Thresholds for the Identification of Wireless SAW RFID-Tags with ASK

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1 Thresholds for the Identification of Wireless SAW RFID-Tags with ASK Gustavo Cerda-Villafaña 1, Yuriy S. Shmaliy 2 Electronics Department, Guanajuato University, Salamanca, 36885, Mexico 1 gcerdav@salamanca.ugto.mx 2 shmaliy@salamanca.ugto.mx Abstract The generic identification (ID) error probability and optimum thresholds are found employing the Marcum Q-function of first order for passive wireless surface acoustic wave (SAW) IDtags with binary amplitude shift keying (BASK). As examples, we find optimal thresholds for the 28-slot pulse-burst and the Barker code-based BASK. It is shown that, under the ideal conditions of equal SNRs of 30 db in On-pulses and zeroth in Off-pulses, the identification error probability lesser 1% (one slip per 100 readings) can be provided employing ASK in time position encoding having any reasonable number of slots. The error lesser 0.1% (1 slip per 1000 readings) can be achieved with seven or lesser slots. Otherwise, ASK needs to be combined with other kinds of encoding. I. INTRODUCTION In wireless sensor networks and smart systems, the identification (ID) of individual sensors has become an integral part of the design. The currency has also gained a wireless radio frequency (RF) ID tag (RFID-tag) in which the inherent sensor function is removed. In a family of such devices a special place occupy passive surface acoustic wave (SAW) coded sensors and delay-line identifiers, called RFID-tags, operating in the gigahertz frequency range. Their basic component is a piezoelectric plate (quartz, lithium-niobate, etc.) with the interdigital transducer (IDT) and a series of code reflectors. The latter are combined with one or several strips and placed at the precisely determined positions. Principles of encoding implemented in modern SAW RFID-tags as outlined in [1]. The time position encoding is most widely used in commercial SAW tags [1]. Here, the total time delay is divided into slots of certain duration. The slots are united into groups of several slots with one slot empty between the groups. In early designs [2], a single group of up to 32 slots was used to organize binary coding. In [3], [4], four slots per group were exploited and investigated with the same aim. In [5], [6], decimal groups have been used. In each of these designs, every reflected pulse can have an individual amplitude, frequency, and phase. Therefore, different methods of encoding and decoding [1], [7] [9] have been developed. A simplest way to identify the SAW RFID tag is to consider the received pulse-burst as an amplitude shift-keying (ASK) signal [10]. Fig. 1 sketches the operation principle of the SAW RFID-tags with binary ASK (BASK) and we notice that the envelope response picture qualitatively remains the same for Normalized envelope at the receiver detector Interrogator (a) Transmitted request signal Time (c) (c) (e) Antenna Identification threshold ITD Echo traces Received request signal Reflectors (b) Environmental echoes Time (d) Time 10-6 s (e) Fig. 1. Operation principle of identification of wireless SAW IDtags: (a) interrogator, (b) SAW ID-tag label coded with 28 bits: , (c) transmitted request signal, (d) received request signal with environmental echoes [7], and (e) response of the ID-tag at the receiver detector [2]. other methods of time position encoding. The difference exists in the number of pulses and empty spaces. The interrogator (a) transmits a radio frequency (RF) impulse (c) toward a tag. The RF impulse is received by the antenna of the IDtag (b) and converted by the interdigital transducer (ITD) to the SAW. About a half of its energy is then distributed toward the reflectors. Owing to the environmental echoes and some other factors, received by a tag the RF impulse (d) is accompanied with a number of additional impulses attenuated and propagated with different delays. It is supposed that each of the reflectors redirects the attenuated SAW (6 db/µs delay time [2]) toward the IDT and that the latter reconverts the SAW to a single RF pulse and retransmits it to the interroga /10/$ IEEE 985

2 tor. In practice, the read out impulse and the retransmitted pulse are also accompanied with the environmental echoes. Moreover, the pulse-burst at the receiver detector is typically composed with the ON-bits (On-pulses) having different peakmagnitudes, mostly due to design problems. In turn, instead of the zero OFF-bits (Off-pulses), there can appear echo traces in the presence of noise characterized in [6]. A typical received 28-bits pulse-burst measured in [2] is shown in Fig. 1e. It has to be remarked now that a rigorous statistical analysis of errors has been provided in the literature only for passive SAW sensors with phase measurement [11] [14]. Regarding the SAW tags, we meet only a few works [6], [15], [16]. The only work [16] discusses the identification error probability. It is therefore still unclear if simple and low-cost method of identification employing ASK is too rough and other methods of encoding must be used. In this paper, we consider the identification error probability of the SAW RFID-tags with BASK and find optimum thresholds in the sense of the minimum error probability. The results are obtained in the form suitable for any design employing the time position encoding. II. SIGNAL MODEL AND PROBLEM FORMULATION Suppose that the reader interrogates the tag with a linear frequency modulated (LFM) RF impulse request signal [6] (Fig. 1c) x(t) = ) 2Sa(t) cos (2πf 0 t + αt2 2 + θ 0, (1) where 2S and θ 0 are the peak-power and initial phase, respectively, f 0 is the initial carrier frequency, and t is the current time. The LFM pulse has a near rectangular normalized waveform a(t) of duration T such that α = ω/t, where ω is a required angular frequency deviation [10], overlapping all the tag responses. At the IDT, the request signal appears with the environmental echoes (Fig. 1b), although noise does not perturb it here substantially. Owing to various factors, the reflected On-pulses have different peak magnitudes and Off-pulses are not always zeroth at the receiver. Assuming Gaussian envelope in each of the slot pulses, the received K-bit pulse-burst with a time-step τ (Fig. 1e) can thus be modeled as s(t) = K s k (t) k=1 = 2S (2a) K β k e b2 (t t d kτ) 2 cos[2πf k t + ϑ k (t)], k=1 (2b) where β k is the coefficient affected by the design problems, attenuation, and echoes 1. Ideally, it is implied that β k = 1 for all On-pulses and β k = 0 for all Off-pulses. Here, 2Sβ k, f k, and ϑ k (t) are the peak-magnitude, frequency, and phase 1 Effect of β k is similar to fading in wireless communication channels. of the kth reflected pulse at the receiver and t d is the time delay caused by RF signal and SAW propagations. At the receiver, each of the RF pulses s k in (2a) is contaminated by the zero-mean additive stationary narrowband Gaussian noise n(t) with the known variance σ 2, so that we have a mixture y k (t) = s k (t) + n(t) (3a) = V k (t) cos[2πf k t + θ k (t)], (3b) in which V k (t) 0 is the positive valued envelope representing either the On-pulse or the Off-pulse in the K-bit burst and θ k is the modulo 2π random phase that does not play any substantial role in coding with ASK, unlike the PSK case [10], and will further be omitted. The received pulse-burst can thus be modeled as K y(t) = y k (t), (4) k=1 where y k (t) is specified with (3b). Following Rice [17], the instantaneous envelope V k in (3b) is distributed at t = t d + τ k with the probability density function 2 (pdf) p(z k γ k ) = 2z k e z2 k γ k I 0 (2 γ k z k ), (5) where the normalized envelope is z k = V k σ 2, (6) the instantaneous SNR in the kth pulse is calculated by and γ k = Sβ2 k σ 2, (7) I 0 (x) = 1 π e x cos ϕ dϕ (8) 2π π is the modified Bessel function of the first kind and zeroth order. The measurement (Fig. 1e) suggests that the pdf (5) is conditional on the given variable γ k playing a substantial role in choosing the identification threshold depicted in Fig. 1e. The problem now formulates as follows. Allowing z k and γ k to be random with supposedly known distributions, we would like to find an optimal threshold for the K-bit pulse-burst (4) with BASK in order to provide the minimum error probability of the SAW ID-tag identification. III. OPTIMAL THRESHOLD While identifying the coded pulse-bursts with BASK, a threshold can logically be located equidistantly between the sets of On- and Off-pulses. Owing to noise, such a strategy does not allow for a minimum identification error, although it gives a reliable effect when the SNR is large. Otherwise, a threshold must be found at the point where the identification error probability reaches a minimum. 2 Rice derived (5) for the constant magnitude of a signal. Shmaliy proved in [18], [19] that the Rice pdf (5) is also valid for time-varying magnitudes typical for SAW ID-tags. 986

3 A. Identification Error Probability Let us specify the identification error probability for the SAW ID-tag with BASK having K-bits. It is known [16] that the location of On- and Off-pulses does not play any role in the determination of the error probability with BASK. Therefore, we first denote M On-pulses, I 1, I 2,..., I M, formed by the reflectors and then K M Off-pulses, O M+1, O M+2,..., O K, representing zeros. For instance, Fig. 1e sketches the 28-bits label, K = 28, composed with 14 On-pulses, M = 14, and 14 Off-pulses, K M = 14. If we now introduce a separating (detection) threshold between the On- and Off-pulses, we then can denote the probability of each of the On-pulses as P I1 (), P I2 (),..., P IM () and the relevant error probabilities as P I1 () = 1 P I1 (), P I2 () = 1 P I2 (),... P IM () = 1 P IM (). For the Off-pulses, we respectively have P O(M+1) (), P O(M+2) (),..., P OK () and P O(M+1) () = 1 P O(M+1) (), P O(M+2) () = 1 P O(M+2) (),..., P OK () = 1 P OK (). The tag cannot be identified correctly if at least one of the On- or Off-pulses fails. Because all of the pulses appear independently, the identification error probability P E () can thus be specified by the probability of the mutually exclusive failures in each of the pulses. Following [20], [21], P E () for the independent pulses can hence be written as P E () = P I1+I2+ +IM+O(M+1)+O(M+2)+ +OK () = 1 P I1 I2...IM O(M+1) O(M+2)...OK () = 1 P I1 ()... P IM ()P O(M+1) ()... P OK () = 1 P Ii () P Oj () = 1 [1 P Ii ()] [1 P Oj ()]. (9) In fact, if any of P Ii () and P Oj () becomes unity (one of the pulses fails), then the whole error probability P E () also becomes unity (the tag fails). Otherwise, if all of P Ii () and P Oj () occur to be zeroth (an ideal case), P E () also reaches zero and the identification is provided precisely. Given (5), the error probability in the ith On-pulse and jth Off-pulse can be calculated as, respectively, P Ii ( ˆγ i ) = 1 P Oj ( ˇγ j ) = p(x ˆγ i )dx, (10) p(x ˇγ j )dx. (11) Referring to (9), (10) and (11), the generic conditional error probability of the SAW RFID-tag identification becomes = 1 = 1 + P E ( ˆγ 1... ˆγ M ˇγ M+1... ˇγ K ) p(x ˆγ i )dx 1 p(x ˆγ i )dx p(x ˆγ i )dx. p(x ˇγ j ) dx p(x ˇγ j )dx (12a) (12b) If to substitute (5) to (12b), P E ( ˆγ 1... ˆγ M ˇγ M+1... ˇγ K ) will attain its most compact form of where = 1 + P E ( ˆγ 1... ˆγ M ˇγ M+1... ˇγ K ) Q( 2ˆγ i, 2) Q( 2ˇγ j, 2) Q( 2ˆγ i, 2), (13) Q(a, b) = b xe x2 +a 2 2 I 0 (ax)dx (14) is the generalized Marcum Q-function of first order commonly used in radar signal detection [22]. 1) Equal SNRs in On- and Off-pulses: A particular situation may occur when the SNRs in the On-pulses and Off-pulses are near equal; that is, ˆΥ = ˆγ i for all i and ˇΥ = ˇγ j for all j. One can also substitute ˆγ i and ˇγ j with their average values ˆΥ = 1 M ˇΥ = M ˆγ i, (15) 1 K M K M j=1 ˇγ j (16) in order to find P E approximately. In both these cases, (13) attains the form of P E ( ˆΥ ˇΥ) = 1 + Q M ( 2 ˆΥ, 2)Q K M ( 2 ˇΥ, 2) Q M ( 2 ˆΥ, 2). (17) It is known that the Q-function cannot be expressed in simple functions and it is usually recommended to use asymptotic formulas [22], [23]. We, however, still do not know exact criteria for the approximations in the SAW ID-tags and 987

4 p z 2 p 0 z 1 ˆ = 0 ˆ = 9 0 z 2 1 z 1 Identification error probability Optimal threshold Fig. 2. The identification error probability (dashed area) of the ID-tag with 2- bit BASK composed by the On-pulse ( 1 ) and Off-pulse ( 0 ) having ˆγ = 9 and ˇγ = 0, respectively. postpone these investigations to further studies preferring in this paper nonasymptotic numerical estimates. Now, to find an optimum value of, the error (13) or (17) needs to be minimized. B. Optimal Threshold The optimal threshold can be found if we minimize (13) by equating to zero the gradient of P E ( ˆγ 1... ˆγ M ˇγ M+1... ˇγ K ) with respect to. That means solving the equation d d [ M Q( 2ˆγ i, 2) = d d i=m+1 Q( 2ˇγ i, 2) Q( 2ˆγ i, 2) (18) with respect to =. 1) Equal SNRs in On- and Off-pulses: In the particular case represented with (17), the optimum threshold can be ascertained by solving the following equation d [ Q M ( 2 d ˆΥ, 2)Q K M ( 2 ˇΥ, ] 2) = d d QM ( 2 ˆΥ, 2). (19) Since exact analytical solutions of (18) and (19) are commonly not available for the Q-function (14) even in simple particular cases, numerical solutions are preferable. 2) Example: SAW ID-tag with 2-bit BASK: Given the SAW ID-tag with 2-bit BASK composed by the On-pulse ( 1 ) and Off-pulse ( 0 ) as shown in Fig. 2 for the pdf (5) with ˆγ = 9 and ˇγ = 0, respectively. The error probabilities in the On- and Off-pulses are defined in Fig. 2 by dashed areas. For the given threshold and above-specified constant values of ˆγ and ˇγ, the identification error probability is defined, by (13), as P E () = 1 + [Q(0, 2) 1]Q( 18, 2). (20) Figure 3 sketches (20) for several values of γ = ˆγ with ˇγ = 0. The left branch of each curve represents the error ] Fig. 3. The identification error probability of the ID-tag with 2-bit BASK (Fig. 2) for ˆγ = var and ˇγ = 0. The optimal threshold is represented with a dashed curve. TABLE I OPTIMUM THRESHOLDS FOR SAW ID-TAGS WITH 2-BIT BASK AND DIFFERENT SNRS ˆγ, units / ˆγ probability of the Off-pulse and the right one the On-pulse. The errors minima correspond to the optimal threshold represented in Fig. 3 with a dashed curve. Following (18) and utilizing (20), the optimal threshold function can be found by solving the equation d d [ Q(0, 2)Q( 18, 2) / ] = d d Q( 18, 2). (21) Table I gives us the optimal values for points depicted in Fig. 3 with circles and found numerically by solving (21). The identification threshold should not obligatorily be optimum, if P E is limited with some value. In fact, allowed P E = 1% for γ = 50, the region for is Fig. 4 sketches the relevant allowed regions as functions of the SNR, assuming P E = 5%, P E = 1%, and P E = 0.5%. IV. APPLICATIONS Based upon the above-given analysis, below we consider two applications for the identification error probability (13) and optimal threshold obtained by solving (18). A. Optimal Threshold for the 28-Bit SAW ID-Tag (Fig. 1e) Let us return to Fig. 1e and find an optimal threshold for the identification of the measured pulse-burst. The noise standard deviation in the absence of signal is determined here by the noise envelope level to be σ = Then the measured peak envelope V of the On- or Off-pulse allows computing the SNR with γ = V 2 2σ 2. (22) 988

5 TABLE II ESTIMATES OF THE SNRS IN THE ON-PULSES, ˆγ i, AND OFF-PULSES, ˇγ j, SHOWN IN FIG. 1E. i, j average ˆγ i ˇγ j Threshold P 5% 1% E Regions of the allowed thresholds 0.5% Optimal threshold Error probability K2 K3 K4 K5 K7 K11 K , units Fig. 4. Thresholds for the ID-tag with 2-bit BASK (Fig. 2) for γ 1 = var and γ 2 = 0. TABLE III BARKER CODES OF LENGTH K. Code Code elements M K K2 01 or 00 1 or 0 2 K K or K K K K Table II gives us the estimates of the SNRs, by (22), in the On-pulses as ˆγ i and in the Off-Pulses as ˇγ j. Knowing these values and letting M = 14 and K = 28, the optimal threshold is determined, by solving (18), to be = B. Optimal Thresholds for the Barker-Coded SAW ID-Tags The Barker codes listed in Table III [10] are often employed in the interrogating radar systems to obtain the most narrow correlation function, thereby providing a reliable identification. We consider below the most typical case of γ = 30 db and σ = assuming all Off-pulses zeroth and all On-pulses unities [16]. Basically, (13) gives us the error probability for any code, if we take M and K from Table III. Numerically calculated, this error is exhibited in Fig. 5 in the threshold region of 0 1, allowing for the following generalizations: The optimal threshold ranges at about 0.55, although it varies for different codes Threshold Fig. 5. Error probability of the identification of the SAW ID-tags with the Barker codes listed in Table III for γ = 30 db and σ = Beyond, the error probability rises dramatically with the slope of about two decades per unity. Fixed the error probability, the region of the allowed thresholds is narrowed, similarly to Fig. 4, by increasing the number of pulses in the coded BASK burst. If we allow the error probability to be 1% using a simplest Barker code K2, then the region for the allowed thresholds will be from 0.39 to In turn, for the longest code K13, it is narrowed to 0.48 and 0.65, respectively. For the error probability of 0.1%, the allowed threshold region narrows substantially. In fact, it can be shown that the codes K2, K3, K4, and K5 require 0.48 < < 0.62, 0.50 < < 0.61, 0.50 < < 0.62, and 0.51 < < 0.63, respectively. It also follows from Fig. 5 that the SAW IDtags coded with K > 13 cannot be identified with the error probability of 0.1%, if these tags are interrogated with γ = 30 db. For these codes, the optimal threshold must be taken exactly from the minima of the error probabilities or calculated by (15). For engineering applications, the diagram shown in Fig. 4 seems to be the most useful. We provide such a diagram for the Barker code K5 in Fig. 6 for several error probabilities. Because each of the functions is characterized with the crosspoint ( γ, ) and the allowed region for the thresholds when P E and γ are fixed, we also give in Table IV the coordinates of the points (circled in Fig. 6) for another Barker codes. Observing Table IV and Fig. 5, one arrives at the following most common conclusion: the code length K substantially narrows the region for the allowed thresholds, although the 989

6 TABLE IV OPTIMAL AND ALLOWED THRESHOLDS FOR THE BARKER CODES OF LENGTH K WITH P E = 0.1%. Optimal threshold Code γ, db for γ = 30 db K K K K K K K SNR, db Fig. 6. Optimal and allowed thresholds for the Barker code K5 with different error probabilities (in %) as a function of the SNR. average optimal threshold still remains at the level of V. CONCLUDING REMARKS An overall conclusion is the following. Under the ideal conditions when all On-pulses have equal SNRs of 30 db and all Off-pulses zeroth SNRs, the identification error probability lesser 1% (one slip per 100 readings) can be provided employing ASK in time position encoding having any reasonable number of slots. The error lesser 0.1% (1 slip per 1000 readings) can be provided with no more than seven slots. Otherwise, ASK must be combined with encoding in phase or frequency as observed, for example, in [1]. 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