Code reading error probability estimation for SAW tag systems with pulse position coding V. Plessky and Y.S. Shmaliy The code reading error probability is estimated in SAW tag systems employing peak-pulse detection with pulse position coding assuming M groups having N slots each. The error is found as a function of the signal-to-noise ratio, proving an extremely reliable deciphering of the tag code for SNR . 20 dB, meaning less than 1 false per billion of reading.
Introduction: The truly passive wireless SAW tags can be read at a distance of more than 10 m with the reader radiating , 10 mW of the electromagnetic power [1, 2]. The SAW tags are cheap, robust, can work in harsh environments such as high temperatures or high levels of ionising radiation, and demand no maintenance. In many applications the record reading distance is not demanded, while the low rate of the reading errors is of primary importance. A probabilistic analysis of such errors to exceed a threshold has recently been provided in [3] for the SAW tag systems with pulse position coding (PPC). In modern designs [2], the PPC is implemented to have M groups with one reflector per group with N slots in each of the groups. For such a design, the identification of a unique pulse in each of the groups is obtained by the peak-pulse detection that requires a new formula for the code reading error probability.
with ‘0’. Supposing that noise induced by the receiver is white 1, C 2 · · · , C N −1 are independent and Gaussian and thus the events C uncorrelated, we can write this error as: 1, C 2 · · · , C N−1 |z)]P(z|g) P˜ Em (z, g) = P(C˜ m |z)P(z|g) = [1 − P(C (2) N −1 n |z) P(z|g) = 1− P(C n=1
On the other hand, the probability of normal functioning of the nth slot n |z) = 1 − P(Cn |z), where P(Cn |z) is the failure probwith ‘0’ is P(C ability in the nth slot. If the probability density function (pdf) p(z|g) of z is known and g is given, then (2) can finally be rewritten as z+Dz N −1 ˜PEm (z, g) = 1 − [1 − P(Cn |z)] p(z|g)dz n=1
z
N −1 [1 − P(Cn |z)] p(z|g)Dz 1−
(3)
n=1
where Dz is assumed to be a small value. So long as z occurs at random with different probabilities, it is to get rid of z. By integrating over z, we thus arrive at the conditional error probability in the mth group: 1
PEm (g) =
1−
N−1
[1 − P(Cn |z)] p(z|g)dz
(4)
n=1 0
Coding algorithm: Let us consider the principle of time position coding implemented in most of the commercially available SAW tag systems [2]. In the time domain, equal slots comparable to 1/B (B is the used frequency band) are allocated for responses. Only one response is allowed in a group of slots. For example, we will consider the groups of 16 slots, one response in each group provides 4 bits of codes. With 6 groups, the tag can have 24 bits of (about 1.7 × 106) different codes. The groups can be separated by additional ‘guard’ slots which are never occupied by any response. We shall suppose the following: † All responses are independent and their peak-envelopes are equal. † The response has no sidelobes and does not induce any signal to the neighbouring slots. † The reader is automatically calibrated to avoid temperature shifts and some other positioning errors, so that the responses are supposed to be centred inside the slots. † The transducer of the SAW tag is relatively narrowband, even in the presence of noise it allows for a smooth envelope of the response. Error probability: Generally, we assume that the tag has M groups with N slots in each. Each group has one slot with ‘1’ and the remaining slots have ‘0’. It is supposed that the signal has power 2S and the narrowband Gaussian noise has the variance s2 . Accordingly, the signal-to-noise 2 ratio (SNR) is specified with √ g = S/s and the normalised signal envelope V with z = V / 2s2 . We would like to find the probability that at least in one group and at least in one slot with ‘0’ the noise envelope exceeds the peak-value in the slot with ‘1’. We shall call such an error the code reading error probability or just the error probability. We shall suppose equal SNRs in each of the slots with ‘1’ and zero SNR in all of the slots with ‘0’. To derive the error probability, let us start with the mth group of slots. Since the error probability does not depend on the positioning of ‘1’, we suppose that N-1 first slots have zeros and the last Nth slot has ‘1’. Normal functioning of such a group is implied when the envelopes in all of the slots with ‘0’ do not exceed the signal peak-value z in the slot with ‘1’. Then define the evens of the normal functioning of the 1, C 2 · · · , C N −1 . The probability of the simultaneous slots with ‘0’ as C normal functioning of all these slots under the condition that z is given is 2 · · · , C N −1 |z). Then the failure probability when 1, C specified with P(C at least one of the slots with ‘0’ falsies (an event C˜ m ) can be defined by: 1, C 2 · · · , C N −1 |z) P(C˜ m |z) = 1 − P(C
(1)
With the given SNR g, the probability that the peak-value of a signal in the slot with ‘1’ falls within the interval z + Dz can be specified as P(z|g), and then the error probability in the mth group defined by the simultaneous occurring of a signal in z + Dz and a failure in the slots
Since signals in the wireless SAW tag system are essentially narrowband processes [4], p(z|g) can be defined by the conditional Rice distribution and P(Cn |z) by integrating the conditional Rayleigh pdf, respectively √ I0 (2z g)
(5)
2xe−x dx = e−z
(6)
p(z, g) = 2ze−z
2
1
P(Cn |z) =
−g
2
2
z
where I0 (x) is a modified Bessel function of the first kind and zeroth order. Referring to (5) and (6), the probability (4) finally becomes: 1 2 2 √ PEm (g) = 2e−g [1 − (1 − e−z )N −1 ] ze−z I0 (2z g)dz
(7)
0
m , m = 1, . . . , M , is the Now consider the whole tag structure. Let A event that the mth group operates normally. Then the probability that all of the groups are simultaneously in normal operation can be 2 · · · A M ). Since P(A m ) = 1 − P(Am ), where Am rep1A defined as P(A resents a false work, the code reading error probability can finally be specified for mutually independent events A1 , A2 , · · · , AM with 2 · · · A 1A M ) = 1 − PE (g) = 1 − P(A
=1−
⎧ ⎨ ⎩
M
[1 − PEm (g)]
m=1
1 − 2e
−g
1
[1 − (1 − e
−z2 N −1
)
−z2
] ze
⎫M √ ⎬ I0 (2z g)dz ⎭
0
(8) Numerical example: The lowest error probability is naturally obtained in such kind of tags when M ¼ 1 and N ¼ 2. For this special case, (8) simplifies to: 1 2 √ PE (g) = 2e−g ze−2z I0 (2z g)dz
(9)
0
One of the recently designed SAW tags has been manufactured to have 6 groups (M ¼ 6) with 16 slots (N ¼ 16) (one slot with ‘1’ and 15 slots with ‘0’). Fig. 1 shows the code reading error probability against the SNR g in the logarithmic measure 10 log(g), dB. As can be seen, the error of 0.1% is achieved with the SNR exceeding 11 dB in a limiting case of M ¼ 1 and N ¼ 2, provided (9). In a real case of M ¼ 6 and N ¼ 16, the same error is achieved with the SNR more than 14 dB.
ELECTRONICS LETTERS 14th October 2010 Vol. 46 No. 21
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would hide the first error so that the reader would declare the false reading as true is many orders smaller and can be estimated as about square of the probability of one error. This result proves an extremely reliable deciphering of the tag code for SNR . 20 dB, meaning less than 1 false per billion of reading.
code reading error probability
10–1 M = 6, N = 16 10
–2
M = 1, N = 2
# The Institution of Engineering and Technology 2010 10 August 2010 doi: 10.1049/el.2010.2207
10–3
10–4
10
V. Plessky (GVR Trade SA, Ch. du Vignoble 31C, 2022 Bevaix, Switzerland)
–5
E-mail:
[email protected] Y.S. Shmaliy (Department of Electronics, Guanajuato University, Salamanca 36855, Mexico)
10–6 0
2
4
6
8
10
12
14
16
18
20
10log(g)
Fig. 1 Code reading error probability of SAW tag with PPC (M groups having N slots in each) employing peak-pulse detection
Conclusions: In this Letter, we proposed a formula for the code reading error probability estimation in the commercially available passive wireless SAW tag systems employing pulse position coding with M groups having N slots each and the peak-pulse detection. We showed that the limiting error does not exceed 0.1% with the SNR of about 11 dB for M ¼ 1 and N ¼ 2 and with the SNR more than 14 dB in real designs with M ¼ 6 and N ¼ 16. We note that an additional group of 16 slots is used in such designs as the check sum. The probability for the reader to have simultaneously one more error in the check sum, which
References 1 Plessky, V.P., Kondratiev, S.N., Nyffeler, F., and Stierlin, R.: ‘SAW-tags; new ideas’. IEEE Ultrasonics Symp., Seattle, WA, USA, 1995, pp. 117– 120 2 Plessky, V., and Reindl, L.: ‘Review on SAW RFID Tags’, IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 2010, 57, (3), pp. 654 –668 3 Cerda-Villafana, G., and Shmaliy, Y.S.: ‘Thresholds for the identification of wireless SAW RFID-tags with ASK’. Proc. 15th IEEE Mediterranian Electromechanical Conf. (MELECON-2010), Grand Hotel Excelsior, Valletta, Malta, 25– 28 April 2010, pp. 985–990 4 Shmaliy, Y.S.: ‘On the multivariate conditional probability density of a signal perturbed by Gaussian noise’, IEEE Trans. Inf. Theory, 2007, 53, (12), pp. 4792–4797
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