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DSTBC Impulse Radios with Autocorrelation Receiver in ISI-Free UWB Channels Qi Zhang, Student Member, IEEE, and Chun Sum Ng, Member, IEEE

Abstract— For transmitted-reference impulse radio, it has been experimentally shown that multiple transmit antennas can provide an energy boost in the received signal. Here, we propose a 2-transmit and Q-receive differential space-time block coded impulse radio with autocorrelation receiver. A mathematical model for predicting the system bit-error-rate (BER) performance is derived for intersymbol interference-free transmission over an ultra-wideband channel, and validated with computer simulated results. From the results, it is observed that in migrating from 1-transmit to 2-transmit antennas, a signal-to-noise ratio gain of 3dB, 2dB and 1dB at BER = 10−5 can be achieved, respectively, for the case of 1-, 2- and 4-receive antennas. Index Terms— Autocorrelation receiver (AcR), impulse radio (IR), differential space-time block codes (DSTBC), ultrawideband (UWB).

I. I NTRODUCTION

T

RANSMITTED-REFERENCE impulse radios (including differential impulse radios) with autocorrelation receiver (TR-IR/AcR) have been proposed in [1]-[10]. The idea behind TR-IR/AcR is to exploit multipath diversity in slowly timevarying channels by coupling one or more data modulated pulses with one or more unmodulated reference (or pilot) pulses. The received pilot pulses are then used to form the correlator template for symbol detection. For fading channels, the advantages of space-time codes have been well documented in multiple-input-multiple-output (MIMO) literature [11]-[15]. On one hand, ultra-wideband (UWB) channels are known to exhibit little to no fading when the multipath energy is harnessed. In this case, multiple receive antennas will still provide an energy boost and thus improve system performance, whereas the amount of diversity gain that can be derived from multiple transmit antennas may not be significant. On the other hand, Qiu et al [16] have experimentally shown that by using four fixed transmit antennas in their time-reversal UWB scheme, the channel impulse response measured by a mobile receive antenna exhibited a peak amplitude that was twice that of an equivalent system with only one transmit antenna. The UWB pulse used had a 10dB-bandwidth spanning for 700 MHz to 1.6 GHz, and an antenna array with elements spaced at 20 cm was found to be sufficient to ensure no correlation between the elements. Qiu et al’s time-reversal scheme may be viewed as a special case of transmitted-reference signaling with autocorrelation

Manuscript received September 26, 2006; revised February 12, 2007 and April 13, 2007; accepted April 13, 2007. The associate editor coordinating the review of this paper and approving it for publication was W. Zhuang. The authors are with the Department of Electrical and Computer Engineering, National University of Singapore, 4 Engineering Drive 3, 117576, Singapore (e-mail: {zhangqi, elengcs}@nus.edu.sg). Digital Object Identifier 10.1109/TWC.2008.060753.

receiver, where the coupled data- and reference-pulse form a time-reversed pair. While MIMO UWB transmission schemes have been reported elsewhere, for example in [17]-[21], to the best of our knowledge, work on TR-IR/AcR based MIMO UWB systems is missing in spite of Qiu et al’s [16] experimental evidence. In this letter, we examine the bit-error-rate (BER) of differentially encoded IR/AcR systems in UWB channels [23] with transmit and receive diversity. Differential spacetime block codes (DSTBC) similar to that described in [14] are used to provide additional diversity and coding gain [21]. To simplify our analysis, we restrict our study to systems with 2-transmit antennas, which is a practical configuration for small portable communication devices. Further, the symbol duration is chosen to be equal to the channel maximum excess delay plus the support length of the monocycle, with which intersymbol interference (ISI)-free transmission may be assumed. For example, a symbol duration of about 40.0 ns is found to achieve near ISI-free transmission for the channel model (CM) 2, 0 ∼ 4 meters’ range non-line-of-sight (NLOS) model [23]. This amounts to a maximum symbol rate of about 25 Mbits/s. The rest of the paper is organized as follows. Section II describes the proposed DSTBC IR/AcR system. In Section III, a mathematical model is derived for predicting the system BER performance. BER results are presented in Section IV to validate the accuracy of the BER predictor and also to examine the effectiveness of differential space-time block codes on the BER performance of IR/AcR systems with 2-transmit antennas. Section V concludes the paper. II. S YSTEM D ESCRIPTION In this paper, we consider a peer-to-peer impulse radio system with 2-transmit and Q-receive antennas in quasi-static UWB environment. It is assumed that the elements of the transmit- and receive-antenna arrays are spaced more than the coherent distance apart so that the channel coefficients for different transmit-receive antenna pairs are statistically independent. A. The Transmitter The code Ω = {U(1) , U(2) , U(3) , U(4) } used in the system is an orthogonal equivalence of Hughes [14] unitary code and has the form 1 0 0 1 (1) (2) U = U = 1 0 0 −1 (1) −1 0 0 −1 U(3) = U(4) = . 0 −1 1 0

c 2008 IEEE 1536-1276/08$25.00

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The transmission block Dk for the k th message matrix Uk ∈ Ω is obtained by Dk = Dk−1 Uk . The initial (or reference) transmission block, D0 , is chosen to be a 2 × 2 matrix with elements belonging to the set {−1, 1}, and satisfies D0 D†0 = 2I, where † denotes the transpose operation, and I the 2 × 2 identity matrix. Because Uk : k = 1, 2, · · · are square orthogonal matrices with only one non-zero entry, (+1 or − 1), per row and column, it is easy to see that Dk : k = 1, 2, · · · will also be 2 × 2 matrices with elements belonging to the set {−1, 1}, and satisfy Dk D†k = 2I, k = 1, 2, · · ·. The columns of Dk , are referred to as spacetime symbols. Each element in Dk is used to amplitude modulate a causal wideband monocycle, ω (t), which begins at t = 0 and has a support length of Tω ( T ) where T is the symbol duration. Letting dp,2k+n−1 denote the pth -row and nth -column of Dk , the transmitted signal from the pth antenna can then be written as sp (t) = =

∞ 2

dp,2k+n−1 ω (t − (2k + n − 1) T )

k=0 n=1 ∞

dp,j ω (t − jT ),

(2)

j=0

807

T

r1 (t + 2kT )

0

r2 (t + 2kT )

0

r1 (t + 2kT + T )

0

r2 (t + 2kT + T )

0

r1 (t + 2kT − 2T )

0

r2 (t + 2kT − 2T )

0

r1 (t + 2kT − T )

0

r2 (t + 2kT − T )

0

T

R(1 1) ,

k

T T

R(2 2) ,

k

T T

R(2 1) ,

k

T T

R(1 2) ,

k

Fig. 1. AcR block diagram for system with 2-transmit and 2-receive antennas.

where j = 2k + n − 1. the system lowpass filter. The signal-to-noise ratio, Eb /No , of T the system is defined as 2 0 ω ω 2 (t) dt No .

B. The Channel Model Following [18] and [20], we consider a quasi-static dense multipath fading environment and assume the channel coefficients for different transmit-receive antenna pairs to be statistically independent. The random channels are generated according to [23]. We model the impulse response of the channel between the pth transmit antenna and the q th receive antenna as Lpq gpq (t) = αpq,l δ (t − τpq,l ) (3) l=1

where Lpq denotes the number of propagation paths, αpq,l and τpq,l are the random amplitude and delay, respectively, associated with the lth path. The signals arriving at the receiver are assumed to be perfectly synchronized and the path delays are normalized so that min [τpq,1 ] = 0. Let hpq (t) be the overall channel response to the monocycle, ω (t), given by hpq (t) = ω (t) ⊗ gpq (t) =

Lpq

αpq,l ω (t − τpq,l )

(4)

l=1

where ⊗ denotes convolution. The received signal at the q th receiver antenna then has the form rq (t) =

∞ 2

C. The Receiver The receiver structure is illustrated in Fig. 1. To implement the autocorrelation receiver, we use rq (t) in (5) to form the matrix ⎡ ⎤ r1 (t + 2kT ) r1 (t + 2kT + T ) ⎢ r2 (t + 2kT ) r2 (t + 2kT + T ) ⎥ ⎥ (7) Yk (t) = ⎢ ⎣ ⎦ ··· ··· rQ (t + 2kT ) rQ (t + 2kT + T ) where Yk (t) is treated as a matrix of continuous time waveforms. The decision statistics, Rk , for detecting the k th message block, Uk , is then obtained as Rk =

(5)

p=1 j=0

where nq (t) is a bandlimited additive white Gaussian noise (AWGN) process with autocorrelation function Rn,q (τ ) = No W sinc (W τ ) .

(6)

In (6), No /2 and W ( 1/Tω 1/T ) are, respectively, the power spectral density of the AWGN and the bandwidth of

Yk† (t) Yk−1 (t) dt

(8)

where Rk is an 2 × 2 matrix, whose uth -row and v th -column element is given by (u,v)

dp,j hpq (t − jT ) + nq (t)

0

T

Rk

=

Q q=1

0

T

rq (t + 2kT + (u − 1) T )

rq (t + 2 (k − 1) T + (v − 1) T ) dt.

(9)

Following [14], the receiver computes k = arg max Tr (URk ) U U∈Ω

where “Tr (·)” denotes the trace of a matrix.

(10)

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 3, MARCH 2008

⎛

Q

⎜ ⎜ ⎜ Sk = ⎜ ⎜ ⎝

E1q +

q=1

Q

E2q ± 2

q=1

±

Q

Q q=1

E1q ∓

q=1

T

0

Q

h1q (t)h2q (t)dt Q

E2q

q=1

q=1

III. BER A NALYSIS In this section, we formulate an equivalent system model and derive a BER expression for the case of ISI-free transmission, which occurs when the symbol duration is equal to or longer than the memory length of the channel monocycle response. Under ISI-free condition, the received signal in (5) can be expressed as rq (t + (2k + n − 1) T ) p˜ (t) =

2

dp,2k+n−1 hpq (t)

p=1

+ nq (t + (2k + n − 1) T ) p˜ (t) (11)

1; 0 ≤ t < T where p˜ (t) = . Using (11), we may now 0; otherwise (u,v) (u,v) (u,v) = Sk +Nk , where express (9) as Rk 2 Q T (u,v) Sk = dp,2k+u−1 hpq (t) × q=1

0

p=1

2

dp,2(k−1)+v−1 hpq (t)

dt

p=1

2 2

=

dp ,2k+u−1 dp ,2(k−1)+v−1

p =1 p =1

Q q=1

and

(u,v)

Nk

T

0

hp q (t) hp q (t) dt

= A1 + A2 + A3

(12)

(13)

in which A1 =

2

dp,2(k−1)+v−1

p=1

A2 =

2

q=1

q=1

Q q=1

0

T

hpq (t)

nq (t + 2kT + (u − 1) T ) dt, Q T dp,2k+u−1 hpq (t)

p=1

A3 =

Q

0

(14)

0

nq (t + 2 (k − 1) T + (v − 1) T ) dt, (15) T

nq (t + 2kT + (u − 1) T ) nq (t + 2 (k − 1) T + (v − 1) T ) dt. (16)

(u,v)

(u,v)

With Sk and Nk forming the uth -row and v th -column elements of Sk and Nk , respectively, we may express the decision statistics Rk in (8) as Rk = Sk + Nk .

±

(17)

E1q +

Q

Q

E1q ∓

q=1

E2q ∓ 2

q=1

Q

⎞

q=1

0

⎟ ⎟ ⎟ ⎟ ⎟ h1q (t)h2q (t)dt ⎠

E2q

q=1 Q T

(18)

Suppose U(1) is transmitted. Sk in (17) can then be expressed as (18) shown on the top of the page, where Epq = T 2 hpq (t) dt. We note that when (10) is used to demodulate, 0 the cross terms in (18) will be cancelled completely. Therefore, we may omit those p = p terms in (12) to obtain Q 2 (u,v) Sk = Epq . dp,2k+u−1 dp,2(k−1)+v−1 (19) p=1

q=1

The signal matrix, Sk , thus reduces to Sk = D†k HDk−1 , where Q Q E1q , E2q . (20) H = diag q=1

q=1

(u,v) Nk

in (13), we recall that nq (t) To simplify our analysis of is a wideband AWGN process with bandwidth W 1/Tω 1/T and hence Rn,q (τ ) = 0; |τ | ≥ Tω

(21)

can be reasonably assumed. The correlation of the noise terms found in (14)-(16), can then be generally written as nq (t + 2 (k − 1) T + (v − 1) T ) E × nq (t + 2kT + (u − 1) T ) 0; q = q = (22) Rn,q (|2 + u − v| T ) ; q = q = q where E [·] denotes the expected value operator. When q = q = q, 1 ≤ u ≤ 2 and 1 ≤ v ≤ 2, the correlation lag |2 + u − v| T in (22) is lower bounded by T , which yields Rn,q (|2 + u − v| T ) = 0 according to (21), causing nq (t + 2 (k − 1) T + (v − 1) T ) and nq (t + 2kT + (u − 1) T ) to become uncorrelated. Clearly, A1 and A2 are Gaussian random variables when conditioned on the set ζ = {hpq (t)}. When W T is large, A3 can also be seen from the central limit theorem as approximately Gaussian distributed. With these observations, we simplify our noise analysis by treating A1 , A2 and A3 , conditioned on ζ, as uncorrelated jointly Gaussian random variables. The (u,v) are, respectively, conditional mean and variance of Nk (u,v) given by E Nk ζ = 0 and (u,v) Var Nk ζ = E A21 ζ + E A22 ζ + E A23 (23) where Var [·] denotes variance value operator. Using (14) and (15), the conditional variances of A1 and A2 can be expressed as Q T T 2 hpq (t ) hpq (t ) E A22 ζ = E A21 ζ = p=1 q=1

0

0

Rn,q (t − t ) dt dt . (24)

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As a second, and coarser, approximation, we let No δ (τ ) , (25) Rn,q (τ ) 2 on the basis that Rn,q (τ ) appears almost impulse like for sufficiently large bandwidth-to-symbol rate ratio, WT. Applying (25) in (24), it is straightforward to show that No E A21 ζ = E A22 ζ 2

Q 2

Epq .

Pr (−φ4 < Φ2 < φ2 |ζ ) =

φ2

−φ4

1

" exp 2 4πσN |ζ

−φ2 2 4σN |ζ

dφ (34)

where

Q

E1q +

Q

q=1

E2q .

(35)

q=1

By letting

(u,v)

Therefore, each element Rk of the decision statistics Rk then has a conditional probability density function given by ! (u,v) f R(u,v) ζ Rk k ⎛ !2 ⎞ (u,v) (u,v) − S − R k k 1 ⎟ ⎜ = " exp ⎝ ⎠ . (28) 2 2 2σ 2πσ N |ζ N |ζ

Suppose we form the set

(29)

k , as defined in (10), be the detected message block. and let U Given ζ, the probability of U(ξ) detected as U(l) is transmitted during the k th transmission block is given by ! k = U(l) U(ξ) , ζ Pr U ! (30) = Pr max {Φ1 , Φ2 , Φ3 , Φ4 } = Φl |ζ ∞ ∂FΦ1 Φ2 Φ3 Φ4 (φ1 , φ2 , φ3 , φ4 |ζ ) = dφl . ∂φl −∞ φk =φl ∀k=l where Fχ (·) denotes the probability distribution function of χ. Averaging (30) over ζ, the corresponding average probability of U(ξ) detected as U(l) is then computed by ! ∞ ∞ ! (l) (ξ) k = U(l) U(ξ), ζ = ··· Pr U Pr Uk = U U 0

fE11 (ε11 ) fE12 (ε12 ) · · · fE2Q (ε2Q ) dε11 dε12 · · · dε2Q . (31) From (1) and (29), we have Φ1 = −Φ3 and Φ2 = −Φ4 . Since Φ1 and Φ2 are independent, FΦ1 Φ2 Φ3 Φ4 (φ1 , φ2 , φ2 , φ4 |ζ ) in (30) can be expressed as FΦ1 Φ2 Φ3 Φ4 (φ1 , φ2 , φ3 , φ4 |ζ) = Pr (−φ3 < Φ1 < φ1 , −φ4 < Φ2 < φ2 |ζ) (32) = Pr (−φ3 < Φ1 < φ1 |ζ) Pr (−φ4 < Φ2 < φ2 |ζ) . From (28) and (29), with ζ given, Φ1 and Φ2 are Gaussian. Therefore, we obtain Pr (−φ3 < Φ1 < φ1 |ζ ) φ1 2 1 − (φ − 2E) " = exp dφ 2 2 4σN −φ3 4πσN |ζ |ζ

(26)

For A3 given (16), using the same approach as in [5], in we have E A23 = QNo2 W T /2. Based on the approximate analysis above, the elements of Nk have identical conditional variances given by Q 2 QNo2 W T (u,v) . (27) σ 2N |ζ = Var Nk ζ = No Epq + 2 p=1 q=1

0

and

E=

p=1 q=1

! Φξ = Tr U(ξ) Rk ; ξ = 1, 2, 3, 4

809

(33)

fE (ε) = fE11 (ε11 ) ⊗ fE12 (ε12 ) ⊗ · · · ⊗ fE2Q (ε2Q )

(36)

and using (30) and (31), the probability of U(1) detected as U(3) is 2 ! ∞ 0 1 −(φ−2ε) k = U(3) U(1) = " Pr U exp 2 4σN 0 −∞ 4πσ 2 |ζ N |ζ ⎛ ⎛ ⎞⎞ ⎝1−2Υ ⎝ " |φ| ⎠⎠ fE (ε) dφ dε (37) 2 2σN |ζ # √ $∞ where Υ(z) = 1/ 2π z exp(−y 2 /2)dy . Using the same method, we can compute the probability of erroneously detecting U(1) as U(2) or U(4) as follows ! ! k = U(4) U(1) k = U(2) U(1) = Pr U Pr U ⎡ ⎛ ⎞ ⎛ ⎞⎤ ∞ ∞ −φ − 2ε φ − 2ε ⎣Υ ⎝ " ⎠ −Υ ⎝ " ⎠⎦ (38) = 2 2 0 0 2σN 2σ |ζ N |ζ 2 1 −φ " exp fE (ε) dφ dε. 2 2 4σ 4πσ N |ζ N |ζ

The binary information bits are Gray coded such that “00”→ U(1) ; “01”→ U(2) ; “11”→ U(3) and “10”→ U(4) . Because of Gray coding and symmetric property of the channel, the system BER is given by ! ! k = U(2) U(1) . k = U(3) U(1) + Pr U BER = Pr U (39) IV. R ESULTS AND D ISCUSSION In this section, we present and examine the BER performance of the proposed DSTBC-IR/AcR system using the UWB channel model in [23] as well as measurements data from [24]. In all cases, the symbol duration T is set to 40.0 ns. The sampling interval is chosen to be 0.05 ns. The bandwidth of the lowpass filter is 10 GHz. As in [22], we select the shape of the monocycle ω(t) to be the second derivative of Gaussian function 1 − 4π(t/τm )2 exp[−2π(t/τm )2 ], where τm = 0.2877 ns. A. UWB Channel Model Here, the random channels are generated according to [23] based on CM 2, 0 ∼ 4 meters’ range non-line-of-sight (NLOS) model. As in [23], the energy of the channel response of ω(t) is characterized by Tω ω 2 (t)dt (40) Epq = 10X/10 · 0

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10

10

P=1, Q=1 (Diff) P=2, Q=1 (DSTBC) P=1, Q=2 (Diff) P=2, Q=2 (DSTBC) P=1, Q=4 (Diff) P=2, Q=4 (DSTBC)

−1

−2

BER

10

0

10

10

−3

−4

8

10

12

14

16 18 Eb/No (dB)

20

22

24

26

Fig. 2. BER of DSTBC- and Diff-IR/AcR systems with P -transmit and Q-receive antennas using UWB channel model CM 2. 10

10

P=1, Q=1, LOS (Diff) P=2, Q=1, LOS (DSTBC) P=1, Q=1, NLOS (Diff) P=2, Q=1, NLOS (DSTBC)

−1

V. C ONCLUSION In this letter, we proposed a DSTBC-IR/AcR system with 2-transmit and Q-receive antennas for ISI-free UWB transmission. The DSTBC scheme was instrumental in causing the channel matrix, H, to be diagonal in an ISI-free environment, which facilitated the straightforward derivation of a theoretical system BER predictor. Computer simulated BER results were found to coincide with theory. It was shown that the proposed system could exploit transmit diversity even if the channel’s natural multipath energies were harnessed.

−2

BER

10

0

10

10

the proposed DSTBC-IR/AcR systems for the case of Q = 1. Both the line-of-sight (LOS) and non-LOS (NLOS) situations are considered. For the P = 2 case, the BERs are simulated by averaging the system performance over a permutation of paired-channel measurements made with the transmit antennas spaced at 20 cm apart. Fig. 3 shows that a performance improvement of about 0.17 dB or 0.36 dB at BER=10−5 can be achieved in a LOS or NLOS situation, respectively, when the number of transmit antennas is increased from P = 1 to P = 2 without receive diversity. Comparing these with the results obtained with the CM 2 channel model, we observe a significant reduction in the achievable performance gain. This may be attributed to the sensitivity of the proposed system to channel correlation. Nevertheless, it should be noted that, unlike those in [16] where a mobile receive antenna was used, these measurement data are obtained with a fixed receive antenna and the number of measurement samples is limited. Hence, the BER results obtained with these measurement data may not closely approximate the ensemble average BER performance.

−3

−4

R EFERENCES 8

10

12

14 Eb/No (dB)

16

18

20

Fig. 3. BER of DSTBC- and Diff-IR/AcR systems with P -transmit and Q-receive antennas using UWB channel measurement data.

where X is a Gaussian random variable with zero mean and variance 9. In Fig. 2, we compare the BER performance of differential IR/AcR (Diff-IR/AcR) systems with that of the proposed DSTBC-IR/AcR systems for the case of Q = 1, 2 and 4. For all the cases considered here, it is observed that there is no noticeable difference between theoretical and computer simulated BER results. The results show that in migrating from 1-transmit to 2-transmit antennas (the number of transmit antennas is denoted as “P ” in the legend), a signal-to-noise ratio (SNR) gain of about 3 dB, 2 dB and 1 dB can be achieved at BER= 10−5 with Q = 1, 2 and 4, respectively. The variation of BER with respect to P and Q also appears consistent with the general trend of MIMO system performance. B. UWB Channel Measurement Data Here, the UWB channel impulse response comprises a set of measurement data obtained from [24]. In Fig. 3, we compare the BER performance of Diff-IR/AcR systems with that of

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[13] B. M. Hochwald and T. L. Marzetta, “Unitary space-time modulation for multiple-antenna communications in rayleigh flat fading,” IEEE Trans. Inf. Theory, vol. 46, pp. 543-563, Mar. 2000. [14] B. L. Hughes, “Differential space-time modulation,” IEEE Trans. Inf. Theory, vol. 46, pp. 2567-2578, Nov. 2000. [15] B. M. Hochwald and W. Sweldens, “Differential unitary space-time modulation,” IEEE Trans. Commun., vol. 48, pp. 2041-2052, Dec. 2000. [16] R. C. Qiu, C. Zhou, N. Guo, and J. Q. Zhang, “Time Reversal With MISO for Ultrawideband Communications: Experimental Results,” IEEE Antennas Wireless Propag. Lett., vol. 5, pp. 269-273, 2006. [17] M. Weisenhorn and W. Hirt, “Performance of binary antipodal signaling over the indoor UWB MIMO channel,” in Proc. ICC 2003, pp. 28722878. [18] L. Yang and G. B. Giannakis, “Analog space-time coding for multiantenna ultra-wideband transmissions,” IEEE Trans. Commun., vol. 52, pp. 507-517, March 2004. [19] E. Baccarelli, M. Biagi, C. Pelizzoni, and P. Bellotti, “A novel multi-

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antenna impulse radio UWB transceiver for broadband high-throughput 4G WLANs,” IEEE Commun. Lett., vol. 8, pp. 419-421, July 2004. [20] W. P. Siriwongpairat, W. Su, M. Olfat, and K. J. R. Liu “MultibandOFDM MIMO coding framework for UWB communication systems,” IEEE Trans. Signal Processing, vol. 54, pp. 214-224, Jan. 2006. [21] Q. Zhang and C. S. Ng, “Differential space-time coded impulse radio systems using MIMO autocorrelation reveivers in UWB channel,” in Proc. VTC 2006-Fall, pp. 1-5. [22] M. Z. Win and R. A. Scholtz, “Ultra-wide bandwidth time-hopping spread-spectrum impulse radio for wireless multiple-access communications,” IEEE Trans. Commun., vol. 48, pp. 679-691, Apr. 2000. [23] IEEE, “Channel modeling sub-committee report final,” IEEE P802.1502/368r5-SG3a, Sept. 2003. [24] IMST-UWBW, “1-11 GHz UWB indoor radio channel measurements,” http://www.imst.de/de/funk wel dow.php.

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DSTBC Impulse Radios with Autocorrelation Receiver in ISI-Free UWB Channels Qi Zhang, Student Member, IEEE, and Chun Sum Ng, Member, IEEE

Abstract— For transmitted-reference impulse radio, it has been experimentally shown that multiple transmit antennas can provide an energy boost in the received signal. Here, we propose a 2-transmit and Q-receive differential space-time block coded impulse radio with autocorrelation receiver. A mathematical model for predicting the system bit-error-rate (BER) performance is derived for intersymbol interference-free transmission over an ultra-wideband channel, and validated with computer simulated results. From the results, it is observed that in migrating from 1-transmit to 2-transmit antennas, a signal-to-noise ratio gain of 3dB, 2dB and 1dB at BER = 10−5 can be achieved, respectively, for the case of 1-, 2- and 4-receive antennas. Index Terms— Autocorrelation receiver (AcR), impulse radio (IR), differential space-time block codes (DSTBC), ultrawideband (UWB).

I. I NTRODUCTION

T

RANSMITTED-REFERENCE impulse radios (including differential impulse radios) with autocorrelation receiver (TR-IR/AcR) have been proposed in [1]-[10]. The idea behind TR-IR/AcR is to exploit multipath diversity in slowly timevarying channels by coupling one or more data modulated pulses with one or more unmodulated reference (or pilot) pulses. The received pilot pulses are then used to form the correlator template for symbol detection. For fading channels, the advantages of space-time codes have been well documented in multiple-input-multiple-output (MIMO) literature [11]-[15]. On one hand, ultra-wideband (UWB) channels are known to exhibit little to no fading when the multipath energy is harnessed. In this case, multiple receive antennas will still provide an energy boost and thus improve system performance, whereas the amount of diversity gain that can be derived from multiple transmit antennas may not be significant. On the other hand, Qiu et al [16] have experimentally shown that by using four fixed transmit antennas in their time-reversal UWB scheme, the channel impulse response measured by a mobile receive antenna exhibited a peak amplitude that was twice that of an equivalent system with only one transmit antenna. The UWB pulse used had a 10dB-bandwidth spanning for 700 MHz to 1.6 GHz, and an antenna array with elements spaced at 20 cm was found to be sufficient to ensure no correlation between the elements. Qiu et al’s time-reversal scheme may be viewed as a special case of transmitted-reference signaling with autocorrelation

Manuscript received September 26, 2006; revised February 12, 2007 and April 13, 2007; accepted April 13, 2007. The associate editor coordinating the review of this paper and approving it for publication was W. Zhuang. The authors are with the Department of Electrical and Computer Engineering, National University of Singapore, 4 Engineering Drive 3, 117576, Singapore (e-mail: {zhangqi, elengcs}@nus.edu.sg). Digital Object Identifier 10.1109/TWC.2008.060753.

receiver, where the coupled data- and reference-pulse form a time-reversed pair. While MIMO UWB transmission schemes have been reported elsewhere, for example in [17]-[21], to the best of our knowledge, work on TR-IR/AcR based MIMO UWB systems is missing in spite of Qiu et al’s [16] experimental evidence. In this letter, we examine the bit-error-rate (BER) of differentially encoded IR/AcR systems in UWB channels [23] with transmit and receive diversity. Differential spacetime block codes (DSTBC) similar to that described in [14] are used to provide additional diversity and coding gain [21]. To simplify our analysis, we restrict our study to systems with 2-transmit antennas, which is a practical configuration for small portable communication devices. Further, the symbol duration is chosen to be equal to the channel maximum excess delay plus the support length of the monocycle, with which intersymbol interference (ISI)-free transmission may be assumed. For example, a symbol duration of about 40.0 ns is found to achieve near ISI-free transmission for the channel model (CM) 2, 0 ∼ 4 meters’ range non-line-of-sight (NLOS) model [23]. This amounts to a maximum symbol rate of about 25 Mbits/s. The rest of the paper is organized as follows. Section II describes the proposed DSTBC IR/AcR system. In Section III, a mathematical model is derived for predicting the system BER performance. BER results are presented in Section IV to validate the accuracy of the BER predictor and also to examine the effectiveness of differential space-time block codes on the BER performance of IR/AcR systems with 2-transmit antennas. Section V concludes the paper. II. S YSTEM D ESCRIPTION In this paper, we consider a peer-to-peer impulse radio system with 2-transmit and Q-receive antennas in quasi-static UWB environment. It is assumed that the elements of the transmit- and receive-antenna arrays are spaced more than the coherent distance apart so that the channel coefficients for different transmit-receive antenna pairs are statistically independent. A. The Transmitter The code Ω = {U(1) , U(2) , U(3) , U(4) } used in the system is an orthogonal equivalence of Hughes [14] unitary code and has the form 1 0 0 1 (1) (2) U = U = 1 0 0 −1 (1) −1 0 0 −1 U(3) = U(4) = . 0 −1 1 0

c 2008 IEEE 1536-1276/08$25.00

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The transmission block Dk for the k th message matrix Uk ∈ Ω is obtained by Dk = Dk−1 Uk . The initial (or reference) transmission block, D0 , is chosen to be a 2 × 2 matrix with elements belonging to the set {−1, 1}, and satisfies D0 D†0 = 2I, where † denotes the transpose operation, and I the 2 × 2 identity matrix. Because Uk : k = 1, 2, · · · are square orthogonal matrices with only one non-zero entry, (+1 or − 1), per row and column, it is easy to see that Dk : k = 1, 2, · · · will also be 2 × 2 matrices with elements belonging to the set {−1, 1}, and satisfy Dk D†k = 2I, k = 1, 2, · · ·. The columns of Dk , are referred to as spacetime symbols. Each element in Dk is used to amplitude modulate a causal wideband monocycle, ω (t), which begins at t = 0 and has a support length of Tω ( T ) where T is the symbol duration. Letting dp,2k+n−1 denote the pth -row and nth -column of Dk , the transmitted signal from the pth antenna can then be written as sp (t) = =

∞ 2

dp,2k+n−1 ω (t − (2k + n − 1) T )

k=0 n=1 ∞

dp,j ω (t − jT ),

(2)

j=0

807

T

r1 (t + 2kT )

0

r2 (t + 2kT )

0

r1 (t + 2kT + T )

0

r2 (t + 2kT + T )

0

r1 (t + 2kT − 2T )

0

r2 (t + 2kT − 2T )

0

r1 (t + 2kT − T )

0

r2 (t + 2kT − T )

0

T

R(1 1) ,

k

T T

R(2 2) ,

k

T T

R(2 1) ,

k

T T

R(1 2) ,

k

Fig. 1. AcR block diagram for system with 2-transmit and 2-receive antennas.

where j = 2k + n − 1. the system lowpass filter. The signal-to-noise ratio, Eb /No , of T the system is defined as 2 0 ω ω 2 (t) dt No .

B. The Channel Model Following [18] and [20], we consider a quasi-static dense multipath fading environment and assume the channel coefficients for different transmit-receive antenna pairs to be statistically independent. The random channels are generated according to [23]. We model the impulse response of the channel between the pth transmit antenna and the q th receive antenna as Lpq gpq (t) = αpq,l δ (t − τpq,l ) (3) l=1

where Lpq denotes the number of propagation paths, αpq,l and τpq,l are the random amplitude and delay, respectively, associated with the lth path. The signals arriving at the receiver are assumed to be perfectly synchronized and the path delays are normalized so that min [τpq,1 ] = 0. Let hpq (t) be the overall channel response to the monocycle, ω (t), given by hpq (t) = ω (t) ⊗ gpq (t) =

Lpq

αpq,l ω (t − τpq,l )

(4)

l=1

where ⊗ denotes convolution. The received signal at the q th receiver antenna then has the form rq (t) =

∞ 2

C. The Receiver The receiver structure is illustrated in Fig. 1. To implement the autocorrelation receiver, we use rq (t) in (5) to form the matrix ⎡ ⎤ r1 (t + 2kT ) r1 (t + 2kT + T ) ⎢ r2 (t + 2kT ) r2 (t + 2kT + T ) ⎥ ⎥ (7) Yk (t) = ⎢ ⎣ ⎦ ··· ··· rQ (t + 2kT ) rQ (t + 2kT + T ) where Yk (t) is treated as a matrix of continuous time waveforms. The decision statistics, Rk , for detecting the k th message block, Uk , is then obtained as Rk =

(5)

p=1 j=0

where nq (t) is a bandlimited additive white Gaussian noise (AWGN) process with autocorrelation function Rn,q (τ ) = No W sinc (W τ ) .

(6)

In (6), No /2 and W ( 1/Tω 1/T ) are, respectively, the power spectral density of the AWGN and the bandwidth of

Yk† (t) Yk−1 (t) dt

(8)

where Rk is an 2 × 2 matrix, whose uth -row and v th -column element is given by (u,v)

dp,j hpq (t − jT ) + nq (t)

0

T

Rk

=

Q q=1

0

T

rq (t + 2kT + (u − 1) T )

rq (t + 2 (k − 1) T + (v − 1) T ) dt.

(9)

Following [14], the receiver computes k = arg max Tr (URk ) U U∈Ω

where “Tr (·)” denotes the trace of a matrix.

(10)

808

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 3, MARCH 2008

⎛

Q

⎜ ⎜ ⎜ Sk = ⎜ ⎜ ⎝

E1q +

q=1

Q

E2q ± 2

q=1

±

Q

Q q=1

E1q ∓

q=1

T

0

Q

h1q (t)h2q (t)dt Q

E2q

q=1

q=1

III. BER A NALYSIS In this section, we formulate an equivalent system model and derive a BER expression for the case of ISI-free transmission, which occurs when the symbol duration is equal to or longer than the memory length of the channel monocycle response. Under ISI-free condition, the received signal in (5) can be expressed as rq (t + (2k + n − 1) T ) p˜ (t) =

2

dp,2k+n−1 hpq (t)

p=1

+ nq (t + (2k + n − 1) T ) p˜ (t) (11)

1; 0 ≤ t < T where p˜ (t) = . Using (11), we may now 0; otherwise (u,v) (u,v) (u,v) = Sk +Nk , where express (9) as Rk 2 Q T (u,v) Sk = dp,2k+u−1 hpq (t) × q=1

0

p=1

2

dp,2(k−1)+v−1 hpq (t)

dt

p=1

2 2

=

dp ,2k+u−1 dp ,2(k−1)+v−1

p =1 p =1

Q q=1

and

(u,v)

Nk

T

0

hp q (t) hp q (t) dt

= A1 + A2 + A3

(12)

(13)

in which A1 =

2

dp,2(k−1)+v−1

p=1

A2 =

2

q=1

q=1

Q q=1

0

T

hpq (t)

nq (t + 2kT + (u − 1) T ) dt, Q T dp,2k+u−1 hpq (t)

p=1

A3 =

Q

0

(14)

0

nq (t + 2 (k − 1) T + (v − 1) T ) dt, (15) T

nq (t + 2kT + (u − 1) T ) nq (t + 2 (k − 1) T + (v − 1) T ) dt. (16)

(u,v)

(u,v)

With Sk and Nk forming the uth -row and v th -column elements of Sk and Nk , respectively, we may express the decision statistics Rk in (8) as Rk = Sk + Nk .

±

(17)

E1q +

Q

Q

E1q ∓

q=1

E2q ∓ 2

q=1

Q

⎞

q=1

0

⎟ ⎟ ⎟ ⎟ ⎟ h1q (t)h2q (t)dt ⎠

E2q

q=1 Q T

(18)

Suppose U(1) is transmitted. Sk in (17) can then be expressed as (18) shown on the top of the page, where Epq = T 2 hpq (t) dt. We note that when (10) is used to demodulate, 0 the cross terms in (18) will be cancelled completely. Therefore, we may omit those p = p terms in (12) to obtain Q 2 (u,v) Sk = Epq . dp,2k+u−1 dp,2(k−1)+v−1 (19) p=1

q=1

The signal matrix, Sk , thus reduces to Sk = D†k HDk−1 , where Q Q E1q , E2q . (20) H = diag q=1

q=1

(u,v) Nk

in (13), we recall that nq (t) To simplify our analysis of is a wideband AWGN process with bandwidth W 1/Tω 1/T and hence Rn,q (τ ) = 0; |τ | ≥ Tω

(21)

can be reasonably assumed. The correlation of the noise terms found in (14)-(16), can then be generally written as nq (t + 2 (k − 1) T + (v − 1) T ) E × nq (t + 2kT + (u − 1) T ) 0; q = q = (22) Rn,q (|2 + u − v| T ) ; q = q = q where E [·] denotes the expected value operator. When q = q = q, 1 ≤ u ≤ 2 and 1 ≤ v ≤ 2, the correlation lag |2 + u − v| T in (22) is lower bounded by T , which yields Rn,q (|2 + u − v| T ) = 0 according to (21), causing nq (t + 2 (k − 1) T + (v − 1) T ) and nq (t + 2kT + (u − 1) T ) to become uncorrelated. Clearly, A1 and A2 are Gaussian random variables when conditioned on the set ζ = {hpq (t)}. When W T is large, A3 can also be seen from the central limit theorem as approximately Gaussian distributed. With these observations, we simplify our noise analysis by treating A1 , A2 and A3 , conditioned on ζ, as uncorrelated jointly Gaussian random variables. The (u,v) are, respectively, conditional mean and variance of Nk (u,v) given by E Nk ζ = 0 and (u,v) Var Nk ζ = E A21 ζ + E A22 ζ + E A23 (23) where Var [·] denotes variance value operator. Using (14) and (15), the conditional variances of A1 and A2 can be expressed as Q T T 2 hpq (t ) hpq (t ) E A22 ζ = E A21 ζ = p=1 q=1

0

0

Rn,q (t − t ) dt dt . (24)

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 3, MARCH 2008

As a second, and coarser, approximation, we let No δ (τ ) , (25) Rn,q (τ ) 2 on the basis that Rn,q (τ ) appears almost impulse like for sufficiently large bandwidth-to-symbol rate ratio, WT. Applying (25) in (24), it is straightforward to show that No E A21 ζ = E A22 ζ 2

Q 2

Epq .

Pr (−φ4 < Φ2 < φ2 |ζ ) =

φ2

−φ4

1

" exp 2 4πσN |ζ

−φ2 2 4σN |ζ

dφ (34)

where

Q

E1q +

Q

q=1

E2q .

(35)

q=1

By letting

(u,v)

Therefore, each element Rk of the decision statistics Rk then has a conditional probability density function given by ! (u,v) f R(u,v) ζ Rk k ⎛ !2 ⎞ (u,v) (u,v) − S − R k k 1 ⎟ ⎜ = " exp ⎝ ⎠ . (28) 2 2 2σ 2πσ N |ζ N |ζ

Suppose we form the set

(29)

k , as defined in (10), be the detected message block. and let U Given ζ, the probability of U(ξ) detected as U(l) is transmitted during the k th transmission block is given by ! k = U(l) U(ξ) , ζ Pr U ! (30) = Pr max {Φ1 , Φ2 , Φ3 , Φ4 } = Φl |ζ ∞ ∂FΦ1 Φ2 Φ3 Φ4 (φ1 , φ2 , φ3 , φ4 |ζ ) = dφl . ∂φl −∞ φk =φl ∀k=l where Fχ (·) denotes the probability distribution function of χ. Averaging (30) over ζ, the corresponding average probability of U(ξ) detected as U(l) is then computed by ! ∞ ∞ ! (l) (ξ) k = U(l) U(ξ), ζ = ··· Pr U Pr Uk = U U 0

fE11 (ε11 ) fE12 (ε12 ) · · · fE2Q (ε2Q ) dε11 dε12 · · · dε2Q . (31) From (1) and (29), we have Φ1 = −Φ3 and Φ2 = −Φ4 . Since Φ1 and Φ2 are independent, FΦ1 Φ2 Φ3 Φ4 (φ1 , φ2 , φ2 , φ4 |ζ ) in (30) can be expressed as FΦ1 Φ2 Φ3 Φ4 (φ1 , φ2 , φ3 , φ4 |ζ) = Pr (−φ3 < Φ1 < φ1 , −φ4 < Φ2 < φ2 |ζ) (32) = Pr (−φ3 < Φ1 < φ1 |ζ) Pr (−φ4 < Φ2 < φ2 |ζ) . From (28) and (29), with ζ given, Φ1 and Φ2 are Gaussian. Therefore, we obtain Pr (−φ3 < Φ1 < φ1 |ζ ) φ1 2 1 − (φ − 2E) " = exp dφ 2 2 4σN −φ3 4πσN |ζ |ζ

(26)

For A3 given (16), using the same approach as in [5], in we have E A23 = QNo2 W T /2. Based on the approximate analysis above, the elements of Nk have identical conditional variances given by Q 2 QNo2 W T (u,v) . (27) σ 2N |ζ = Var Nk ζ = No Epq + 2 p=1 q=1

0

and

E=

p=1 q=1

! Φξ = Tr U(ξ) Rk ; ξ = 1, 2, 3, 4

809

(33)

fE (ε) = fE11 (ε11 ) ⊗ fE12 (ε12 ) ⊗ · · · ⊗ fE2Q (ε2Q )

(36)

and using (30) and (31), the probability of U(1) detected as U(3) is 2 ! ∞ 0 1 −(φ−2ε) k = U(3) U(1) = " Pr U exp 2 4σN 0 −∞ 4πσ 2 |ζ N |ζ ⎛ ⎛ ⎞⎞ ⎝1−2Υ ⎝ " |φ| ⎠⎠ fE (ε) dφ dε (37) 2 2σN |ζ # √ $∞ where Υ(z) = 1/ 2π z exp(−y 2 /2)dy . Using the same method, we can compute the probability of erroneously detecting U(1) as U(2) or U(4) as follows ! ! k = U(4) U(1) k = U(2) U(1) = Pr U Pr U ⎡ ⎛ ⎞ ⎛ ⎞⎤ ∞ ∞ −φ − 2ε φ − 2ε ⎣Υ ⎝ " ⎠ −Υ ⎝ " ⎠⎦ (38) = 2 2 0 0 2σN 2σ |ζ N |ζ 2 1 −φ " exp fE (ε) dφ dε. 2 2 4σ 4πσ N |ζ N |ζ

The binary information bits are Gray coded such that “00”→ U(1) ; “01”→ U(2) ; “11”→ U(3) and “10”→ U(4) . Because of Gray coding and symmetric property of the channel, the system BER is given by ! ! k = U(2) U(1) . k = U(3) U(1) + Pr U BER = Pr U (39) IV. R ESULTS AND D ISCUSSION In this section, we present and examine the BER performance of the proposed DSTBC-IR/AcR system using the UWB channel model in [23] as well as measurements data from [24]. In all cases, the symbol duration T is set to 40.0 ns. The sampling interval is chosen to be 0.05 ns. The bandwidth of the lowpass filter is 10 GHz. As in [22], we select the shape of the monocycle ω(t) to be the second derivative of Gaussian function 1 − 4π(t/τm )2 exp[−2π(t/τm )2 ], where τm = 0.2877 ns. A. UWB Channel Model Here, the random channels are generated according to [23] based on CM 2, 0 ∼ 4 meters’ range non-line-of-sight (NLOS) model. As in [23], the energy of the channel response of ω(t) is characterized by Tω ω 2 (t)dt (40) Epq = 10X/10 · 0

810

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10

10

P=1, Q=1 (Diff) P=2, Q=1 (DSTBC) P=1, Q=2 (Diff) P=2, Q=2 (DSTBC) P=1, Q=4 (Diff) P=2, Q=4 (DSTBC)

−1

−2

BER

10

0

10

10

−3

−4

8

10

12

14

16 18 Eb/No (dB)

20

22

24

26

Fig. 2. BER of DSTBC- and Diff-IR/AcR systems with P -transmit and Q-receive antennas using UWB channel model CM 2. 10

10

P=1, Q=1, LOS (Diff) P=2, Q=1, LOS (DSTBC) P=1, Q=1, NLOS (Diff) P=2, Q=1, NLOS (DSTBC)

−1

V. C ONCLUSION In this letter, we proposed a DSTBC-IR/AcR system with 2-transmit and Q-receive antennas for ISI-free UWB transmission. The DSTBC scheme was instrumental in causing the channel matrix, H, to be diagonal in an ISI-free environment, which facilitated the straightforward derivation of a theoretical system BER predictor. Computer simulated BER results were found to coincide with theory. It was shown that the proposed system could exploit transmit diversity even if the channel’s natural multipath energies were harnessed.

−2

BER

10

0

10

10

the proposed DSTBC-IR/AcR systems for the case of Q = 1. Both the line-of-sight (LOS) and non-LOS (NLOS) situations are considered. For the P = 2 case, the BERs are simulated by averaging the system performance over a permutation of paired-channel measurements made with the transmit antennas spaced at 20 cm apart. Fig. 3 shows that a performance improvement of about 0.17 dB or 0.36 dB at BER=10−5 can be achieved in a LOS or NLOS situation, respectively, when the number of transmit antennas is increased from P = 1 to P = 2 without receive diversity. Comparing these with the results obtained with the CM 2 channel model, we observe a significant reduction in the achievable performance gain. This may be attributed to the sensitivity of the proposed system to channel correlation. Nevertheless, it should be noted that, unlike those in [16] where a mobile receive antenna was used, these measurement data are obtained with a fixed receive antenna and the number of measurement samples is limited. Hence, the BER results obtained with these measurement data may not closely approximate the ensemble average BER performance.

−3

−4

R EFERENCES 8

10

12

14 Eb/No (dB)

16

18

20

Fig. 3. BER of DSTBC- and Diff-IR/AcR systems with P -transmit and Q-receive antennas using UWB channel measurement data.

where X is a Gaussian random variable with zero mean and variance 9. In Fig. 2, we compare the BER performance of differential IR/AcR (Diff-IR/AcR) systems with that of the proposed DSTBC-IR/AcR systems for the case of Q = 1, 2 and 4. For all the cases considered here, it is observed that there is no noticeable difference between theoretical and computer simulated BER results. The results show that in migrating from 1-transmit to 2-transmit antennas (the number of transmit antennas is denoted as “P ” in the legend), a signal-to-noise ratio (SNR) gain of about 3 dB, 2 dB and 1 dB can be achieved at BER= 10−5 with Q = 1, 2 and 4, respectively. The variation of BER with respect to P and Q also appears consistent with the general trend of MIMO system performance. B. UWB Channel Measurement Data Here, the UWB channel impulse response comprises a set of measurement data obtained from [24]. In Fig. 3, we compare the BER performance of Diff-IR/AcR systems with that of

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