source signal $(s_{1},s_{2},...,s_{N})$ having range $[0,R_{1}]$, $[0,R_{2}]$, …, $[0,R_{N}]$ is scaled and mapped to a point, $(x_{1},x_{2},...,x_{N})$, on the $N$-dimensional space by $x_{1}=\frac{d}{{R_{1}}}s_{1}$, $x_{2}=s_{2}+\lambda_{2}$, …, $x_{N}=s_{N}+\lambda_{N}{\kern 1.0pt}$. The transmitting signal constraint now generalizes to $\mathop{\prod}\limits_{k=1}^{N-1}L_{k}\cdot d\leq D_{\max}$. Similarly to the 3:1 mapping analysis, the MSE for N:1 mapping can be expressed as,

In the most general case for medium/low SNR, it is possible that the received point is disturbed to another stage (parallel line) or even to the adjacent z-plane in boundary cases (Fig. 3). We call the event when the received point is disturbed to another line (also referred to as stage or level) as a Stage Crossing event. When the SNR is medium to low, the probability of this event cannot be neglected. To compute the MSE, we model the received signal as $x_{r}=x_{s}+n+\nu_{x}$, $y_{r}=y_{s}+\nu_{y}$ and $z_{r}=z_{s}+\nu_{z}$. where the variables $\nu_{x}$, $\nu_{y}$, and $\nu_{z}$ are the values added by the fact that the received point is located at an adjacent stage. There are two types of stage crossing: the point is crossed to the left-hand-side, named Left Stage Crossing (LSC), and the point is crossed to the right-hand-side, named Right Stage Crossing (RSC). In the following, we firstly derive the probabilities of LSC and RSC; then find the distributions of variables $\nu_{x}$, $\nu_{y}$, and $\nu_{z}$. Finally, the MSE of three-dimensional mapping is derived based on these probabilities and distributions. For analog sensing, the source signals are mapped to one point on the AJSCC curve. With frequency modulation and the assumption of the equivalent Gaussian noise channel, the relationship between the sensing source signal $(s_{1},s_{2},s_{3})$ and the recovered source signal $(\tilde{s}_{1},\tilde{s}_{2},\tilde{s}_{3})$ can be written by $\tilde{s}_{1}=s_{1}+\nu_{x}+\frac{{R_{1}L_{1}L_{2}}}{{D_{\max}}}n$, $\tilde{s}_{2}=s_{2}+\lambda_{y}+\nu_{y}$ and $\tilde{s}_{3}=s_{3}+\lambda_{z}+\nu_{z}$. In the following, we characterize the properties of the variables $\nu_{x},\nu_{y}$ and $\nu_{z}$.

Probabilities of Left and Right Stage Crossing: Define the PDF of signal $x_{r}$, normally distributed as $p_{X}(x)=N(x_{s},\sigma_{n}^{2})$, where the mean $x_{s}$ is also a random variable following distribution, ${p_{S}}$ (which is bounded in [$0$, $d$], and can be expressed by the distribution corresponding to $s_{1}$). The LSC event happens when the received signal $x_{r}$ is less than zero, which is expressed by $\Pr\{LSC\}=\int\limits_{0}^{d}{p_{S}(s)\left[{\Pr\left\{{x<0}\right\}}\right]ds}=\int\limits_{0}^{d}{p_{S}(s)\int\limits_{-\infty}^{0}{p_{X}(x)dxds}}$. Note that, the variable $s$ represents the source distribution and the variable $x=s+n$ is the received signal with mean of $s$. Due to the distribution $p_{X}(x)$ following $N(s,\sigma_{n}^{2})$, the LSC probability can be expressed by,

As mentioned above, distribution $p_{S}$ follows the distribution as measured in the source signal. The RSC probability can be similarly written as,

In particular, if the source is Gaussian, the signal will be bounded in the range [0, $d$]. The signal outside of this bound will be limited to the values at the boundaries. This is a unique property if Gaussian source is encoded by AJSCC. This is because, the instantaneous realization of the random variable outside of the bound [0, $d$] is truncated and the random variable will take the value at the upper bound $d$ or the lower bound $0$. This property will result in a new type of signal distribution that is continuous in the range (0, $d$), but has discrete values at the two boundaries, $0$ and $d$. We introduce a new type of distribution, named Discrete-Boundary Gaussian Distribution, which is defined by,

For this type of distribution, the LSC probability can be expressed by

The RSC probability can be expressed in a similar way.

Cases of variables $\nu_{x}$, $\nu_{y}$, and $\nu_{z}$: Based on geometry, as observed from Fig. 3, we partition the stage crossing into three major cases—a), b), and c)—based on the locations of the transmitted points. In case a), the point is located at any stage besides top and bottom stage of any z-plane. In case b), the point is located at the top or bottom stages of z-plane besides the first and last z-plane. In case c), the point is located at the top or bottom stage of the first and last z-plane. In each case, the distribution differs for odd and even number of stages and z-planes, and differs for LSC and RSC and other geometry-specific information. The values of variables $\nu_{x}$, $\nu_{y}$, and $\nu_{z}$ for different cases and sub-cases are listed in Table I along with corresponding parameter conditions.

For the case group a), point $(x_{r},y_{r},z_{r})$ is not located at the top or bottom stage of any z-plane. There are two cases a1) and a2) discussed in the following. a1) For odd-numbered stages: As observed from three-dimensional AJSCC mapping figure (Fig. 3), LSC will lead to one stage down from left-hand side of the received signal point $(x_{r},y_{r},z_{r})$, compared with transmitted signal point $(x_{s},y_{s},z_{s})$. This can be expressed by $x_{r}=\left|{x_{s}+n}\right|$, $y_{r}=y_{s}-\Delta_{y}$ and $z_{r}=z_{s}$. It is to be noted that the value ${x_{s}+n}$ is less than zero for LSC, therefore $\left|{x_{s}+n}\right|=-n-x_{s}$ and is a value greater than zero. Variables $\nu_{x}$, $\nu_{y}$, and $\nu_{z}$ have values as $v_{x}=-n-2x_{s}$, $v_{y}=-\Delta_{y}$ and $v_{z}=0$. The RSC will lead to one stage upward from right-hand side. This can be written as,

a2) For even-numbered stages: As in Fig. 3, LSC will lead to one stage up from left-hand side and RSC will lead to one stage down from right-hand side.

In case group b), point $(x_{r},y_{r},z_{r})$ is located at the top or bottom stage, but not at the 1st or last z-plane. There are four sub-cases of b1) to b4) explained in Table I. In case group c), point $(x_{r},y_{r},z_{r})$ is located at the 1st or last z-plane. The four sub-cases of c1) to c4) are also explained in Table I. Readers can refer to the table for details of these sub-cases.

MSE of three-dimensional mapping can now be written as,

where ${e_{LSC}}$ is the error of LSC and ${e_{RSC}}$ is the error of RSC. The terms can be computed as

and

where $p_{N}$ follows Gaussian distribution $N(0,\sigma_{n}^{2})$ and $p_{S}$ is the source distribution of $x_{s}$. In the above formulation, the source distribution is in general form. For a particular source distribution, the MSE results will have particular outputs. The MSE term can be further expressed as

which contains the errors due to left and right stage crossing compared to the high SNR case (see (2)).

In this subsection, the MSE analysis of high and medium/low SNR is given for digital sensors with low-resolution ADC. The motivation to investigate the low-resolution digital sensing is the high amount of power consumption in case of high-resolution ADC (8 bits to 16 bits). It is thus of practical value to evaluate the performance of a low-resolution ADC (5 bits and less) and compare its performance with that of an analog sensor employing rectangular-type AJSCC. Such a performance comparison provides insights into effective sensor design methodologies. For fair comparison, we compare both analog sensor and digital sensor under the same system setup including frequency modulation, effect of wireless channel and the receiver signal processing and decoding.

We will now present the MSE analysis for digital sensors where the source signal is processed by ADC. In contrast with the analog sensor case, the source signal $s_{1}$ is quantized due to the ADC effect. The sampled and mapped signals are denoted by $\hat{x}_{s},\hat{y}_{s},\hat{z}_{s}$. The expressions of these signals are $\hat{x}_{s}=\frac{d}{{R_{1}}}(s_{1}+\lambda_{x})$, $\hat{y}_{s}=s_{2}+\lambda_{y}$ and $\hat{z}_{s}=s_{3}+\lambda_{z}$. The term $\lambda_{x}$ is the quantization error term due to the ADC effect. The term $\left|{\lambda_{x}}\right|$ can be expressed by $\left|{\lambda_{x}}\right|=\left|{s_{1}-\tilde{s}_{1}}\right|=\left|{s_{1}-round\left({\frac{{s_{1}}}{{\Delta_{x}}}}\right)\cdot\Delta_{x}}\right|$, where the parameter of $\Delta_{x}$ is the ADC quantization spacing. Assume there are $N_{b}$ bits in the ADC to quantize the source signal $s_{1}$, $\Delta_{x}=d/R_{1}1/(2^{N_{b}}-1)$. $\left|{\lambda_{x}}\right|$ is a random variable uniformly distributed in $[0,{\Delta_{x}}/2]$. It is further assumed that the source quantization and AJSCC mapping for $s_{2}$ and $s_{3}$ are chosen so that the quantization levels of $y$ and $z$ are equal to the number of parallel lines $L_{1}$ and $L_{2}$. The intuition is, the source quantization and AJSCC mapping are jointly designed so as to minimize the distortions introduced by the source quantization and AJSCC mapping. The received signal of the x-axis after AJSCC demapping at the receiver is added with a Gaussian noise term as $\hat{x}_{r}=\hat{x}_{s}+n$. The source of this noise is the equivalent noise of the frequency modulation in wireless fading channel along with the fading effect and the additive noise. The recovered source signal of the first dimension can be written as $\hat{s}_{1}=\frac{{R_{1}}}{d}\hat{x}_{r}=s_{1}+\lambda_{x}+\frac{{R_{1}}}{d}n$. The sum MSE of signal recovery with digital sensing with ADC can be expressed by

which can be further simplified to

For digital sensing with ADC, the signal recovery for medium/low SNR scenario can be written as $\hat{s}_{1}=s_{1}+\lambda_{x}+\nu_{x}+\frac{{R_{1}L_{1}L_{2}}}{{D_{\max}}}n$, $\hat{s}_{2}=s_{2}+\lambda_{y}+\nu$ and $\hat{s}_{3}=s_{3}+\lambda_{z}+\nu_{z}$. The MSE of the signal recovery can be written as ${MSE_{3,L,D}}=\left[{\Pr\{LSC\}E\left\{{\left|{e_{LSC}}\right|^{2}}\right\}+}\right.\Pr\{RSC\}E\left\{{\left|{e_{RSC}}\right|^{2}}\right\}+\frac{{R_{1}^{2}L_{1}^{2}L_{2}^{2}}}{{D_{\max}^{2}}}E\{\left|n\right|^{2}\}+E\{{\left|{\lambda_{x}}\right|^{2}}\}+E\{{\left|{\lambda_{y}}\right|^{2}}\}+E\{{\left|{\lambda_{z}}\right|^{2}}\}$, which can be further expressed as,

where the $\Pr\{LSC\}$, $\Pr\{RSC\}$ can be calculated according to the above derived formulations.