Capacity Analysis for Tunnel Diode Amplifier Assisted Ambient Backscatter Communications

Ambient backscatter communication (AmBC) addresses connectivity, cost, and congestion bottlenecks for the Internet of Things (IoT) deployment. AmBC, by avoiding a dedicated power infrastructure and carrier emitter, allows tags to communicate by simply reflecting the ambient radio frequency (RF) signals to the reader. Thus, the tag can operate in the low-maintenance, battery-free mode and with energy harvesting. However, a fundamental bottleneck is the limited communication range. A novel solution is that the tag can use a tunnel diode amplifier to enhance its backscattered signal power and thus extend the communication range. This paper studies the tunnel diode amplifier solution and investigates the resulting capacity performance. Specifically, we first develop the system’s mathematical model, elaborate on the tunnel diode’s amplification mechanism, and derive its reflection gain. Subsequently, we derive the closed-form channel capacity and propose a reader-tag distance adjustment scheme to improve the system’s performance. Finally, simulation results corroborate our theoretical results. They show that the tunnel diode amplifier can significantly increase the system capacity by an order of magnitude when the tag receives less than -30 dBm of incident RF signal power. They also demonstrate a significant increase in the communication range.

function of the reflection gain. Section IV develops the sys-93 tem's capacity analytically and discusses the optimal RF 94 source-to-tag distance of the system. Numerical results are 95 provided in Section V. Finally, Section VI concludes this 96 paper. 98 Consider an AmBC system comprising an RF source (S), 99 a backscatter tag with a tunnel diode amplifier (T), and a 100 reader (R) ( Fig. 1 (a)). Fig. 1 (b) shows the block diagram 101 of the tag. Let d sr , d st , and d tr be the S − R, S − T , and 102 T − R distances, respectively. Let h sr , h st , and h tr represent 103 the S −R, S −T , and T −R channel coefficients, respectively. 104 Following [7], [20], we assume the existence of line-of-105 sight signals, and hence the channel coefficients follow the 106 normalized Rician distribution. We assume channel state information (CSI) is avail-109 able at the reader via pilot-assist channel estimation 110 VOLUME 10, 2022 techniques [7], [21], [22], [23]. Besides, if the CSI is not tag can be written as 127 where P s is the transmit power of the RF source, and α is the 128 large-scale path loss factor.

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The tag will adjust its refection coefficient by dynami-130 cally changing the load impedance Z L to modulate its infor-131 mation. It can be expressed as

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where Z A and Z L are antenna impedance and load impedance, 134 respectively, and | | and θ are the amplitude and phase of the 135 reflection coefficient, respectively. 136 Consequently, the backscattered signal can be expressed as 137 x t (n) = | |e jθ x(n)y t (n).
(4) 138 The noise contribution from the tag is negligible and can be 139 omitted [7], [20]. 140 Denote the antenna impedance by where R A and X A are the real and imaginary parts of Z A .

143
Typically, the tag applies on-off key (OOK) modulation. 144 To achieve it, the tag switches the load impedance Z L between 145 two values, which generates binary OOK modulated signals.

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This condition results in the amplification of the backscatter 159 signals. By exploiting these two options, the tag may or 160 may not amplify the signals depending on practical operating 161 scenarios. We will discuss these next.

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Without loss of generality, the signal received at the reader 163 can be expressed as This section briefly describes the amplification mechanism 171 of the tunnel diode and derives the reflection gain in dB as a 172 function of the input power.

173
A tunnel diode is a two-port device with a heavily doped 174 positive-to-negative (P-N) junction about 10 nm wide. The 175 heavy doping causes a broken band gap, where conduc-176 tion band electron states on the N-side are roughly aligned 177 with valence band hole states on the P-side. The deple-178 tion layer of the tunnel diode is small. So the electrons 179 can directly tunnel across it from the n-side conduction 180 band into the p-side valence band. This quantum tunneling 181 effect can result in a reverse current and thus a negative 182 impedance [18], [28], [29].

183
The negative impedance of a tunnel diode is evident from 184 its I − V curve. Fig. 2 depicts the ideal characteristic I − V 185 curve of a tunnel diode [18]. This curve shows that when the 186 tunnel diode is properly biased, current I decreases with bias 187 voltage V bias (regions II and IV).

188
Therefore, the resistance of a tunnel diode can be set 189 to a negative value. For instance, consider the negative 190 impedance design of backscatter devices. The tag in [15] uses 191 an MSP430 microcontroller to output a voltage and leverages 192 a matching network configured for a specific impedance 193 value of the tunnel diode to bias the output voltage between 194 65 mV and 150 mV, yielding an impedance approximately 195 to −287 .

196
Let the load impedance Z L be expressed in the complex 197 form where R L > 0.

200
The negative impedance characteristic of tunnel diodes, 201 i.e., −R L < 0, can be exploited to design tags with reflec-202 tion amplifiers. When a tunnel diode amplifier is properly 203 matched to the antenna impedance, we have the following 204 property: Therefore, the amplitude of the reflection coefficient | | is 211 greater than one, indicating that the tag can amplify the 212 impinging RF signal.

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The reflection coefficient | | is not necessarily fixed and Equation (11) suggests three cases for the reflection 241 gain G: 242 1) The backscattered signal will be amplified when the 243 input power P in is in the range of [P l , P u ]. We name 244 this range the amplification input range.

245
2) When the input power P in is less than P l , the 246 input signal is too weak to trigger a strong injec-247 tion locking. In such a case, the tag cannot work 248 correctly.

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3) When P in is greater than P u , no amplification is 250 achieved because the mismatch of Z A and Z L results 251 in a reflection coefficient whose amplitude is less than 252 one. The tag may then perform even worse than the 253 traditional tag. To avoid this, the authors of [15] pro-254 pose a switchover mechanism so that the tag can sense 255 the received signal power P in and switch its operating 256 mode from the amplification mode to the standard 257 backscatter mode when P in ≥ P u . Using this mecha-258 nism, we can fix the reflection gain G as a constant η 259 less than or equal to 0 dB.

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Remark 1: The parameters a, b, c, f , P l , and P u in (11) 261 are determined by the specific circuit designs of a tag. The 262 values of these parameters can be determined according to 263 the measured data of the tag.

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Remark 2: A tunneling tag may not be activated if the 265 input power P in is less than its activation threshold. To ensure 266 this does not happen, the tag cannot limit itself to only 267 harvesting energy from the RF source. It may also harvest 268 energy from different sources, such as light, vibrations, etc. 269 If the energy density of the environment is sufficient, the 270 tag charges faster than it consumes power and can operate 271 for several minutes continuously. Thus, ensuring the tag's 272 energy availability makes it possible to connect to an IoT 273 network readily. The potential IoT applications include smart 274 agriculture, smart home, and RFID systems. Overall, the tag 275 can eliminate the need for power-hungry RF emitters and 276 improve communication efficiency.

278
In this section, we analyze the capacity of the AmBC system 279 with a tunneling tag over the binary input and binary output 280 (BIBO) channel. The input is the ideal OOK state of the tag. 281 The output is the detected symbols from the energy detector 282 of the reader.

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A. CAPACITY OF BIBO CHANNEL 284 We assume s(n) ∼ CN (0, 1). The received signal at the 285 reader, (6), is distributed as We define the binary input distribution as Let P j|i denote the conditional probability of output j given 302 input i. The conditional probability P 0|i of the output 0 given 303 the input i can be computed as [7]: where N is the ratio of the data rates of the RF source to the 306 tag, T h is the detection threshold defined as where the the binary entropy function is defined as 318 See Appendix A for the proof. 327 Therefore, we aim to derive the optimal d * st that maximizes 328 C BIBO .
At the same 332 time, it also need meets the condition d st < α P s |h st | 2 /P l 333 to ensure the power P in is greater than the threshold P l thus 334 the tag can backscatter signals. Therefore, we define d min 335 |d tr − d sr |, and d max min d tr + d sr , α P s |h st | 2 /P l .

336
Clearly, as the distance d st varies, the input power P in may 337 fall into one of the following two conditions: (1) P in > P u ; 338 (2) P l < P in ≤ P u . Therefore, we transform the original 339 optimization problem (20) into two optimization problems 340 for different distance intervals, i.e., the interval I 1 without 341 reflection gain, and the interval I 2 with reflection gain. These 342 two intervals can be represented as  where d amp min α P s |h st | 2 /P u is the minimum distance that 346 the tag has a reflection gain. We find the maximum values 347 in the two intervals, respectively, and then select the distance 348 which maximizes the channel capacity.

349
In the interval I 1 , the input power P in is strong and there is 350 no amplification gain. Obviously, the optimal distance in this 351 range is d min .

352
Whereas in the interval I 2 , the capacity varies according to 353 the reflection gain. The derivative of (17) is According to (14), the derivative of P 0|i with respective to 362 d st is computed as      of the animals. We should note that the tag does not consis-408 tently achieve the optimal distance. A suboptimal distance is , 2, 3, 4}. 14: if t k > 2, then 15: if z k 4 − z k 1 < , then 16:     reflection gains according to [19]. Table 2 shows the fitted 419 parameters, and Fig. 3 shows the fitted curves.       power is close to σ 2 1 or σ 2 0 . Therefore, for the system whose 466 value of σ 2 1 is close to that of σ 2 0 , like system 2, system 3, 467 and the traditional system, the capacity changes. Whereas 468 system 1 achieves the best performance because its gain is 469 large enough, the phase shift makes no difference. 470 We next investigate the optimal distance d * st and the cor-  Fig. 7 (a) shows the optimal distance d * st versus the RF 474 power P s . It shows that in the traditional system, the optimal 475 placement of the tag is close to the RF source, while that 476 of the TD-assisted system depends on the RF power level. 477 Of course, the optimal distance of the tunneling tag is much 478 larger than that of the traditional system. Fig. 7 (b) shows 479 the optimal capacities versus the transmit SNR ρ. This fig-480 ure shows that the curves gradually reach 0.05 (bps/Hz) as 481 the transmit SNR ρ increases. The capacities of all three 482 TD-assisted AmBC systems show a precipitous drop beyond 483 a certain distance. This drop occurs because of interference 484 from the RF source to the reader. Fig. 8 shows the searched 485 distance d st and the corresponding capacity for given d st . The 486 stop length is set as = 0.01. Although the searched distance 487 may differ from the optimal distance in Fig. 7 (a), the capacity 488 performance is very close, demonstrating the effectiveness of 489 the search method.

491
This paper investigated the channel capacity of tunnel diode 492 amplifier-assisted AmBC systems. Specifically, we formu-493 lated the system model and developed the mathematical 494 model for the reflection gain of the tag. We then derived 495 the closed-form channel-capacity expression. Finally, we pro-496 vided simulation results to corroborate the theoretical deriva-497 tions. They show that using the tunnel diode amplifier in 498 AmBC can significantly improve the channel capacity and 499 increase the coverage range. Another way to maximize the 500 channel capacity is to optimize the distance between the RF 501 source and the tag. Our study sheds light on this matter 502 too. We also find that the system performance depends on 503 the amplification range and the peak reflection gain of the 504 tunnel diode amplifier to a significant degree. Consequently, 505 to maximize the system gain from its use, the performance of 506 the tunnel diode amplifier should be optimized in terms of its 507 operating point and other factors. where I (x = 0;x) is the mutual information for x = 0 aver-524 aged over the output, and E is a scalar that is greater than 0.

525
Utilizing the Bayes Rule, the mutual information can be 526 express as