On the performance of MIMO RFID backscattering channels
- Chen He^{1}Email author,
- Xun Chen^{1},
- Zhen Jane Wang^{1} and
- Weifeng Su^{2}
DOI: 10.1186/1687-1499-2012-357
© He et al.; licensee Springer. 2012
Received: 25 August 2012
Accepted: 7 November 2012
Published: 29 November 2012
Abstract
In order to increase the reliability of data transmission, using multiple antennas in radio frequency identification (RFID) systems has been investigated by researchers, mainly through measurements and simulations. The multiple-input multiple-output (MIMO) RFID backscattering channel exhibits a special type of cascaded structure rather than that of other better-studied cascaded channels such as the keyhole fading and the double scattering fading. In this article, we analytically study the bit error rate performances of the MIMO RFID channel under two transmission schemes, the identical signaling transmission scheme and the orthogonal space–time coding scheme (OSTBC). We show that the diversity order of the MIMO RFID channel is min(N,L) under the identical transmission scheme, and the diversity order is L under the OSTBC scheme, where L is the number of tag antennas and N is the number of reader receiving antennas. A performance bottleneck is also observed in the MIMO RFID channel. Our results can provide useful guidance on designing an RFID system with multiple antennas.
Keywords
MIMO systems RFID Backscattering channel Cascaded structures STBC Diversity gainIntroduction
Radio frequency identification (RFID) is a wireless communication technology that allows an object to be identified remotely, which has many applications including inventory checking, access control, transport payment, electronic vehicle registration, product tracking, and secure automobile keys[1]. A typical RFID system includes major components such as readers (also known as interrogators) and tags (also known as labels), as well as RFID software or RFID middleware[2]. An RFID tag is a small electronic device which has a unique ID. The tags can be categorized into active and passive tags. An active tag has an RF transmitter and utilizes its internal battery to continuously power its RF communication circuitry, while a passive tag does not have an RF transmitter and it modulates a carrier signal received from an interrogator by its antenna load impedance. Usually, a passive tag does not have its own battery and powers its circuitry by using the carrier signal energy, but it can be battery assisted. For passive RF tags, the range increase caused by multiple RF tag antennas will be limited by the RF tag chip sensitivity[3]. The sensitivity is strongly dependent upon the design of the tag’s RF circuitry[4]. During the last couples of years, it was shown that the improved circuitry design[3, 5, 6] can decrease this limitation.
Most RFID applications deployed today use passive tags because they usually do not require internal batteries and have longer life expectancy. Measurements in[7, 8] showed that the RFID channel in passive systems could be modeled as a cascaded channel with a forward channel and a backscattering channel, and both the sub-channels are Rayleigh distributed in rich scattering environments[9]. This cascaded channel fades deeper than the Rayleigh channel and hence can reduce the data transmission reliability.
To allow reliable data transmission, researchers[7, 9–11] investigated multiple-input multiple-output (MIMO) settings in the RFID backscattering channel and both bit error rate (BER) and reading range improvements were observed. The advantages of the MIMO RFID technology over the single-antenna RFID allow many potential applications in the areas of accurate tracking, identifications, since it can significantly increase the reliability and throughput. For instance, in[12], a tracking problem was investigated for RF tags with multiple antennas and experimental results showed improved accuracy. To achieve good performance, a sufficient separation between tag antennas is needed and hence a very high-frequency band is required for operations of RF tags with multiple antennas due to the small size of an RF tag. Fortunately, the unlicensed frequency band 5.8 GHz is available for backscatter radio applications. This above ultra-high frequency band has several potential advantages for backscatter radio systems such as increased antenna gain, reduced object attachment losses[13]. It has already been used in a passive backscatter radio system for monitoring[14]. To the best of the authors’ knowledge, the current studies about the performances of the MIMO RFID channel were mostly based on the measurements and/or Monte Carlo simulations. The main purpose of this study is to provide a fundamental, analytical study of the behavior of the MIMO RFID channel, and to provide useful guidances on the design of potential RFID systems with multiple antennas.
Channel model
Passive tag signaling model
where Δ_{ l }(t) means the load reflection coefficient of the l th tag antenna at time t. In this case, the RF tag antennas have different load reflection coefficients. A diagonal signaling matrix with unequal load reflection coefficients may result from space–time-coded tag circuit designs, where the reflection coefficients of different tag antennas are pre-designed according to a certain space–time code.
where$\left(\right)close="">{h}_{l}^{f}$’s (l = 1,…,L) represent forward channels of the N × L channel,$\left(\right)close="">{h}_{l,n}^{b}$’s (l = 1,…,L, n = 1,…,N) represent backscattering channels, and$\left(\right)close="">{h}_{l}^{f}$’s and$\left(\right)close="">{h}_{l,n}^{b}$’s are independent complex Gaussian random variables. The total channel gain at the n th receiving antenna is$\left(\right)close="">{h}_{n}=\sum _{l=1}^{L}{h}_{l}^{f}{h}_{l,n}^{b}$. Note that the channel gains h_{ n }’s, for n = 1,2,…,N, contain the common terms$\left(\right)close="">{h}_{1}^{f},\dots ,{h}_{L}^{f}$. It implies that the channel gains at different receiving branches at the reader are not statistically independent with each other.
Difference between the RFID channel and Other types of cascaded channels
The cascaded Rayleigh channel is also found in the propagation scenarios such as the keyhole propagation[15–17], the double scattering propagation[18] and the amplify-and-forward (AF) relay cooperative communication systems. As shown in Figure2 we would like to emphasize that the MIMO RFID channel exhibits a different type of cascaded structure rather than that of the keyhole, double scattering, and AF channels due to the modulation and coding that are done at the tag side (middle way of a cascaded channel) rather than the reader transmission side.
It is clear that here the rank of the above matrix H_{keyhole} is 1. Comparing with the RFID channel matrix in Equation (5) which is generally full rank, we clearly note that the two channels are different in structure, although for both channels each entry of the matrices is double Rayleigh distributed. Their different channel statistics lead to different performances and diversity gains. The structure of the MIMO RFID channel is also different from that of the double scattering channel in[18], and we will see later in the result part that their performances and diversity gains are different.
Another type of cascaded channel is the AF cooperative channel. For only identical signaling scheme, theoretically it may be possible for an AF model to be reduced to the MIMO RFID model if the AF channel contains L relays and has N receiving antennas. A recently published work in[17] discusses the performance of an AF channel with such antenna setting, which is the most similar model to the MIMO RFID channel by far. However, we cannot generalize their results to infer the performance of the MIMO RFID channel. Because in[17], the model is based on a specific transmission scheme in which the relays intentionally forward messages to the destination at different time slots, and the forwarded messages are buffered at the destination, matched filtered, and then combined using maximal ratio combining (MRC). In the MIMO RFID channel, whatever the signaling schemes are used, the signals from the tag antennas (analogy to the relays) arrive at the reader (analogy to the destination) at the same time slot, and this makes the two models significantly different. We will also see differences of these two models from their performances and diversity orders. Other existing references about AF channels either do not consider multiple relays or do not consider multiple receiving antennas and thus cannot be a generalized version of the MIMO RFID channel. Some previous articles consider a single relay with multiple antennas and the model looks similar. However, the model is indeed different (e.g., the relay uses different antennas for transmitting and receiving).
The intention of this study is to analytically derive the performances of the MIMO RFID channel, a specific type of cascaded channel which exists in passive MIMO RFID systems.
Performance analysis of MIMO RFID channel under identical signaling scheme
In this section, we analytically study BER performances and diversity gains of the N × L RFID channel when the identical signaling scheme is employed.
Here α_{ l } is the squared magnitude of the channel gain of the l th receiving branch, N and L are the index of the function$\left(\right)close="">{G}_{N,L}\left(\stackrel{\u0304}{\stackrel{\u0304}{\gamma}}\right)$, and we define$\left(\right)close="">\stackrel{\u0304}{\stackrel{\u0304}{\gamma}}=\frac{g\stackrel{\u0304}{\gamma}}{{sin}^{2}\theta}$, where$\left(\right)close="">\stackrel{\u0304}{\gamma}$ is the average SNR and g is a constant which is modulation dependent. For the coherent transmission case, the function$\left(\right)close="">{G}_{N,L}\left(\stackrel{\u0304}{\stackrel{\u0304}{\gamma}}\right)$ is the moment generating function (MGF) of the MIMO RFID channel with L tag antennas and N receiving antennas. For the non-coherent transmission case, the form of G_{N,L}(·) is required in deriving the BER performance. The function G_{N,L}(·) defined in Equation (10) has the following recursive and asymptotic properties
Proposition 1
Proposition 2
Proposition 3
Proposition 4
In the above propositions, E_{ N }(·) and E_{ L }(·) are the exponential integrals defined as$\left(\right)close="">{E}_{N}(x)={\int}_{t=1}^{\infty}\frac{exp(-\mathit{\text{tx}})}{{t}^{N}}\mathit{\text{dx}}$ and$\left(\right)close="">{E}_{L}(x)={\int}_{t=1}^{\infty}\frac{exp(-\mathit{\text{tx}})}{{t}^{L}}\mathit{\text{dx}}$[22], where N and L are positive integers. The proofs of these propositions can be found in Appendix. With the above properties, we are now ready to derive the exact and asymptotic BER performances and study how the MIMO RFID backscattering channel behaves.
Non-coherent Case
We first investigate the non-coherent transmission case, which applies non-coherent equal gain combining (EGC) at the reader receiver side.
Non-coherent case of the identical signaling scheme: closed-form BER expressions for the N × L RFID channel (Equation 17)
L= 1 | L= 2 | |
---|---|---|
N = 1 | $\left(\right)close="">{e}^{\frac{1}{g\stackrel{\u0304}{\gamma}}}\frac{{E}_{1}(\frac{1}{g\stackrel{\u0304}{\gamma}})}{2g\stackrel{\u0304}{\gamma}}$ | $\left(\right)close="">{e}^{\frac{1}{g\stackrel{\u0304}{\gamma}}}\frac{{E}_{2}(\frac{1}{g\stackrel{\u0304}{\gamma}})}{2g\stackrel{\u0304}{\gamma}}$ |
N = 2 | $\left(\right)close="">\frac{2{e}^{\frac{1}{g\stackrel{\u0304}{\gamma}}}{E}_{2}(\frac{1}{g\stackrel{\u0304}{\gamma}})}{g\stackrel{\u0304}{\gamma}}+$ | $\left(\right)close="">\frac{2(-g\stackrel{\u0304}{\gamma}+{e}^{\frac{1}{g\stackrel{\u0304}{\gamma}}}{E}_{1}(\frac{1}{g\stackrel{\u0304}{\gamma}})+g\stackrel{\u0304}{\gamma})}{g\stackrel{\u0304}{\gamma}}+$ |
$\left(\right)close="">\frac{{(g\stackrel{\u0304}{\gamma})}^{2}-2g\stackrel{\u0304}{\gamma}{e}^{\frac{1}{g\stackrel{\u0304}{\gamma}}}{E}_{1}(\frac{1}{g\stackrel{\u0304}{\gamma}})+g\stackrel{\u0304}{\gamma}-{e}^{\frac{1}{g\stackrel{\u0304}{\gamma}}}{E}_{1}(\frac{1}{g\stackrel{\u0304}{\gamma}})}{4{\left(g\stackrel{\u0304}{\gamma}\right)}^{3}}$ | $\left(\right)close="">\frac{{e}^{\frac{1}{g\stackrel{\u0304}{\gamma}}}{E}_{1}(\frac{1}{g\stackrel{\u0304}{\gamma}})-3{(g\stackrel{\u0304}{\gamma})}^{2}+2{(g\stackrel{\u0304}{\gamma})}^{2}{e}^{\frac{1}{g\stackrel{\u0304}{\gamma}}}{E}_{1}(\frac{1}{g\stackrel{\u0304}{\gamma}})-g\stackrel{\u0304}{\gamma}+4g\stackrel{\u0304}{\gamma}{e}^{\frac{1}{g\stackrel{\u0304}{\gamma}}}{E}_{1}(\frac{1}{g\stackrel{\u0304}{\gamma}})}{4{\left(g\stackrel{\u0304}{\gamma}\right)}^{3}}$ |
It means that the asymptotic diversity order of the N × L RFID channel under non-coherent transmission schemes is determined by the smaller value of N and L. For the case of L = N, compared with the case of L ≠ N, it requires a higher SNR to achieve the diversity order N, because of the logarithm function in the numerator in Equation (18) when N = L. This property means that even the diversity orders are the same the BER performances of the settings with N = L + 1 or L = N + 1 are remarkably better than the performance of the setting with N = L. The BER performance improvements from N = L + 1 to N = L + 2, or from L = N + 1 to L = N + 2, is not significant. These observations generalize the findings about the MISO case in[25].
Coherent case
We now look at the coherent detection case. We assume that the reader knows the channel state information (CSI) and MRC is applied at the reader receiver side.
Coherent case of the identical signaling scheme: MGFs G_{ N , L }( θ ) for the N × L RFID channel (Equation 21)
L= 1 | L= 2 | |
---|---|---|
N = 1 | $\left(\right)close="">{e}^{\frac{{sin}^{2}\theta}{g\stackrel{\u0304}{\gamma}}}\frac{{E}_{1}(\frac{{sin}^{2}\theta}{g\stackrel{\u0304}{\gamma}})}{g\stackrel{\u0304}{\gamma}}$ | $\left(\right)close="">\frac{{e}^{\frac{{sin}^{2}\theta}{g\stackrel{\u0304}{\gamma}}}{E}_{2}(\frac{{sin}^{2}\theta}{g\stackrel{\u0304}{\gamma}}){sin}^{2}\theta}{g\stackrel{\u0304}{\gamma}}$ |
N = 2 | $\left(\right)close="">\frac{{e}^{\frac{{sin}^{2}\theta}{g\stackrel{\u0304}{\gamma}}}{E}_{2}(\frac{{sin}^{2}\theta}{g\stackrel{\u0304}{\gamma}}){sin}^{2}\theta}{g\stackrel{\u0304}{\gamma}}$ | $\left(\right)close="">\frac{-\stackrel{\u0304}{\gamma}{sin}^{4}\theta +{e}^{\frac{{sin}^{2}\theta}{g\stackrel{\u0304}{\gamma}}}{E}_{1}(\frac{{sin}^{2}\theta}{g\stackrel{\u0304}{\gamma}}){sin}^{6}\theta +\stackrel{\u0304}{\gamma}{e}^{\frac{{sin}^{2}\theta}{g\stackrel{\u0304}{\gamma}}}{E}_{1}(\frac{{sin}^{2}\theta}{g\stackrel{\u0304}{\gamma}}){sin}^{4}\theta}{{\left(g\stackrel{\u0304}{\gamma}\right)}^{3}}$ |
As we can see that the asymptotic diversity order is still min(N,L) in the coherent transmission case, and the BER behavior is similar to that of the non-coherent case.
Performance analysis of MIMO RFID channel under OSTBC scheme
In this section, we analytically study BER performances and diversity gains of the N × L RFID channel when the orthogonal space-time coding scheme (OSTBC) is employed. OSTBC is one of the most attractive MIMO schemes with a very simple decoding process based on linear combining at the receiver. Because of its orthogonality, OSTBC achieves full diversity LN for L transmission antennas and N receiving antennas in i.i.d. MIMO Rayleigh fading channels[26, 27]. In this section, we investigate the error rate performance of the MIMO RFID channel using the OSTBC scheme. For MIMO RFID systems, OSTBC can be implemented by applying the signaling matrix in Equation (3) for the tag antennas. We assume that CSI is known by the reader and the channel is quasi-static.
where R means the symbol rate and we define$\left(\right)close="">g=\frac{{\text{log}}_{2}K}{R}$.
Diversity order comparisons between different cascaded structures
Channel and coding scheme | Diversity order | Performance bottleneck |
---|---|---|
RFID (identical) | min(L,N) | |L−N| > 1 |
RFID (OSTBC) | L | N > 2 |
Keyhole channel (OSTBC) | min(L N)[16] | |L−N| > 1 |
Double scattering channel (OSTBC) | MLN/max(M L N)[18] | − |
Independent Double Rayleigh (identical) | N | L > 2 |
Independent Double Rayleigh (OSTBC) | NL[30] | No bottleneck |
Results and discussions
In this section, we compare the error rate performances of the MIMO RFID backscattering channel with other forms of cascaded channels: the keyhole channel, the double scattering channel, and the independent double Rayleigh channel. We also compare the identical signaling scheme and the OSTBC scheme in the MIMO RFID channel and discuss how much improvement can be achieved by the OSTBC scheme under different antennas settings.
The derivations of the above results are given in Appendix.
Diversity comparison and performance bottleneck
Under the identical signaling scheme, one interesting observation is that the diversity order of the studied MIMO RFID channel is min(N,L), as shown in Equation (24). Equation (23) implies that if N − L > 1, solely increasing the number of receiving antennas N does not enhance the BER performance significantly. Similarly, if L − N > 1, solely increasing the number of tag antennas L does not enhance the BER performance significantly either. Here, we refer to this observation as the performance bottleneck. In contrast, for both the Rayleigh channel with independent receiving branches and the double Rayleigh channel with independent receiving branches, the diversity order is N, which means solely increasing the number of receiving antennas can significantly enhance the BER performance. Also there is a significant BER improvement from L = 1 to L = 2 for the double Rayleigh channel with independent receiving branches, as shown in Equation (39). It is worth mentioning that, for the AF cooperative channel in Equation (17), the diversity order is (L + N)[17].
Under the OSTBC scheme, the diversity order of the studied MIMO RFID backscattering channel is the number of tag antennas L, regardless of the number of the reader receiving antennas N, as shown in Equation (34). In contrast, for both the Rayleigh channel with independent receiving branches and the double Rayleigh channel with independent receiving branches, the diversity order is NL. The performance of the MIMO RFID channel is also significantly different from keyhole fading and double scattering fading. For the keyhole channel, the diversity order is min(N,L), and for the double scattering channel the diversity order is MLN/max(M,L,N).
Although the diversity order is always L for the MIMO RFID channel with OSTBC, a significant BER improvement can still be observed from N = 1 to N = 2, because of the logarithm function in the numerator of the BER expression in Equation (33) when N = 1. From N = 2 to N > 2, the BER improvement is not significant, meaning that increasing the number of receiving antennas helps little when N > 2. We consider this the performance bottleneck for the MIMO RFID backscattering channel under the OSTBC scheme. While for both Rayleigh channel with independent receiving branches and double Rayleigh channel with independent receiving branches, solely increasing the number of receiving or transmitting antennas can yield a higher diversity order and a significant BER performance improvement.
The above different behaviors of the MIMO RFID channel are due to the special structure of the MIMO RFID backscattering channel. All the above discussions are summarized in Table3.
Performance improvement by employing OSTBC in RFID backscattering channel
Conclusion
In this article, we analyzed the performance of the MIMO RFID backscattering channel, a cascaded channel that has a special kind of structure. We observe several interesting properties of the channel from our analytical results. First, the diversity order is min(N,L) when employing the identical signaling scheme and the diversity order is L for the OSTBC signaling scheme. Second, the analytical results reveal that there is a performance bottleneck for the channel. More specifically, with the identical signaling scheme, when |N − L| > 1 the BER performance enhancement is not significant by increasing the number of tag antennas or the number of receiving antennas alone. With the OSTBC signaling scheme, the SER performance enhancement is not significant when N > L. These properties of the MIMO RFID channel are significantly different from that of other types of cascaded channels such as keyhole and double scattering. The analytical results and observations presented in this article could provide a useful guidance for the design of RFID systems using multiple antennas.
Appendix
The last step is obtained by the asymptotic property of the generalized exponential integral E_{ L }(·)[22].
where$\left(\right)close="">x=1+\stackrel{\u0304}{\stackrel{\u0304}{\gamma}}\alpha $, and$\left(\right)close="">{x}^{\prime}=\frac{x}{\stackrel{\u0304}{\stackrel{\u0304}{\gamma}}}$. The asymptotic expression is just like that in Proposition 2.
The last step is obtained by changing the index, i.e., k = N−i.
Therefore, Equation (14) is valid for N > L.
Case 3: N < L A similar approach as that of Case 1 can be obtained for this case; therefore, we omit the details here.
Declarations
Acknowledgements
This study was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) under STPGP 364962-08.
Authors’ Affiliations
References
- Want R: An introduction to RFID technology. IEEE Pervasive Comput 2006, 13: 25-33.View Article
- Klaus F: RFID Handbook: Fundamentals and Applications in Contactless Smart Cards and Identification. Wiley, Hoboken, NJ; 2003.
- Nikitin P, Rao KVS: Performance limitations of passive UHF RFID systems. IEEE Antennas and Propagation Society International Symposium 9–14 July 2006.
- Vita GD, Iannaccone G: Design criteria for the RF section of UHF and microwave passive RFID transponders. IEEE Trans. Microw. Theory Tech 2005, 53: 2978-2990.View Article
- Yeager D, Zhang F, Zarrasvand A, Otis BP: A 9.2 uA gen 2 compatible UHF RFID sensing tag with -12 dBm sensitivity and 1.25 uVrms input-referred noise floor. In ISSCC 2010. San Francisco, USA; 7–11 Feb 2010.
- Nikitin PV, Rao KS: Gain measurement of antennas using RFID. IEEE International Symposium on Antennas and Propagation 3–8 July 2011.
- Griffin JD, Durgin GD: Multipath fading measurements for multi-antenna backscatter RFID at 5.8 GHz. IEEE Trans. Antennas Propagat 2010, 58: 3693-3700.View Article
- Kim D, Ingram M, Smith W: Measurements of small-scale fading and path loss for long range RF tags. IEEE Trans. Antennas Propagat 2003, 51: 1740-1749. 10.1109/TAP.2003.814752View Article
- Griffin JD, Durgin GD: Gains for RF tags using multiple antennas. IEEE Trans. Antennas Propagat 2008, 56: 563-570.View Article
- Griffin JD, Durgin GD: Gains For RF tags using multiple antennas. IEEE Commun. Lett 2007, 11: 735-737.View Article
- Ingram M, Demirkol M, Kim D: Transmit diversity and spatial multiplexing for RF links using modulated backscatter. In International Symposium on Signals, Systems, and Electronics. Tokyo, Japan; 24–27 Jul 2001.
- Sarkka S, Huusko M, Viikari VV, Jaakkola K: Phase-based UHF RFID tracking with nonlinear Kalman filtering and smoothing. IEEE Sensors J 2012, 12: 904-910.View Article
- Griffin JD, Durgin GD: Complete link budgets for backscatter radio and RFID systems. IEEE Antennas Propagat. Mag 2009, 51: 11-25.View Article
- Strassner B, Chang K: Passive 5.8-GHz radio-frequency identification tag for monitoring oil drill pipe. IEEE Trans. Microwave Theory Tech 2003, 51: 356-363. 10.1109/TMTT.2002.807832View Article
- Shin H, Lee JH: Effect of keyholes on the symbol error rate of space-time block codes. IEEE Commun. Lett 2003, 7: 27-29.View Article
- Gong Y, Letaief KB: On the error probability of orthogonal space-timeblock codes over keyhole MIMO channel. IEEE Trans. Wirel. Commun 2007, 6: 2402-2409.View Article
- Elkashlan M, Yeoh PL, Louie RHY, Collings I: On the exact and asymptotic SER of receive diversity with multiple amplify-and-forward relays. IEEE Trans. Veh. Technol 2011, 59: 4602-4608.View Article
- Shin H, Win MZ: MIMO diversity in the presence of double scattering. IEEE Trans. Inf. Theory 2008, 54: 2976-2996.MathSciNetView ArticleMATH
- Abramowitz M, Stegun IA: Handbook of Mathematical Functions. National Bureau of Standards, Washington, DC; 1972.MATH
- Simon M, Alouini M: A unified approach to the performance analysis of digital communication over generalized fading channels. Proc. IEEE 1998, 86: 1860-1877. 10.1109/5.705532View Article
- Brennanr DG, Linear diversity combining techniques: Proc. IEEE. 2003, 91: 331-356. 10.1109/JPROC.2002.808163View Article
- Gradsheteyn I, Ryzhik I: Table of Integrals, Series and Product. Academic Press, New York; 1994.
- Proakis J: Digital Communications. McGraw-Hill, New York; 2001.MATH
- Zhu C, Mietzner J, Schober R: On the performance of non-coherent transmission schemes with equal-gain combining in generalized K-fading. IEEE Trans. Wirel. Commun 2010, 9: 1337-1349.View Article
- He C, Wang ZJ: Closed-form BER analysis of non-coherent FSK in MISO double rayleigh fading/RFID channel. IEEE Commun. Lett 2011, 15: 848-850.View Article
- Alamouti SM: A simple transmit diversity technique for wireless communications. IEEE J. Sel. Areas Commun 1998, 16: 1451-1458. 10.1109/49.730453View Article
- Tarokh V, Jafarkhani H, Calderbank A: Space-time block coding for wireless communications: performance results. IEEE J. Sel. Areas Commun 1999, 17: 451-460. 10.1109/49.753730View ArticleMATH
- Sandhu S, Paulraj A: Space-time block codes: a capacity perspective. IEEE Commun. Lett 2000, 10: 384-386.View Article
- Tarokh V, Seshadri N, Calderbank A: Space-time codes for high data rate wireless communication: performance criterion and code construction. IEEE Trans. Inf. Theory 1998, 44: 744-765. 10.1109/18.661517MathSciNetView ArticleMATH
- Uysal M: Diversity analysis of space-time coding in cascaded Rayleigh fading channels. IEEE Commun. Lett 2006, 10: 165-167. 10.1109/LCOMM.2006.1603372View Article
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