IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 12, DECEMBER 2007
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Reduced-Complexity UWB Time-Reversal Techniques and Experimental Results Nan Guo, Member, IEEE, Brian M. Sadler, Fellow, IEEE, and Robert C. Qiu, Senior Member, IEEE
Abstract— This paper presents a reduced-complexity time reversal technique for ultra-wideband (UWB) communications. Time reversal takes advantage of rich scattering environments to achieve signal focusing via transmitter-side processing, which enables the use of simple receivers. The goal of this paper is to demonstrate a UWB time reversal system architecture based on experimental results and practical pulse waveform, taking into account some practical constraints, and to show feasibility of UWB time reversal. Pre-decorrelating in addition to time reversal processing is considered for a downlink multiuser configuration. Multiple transmit antennas are employed to improve the performance. Index Terms— Time reversal, ultra-wideband (UWB), autocorrelation correlation demodulation (ACD), and multiple antennas.
I. I NTRODUCTION TIMULATED by the FCC’s move that allows UWB waveforms to overlay over other systems, UWB radio has received significant attention recently [1]-[12]. Mainly due to potentially low implementation complexity, suboptimal reception strategies, such as transmitted reference (TR) [3][7], autocorrelation demodulation (ACD, it can be viewed as a differentially-encoding version of TR) [6][8] [14] and energy detection [9][10], have gotten increasing attention for complexity and cost constrained UWB applications. These schemes may be enhanced to handle multipath and intersymbol interference (ISI), but at the cost of complexity. The UWB channel impulse response (CIR) contains a large number of resolvable components coming through different paths, especially in indoor environments. However, making good use of these signal components is not straight-forward. In this paper we consider a signal focusing technique called time reversal that can turn multipath into benefit and shift part of receiver complexity burden to the transmitter side [11][13]. Please refer to [13] for the principle of time reversal. Time reversal can be viewed as a simple waveform precoding scheme that maximizes the received signal peak using a
S
Manuscript received May 11, 2006; revised November 15, 2006, June 29, 2007, and September 18, 2007; accepted September 21, 2007. The associate editor coordinating the review of this paper and approving it for publication was A. Nallanathan. This work was supported in part by the Army Research Laboratory through a Short Term Innovative Research (STIR) grant (W911NF-05-1-0468), and by the Army Research Office through a Defense University Research Instrumentation Program (DURIP) grant (W911NF-051-0111). N. Guo and R. C. Qiu are with the Center for Manufacturing Research, Tennessee Technological University (TTU), Cookeville, TN 38505, USA (email:
[email protected],
[email protected]). R. C. Qiu is also an associate professor of the Department of Electrical and Computer Engineering, TTU. B. M. Sadler is with the US Army Research Laboratory, AMSRDARL-CI-CN, 2800 Powder Mill Rd., Adelphi, MD 20783 (e-mail:
[email protected]). Digital Object Identifier 10.1109/TWC.2007.060251.
transmitter-side filter, leading to a condensed equivalent CIR in both time and space. Temporal focusing can soften the impact of ISI. Time reversal combined with antenna array results in spatial focusing that can focus energy at a desired location, enables a number of unique functions such as spatial division multiple access (SDMA) and location-based security enhancement. Combined with ACD, a reduced-complexity time reversal technique is considered in this paper to show how the performance is improved. The effectiveness of antenna array at the transmitter (called multiple input single output or simply MISO) [12] is studied by testing signal focusing and performance for both single-user and multi-user cases. While promising, applying UWB time reversal at present is extremely challenging. The main difficulties come from implementing the pre-filter in the case of such high bandwidth. It is desired for the pre-filter to accurately represent the timereversed waveform. However, high-fidelity signal representation needs high sampling rate and high resolution in magnitude, implying a prohibitively expensive solution. In this paper a digital FIR filter followed by an interpolator is considered as the pre-filter. We found that the requirements for the FIR filter’s tap spacing, coefficient resolution and filter length can be significantly reduced, while still providing good focusing property. A UWB time reversal scheme with consideration of measured channels as well as major practical issues is demonstrated. Note that the typical UWB channel models, such as the IEEE 802.15.3 channel models, do not provide location information and do not account for spatial correlation. Therefore, these models are not suitable for studying some location-related issues such as antenna array effect on the performance. The paper continues with Section II that describes the system in detail. Discrete channel models and BER analysis are provided in Section III. Experimental and numerical results with comments are given in Section IV, followed by concluding remarks in Section V. II. S YSTEM D ESCRIPTION The system under consideration is a pulse based system that uses baseband only. We assume the channel remains static during a data burst (say 100 µs [5]). Let us start with a SISO (single input single output) transmitter-receiver link with CIR h(t). An ideal low-pass filter with one-sided bandwidth W is placed at the receiver’s front-end, where W is chosen such that the impairment on the received signal due to filtering is negligible. The transmitted antipodal-modulation signal is
c 2007 IEEE 1536-1276/07$25.00
Stx (t) = α
∞ j=−∞
dj wtx (t − jTb ),
(1)
2
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 12, DECEMBER 2007
where Tb is the symbol duration, dj ∈ {±1} is j-th transmit bit, α is a power control gain factor that governs the transmit power, and wtx (t) is the transmitted symbol waveform that takes into account the pre-filter’s effect and is defined over [0, Tb ). For the ACD scheme, the original data bit qj ∈ {±1} is differentially encoded, i.e., dj = qj dj−1 . One pulse per symbol is assumed in this paper for simplification. Corresponding to wtx (t), the received non-modulating symbol waveform wrx (t) is given by wrx (t) = h(t) ⊗ wtx (t),
(2)
where “⊗” denotes convolution operation. Without loss of generality, assume the minimal propagation delay is equal to zero. The received noiseless signal is Srx (t) = h(t) ⊗ Stx (t),
(4)
j=−∞
where n(t) is a low-pass additive zero-mean Gaussian noise with one-sided bandwidth W as well as one-sided power spectral density N0 . Now let us derive the transmit symbol waveform wtx (t). Knowledge of the CIR is a key to time reversal. A perfect CIR is not available in a real system. There have been many waveform estimation strategies using transmitted pilots. Without sticking to any specific channel estimator, consider a general model for the received pilot signal. The channel estimator’s outputs can be expressed as p˜rx (t) = prx (t) + n0 (t),
(5)
where prx (t) = h(t) ⊗ ptx (t) is the received pilot signal triggered by a transmitted pilot pulse ptx (t), n0 (t) is a reduced noise modeled as low-pass additive zero-mean Gaussian noise with one-sided bandwidth W and one-sided power spectral density N0 . p˜rx (t) can be viewed as an estimate of the CIR h(t). Further processing on p˜rx (t) is made to reduce complexity of the pre-filter. Mono-bit A/D conversion is the simplest one and is adopted in this paper. After sampling at rate 1/ts , truncating to length Kts and A/D converting, we have a discrete version of the CIR estimate pˆrx (kts ) = sign(˜ prx (kts )), k = 0, 1, 2, · · · , K − 1,
(6)
where sign(·) is the sign function and pˆrx (kts ) takes values ±1. p˜rx (t) can be generated using a simple digital FIR filter with mono-bit coefficients. Practically the FIR filter uses pˆrx ((K −k)ts ), instead of pˆrx (−kts ), to set its K coefficients, implying the FIR filter has an impulse response f (t) =
K−1
Differential Encoder
(7)
k=0
To convert the discrete signal to analog signal while achieving a desired symbol waveform, an interpolating waveform ytx (t)
Antenna
Interpolator
{+1, -1}
PA
Coefficient Estimator
LNA
Conceptual diagram of time-reversal-equipped transmitter.
Fig. 1.
is used at the transmitter, leading to a transmitted symbol waveform, or equivalently, the pre-filter’s impulse response wtx (t) = f (t) ⊗ ytx (t) K−1 = pˆrx ((K − k − 1)ts ) · ytx (t − kts ) . (8) i=0
Therefore, the pre-filter can be implemented using the monobit FIR filter followed by an interpolator. Based on the above discussion, a base station transmitter with time reversal prefilter is proposed and conceptually illustrated in Fig.1. Let Td be the channel delay spread. If the pre-filter’s length in time is set to Kts (≤ Td ), then the k-th received symbol would last Kts +Td seconds (from kTb to kTb +Kts +Td ) and have a peak at time t = kTb +Kts . kTb +Kts −Tb /2 and kTb + Kts + Tb /2 are considered as the lower-end and upper-end symbol boundaries for the k-th received symbol. Denoted by TI the integration window size, then the k-th decision variable at the output of the integrator is given by kTb +Kts +TI /2 1 r(t)r(t − Tb )dt, (9) zk = √ Eb Tb kTb +Kts −TI /2 where Eb is the average per-bit signal energy. The system model described above can be extended to include a MISO downlink multi-user system that relies on time reversal and a simple power control mechanism to separate different data streams. Using index m for transmit antenna element m (1 ≤ m ≤ M ), i and i for user i and i (1 ≤ i, i ≤ I), the transmit signal from antenna element m and the receive noisy signal at user i can be expressed, respectively, as Stx,m (t) =
I
∞
α(i)
i=1
(i)
(i)
dj wtx,m (t − jTb )
(10)
j=−∞
and
r(i ) (t) M ) (i ) h(i (t) = m (t) ⊗ Stx,m (t) + n m=1
= α
∞
(i )
(i ) dj
I
α
(i)
i=1,i=i
M m=1
j=−∞
+ pˆrx ((K − k − 1)ts ) · δ(t − kts ).
FIR Filter
(3)
and the received noisy signal is r(t) = Srx (t) + n(t) ∞ = dj wrx (t − jTb ) + n(t),
Pre-filter
∞
(i) dj
j=−∞
+ n(i ) (t),
I
) h(i m (t)
M
⊗
(i ) wtx,m (t
− jTb )
) h(i m (t)
⊗
(i) wtx,m (t
− jTb )
m=1
(11)
where the summation i=1,i=i is the inter-user interference (IUI) to the i -th user. The receive signal in matrix expression
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 12, DECEMBER 2007
is given by ∞
r(t) =
Wrx (t − jTb )Adj + n(t),
(12)
j=−∞
where dj , n(t) and r(t) are column vectors with elements corresponding to I users, A is a diagonal matrix with power control gain factors α(i) ’s as its diagonal elements, and Wrx (t) (i ,i) is an I × I equivalent channel matrix with wrx (t) = M (i ) (i) m=1 hm (t) ⊗ wtx,m (t) as its element in i -th raw and i-th column.
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∞ using the Q-function Q(x) = √12π x exp(−ν 2 /2)dν. With consideration of differential encoding, totally N + 2 consecutive encoded bits will contribute to one decision variable. For convenience, for any time index k, we use notation d to represent a specific vector dTk−1,k−N = (dk+N1 −1 , · · · , dk−N )T that has length N + 2 and has impact on the k-th decision variable zk . Given wrx (t), or equivalently, given {ptx (t), ytx (t), h(t), n0 (t)}, we have Pb (ptx , ytx , h, n0 ) 1 1 qk = 1) P r(zk < 0|d; = N +2 · { 2 2 d qk =1
III. S INGLE -U SER P ERFORMANCE A NALYSIS With time reversal, a received symbol waveform contains a tail following the peak. That tail covers a number of signal peaks (or symbols) and causes ISI. To count the impact of ISI, define the tail length (in symbols) as Td − Tb /2 . (13) N = ceil Tb After some mathematical manipulation, we have ⎡ ⎤ Kts +TI /2 N 1 ⎣ zk = √ dk−j wrx (t + jTb )⎦ Eb Tb Kts −TI /2 j=0 ⎤ ⎡ N dk−j−1 wrx (t + jTb )⎦ dt + ηk , (14) ·⎣ j=0
M where wrx (t) = m=1 hm (t) ⊗ wtx,m (t) for MISO case, and ηk is a noise term given by ηk = 1 √ Eb T b +
N Kts +TI /2
Kts −TI /2
N
[
dk−j wrx (t + jTb )n(t + (k − 1)Tb )
j=0
dk−j−1 wrx (t + jTb )n(t + kTb )
j=0
+ n(t + kTb )n(t + (k − 1)Tb )]dt, Define a matrix C0 as ⎛ c0,0 c0,1 ⎜ c1,0 c1,1 ⎜ C0 = ⎜ . .. ⎝ .. .
··· ···
c0,N c0,N .. .
cN,1
···
cN,N
cN,0 1 Eb T b = cj,i .
ci,j = √
Kts +TI /2
Kts −TI /2
(15)
⎞ ⎟ ⎟ ⎟, ⎠
wrx (t + iTb )wrx (t + jTb )dt (16)
Let dk1 ,k2 = (dk1 , · · · , dk2 )T , k1 > k2 , then Equation (14) can be rewritten as zk = dTk−1,k−N −1 C0 dk,k−N + ηk .
(17)
A Gaussian approximation of the decision variable, which is rather accurate under the assumption of large time-bandwidth product [14][3] [6][8], is applied to evaluate BER performance. Once we know the decision variable’s mean and variance, BER performance can be calculated approximately
+ =
≈
1 qk = −1)} P r(zk > 0|d; 2
1
d qk =−1
2N +2 1 2N +2
P r{qk zk < 0|d}
d
·
d
⎞ qk dTk−1,k−N −1 C0 dk,k−N ⎠, Q⎝ 2 ση (d, ptx , ytx , h, n0 ) ⎛
(18)
ptx , ytx , h, n0 ) is the noise where qk = dk−1 dk , and ση2 (d, variance that can be approximately calculated using the way developed in [15, page 486], assuming sufficiently large TI W : ptx , ytx , h, n0 ) ση2 (d, N0 ≈ √ [dT C0 dk,k−N + dTk−1,k−N −1 C0 dk−1,k−N −1 ] 2 Eb Tb k,k−N TI W N02 . (19) + 2Tb Eb The last term in (19) is caused by the noisy demodulation reference. This harmful term is in proportion to the integration window size TI (< Tb ), suggesting that integration over a narrow interval can reduce the noise embedded in the decision variable. IV. E XPERIMENTAL AND N UMERICAL R ESULTS Numerical results are obtained using both analysis and simulation. The pilot pulse energy is denoted by Ep and Ep /N0 is used as an SNR indicator for the post-processed pilot pulses. Unless otherwise stated, a pre-filter length of 45 ns, mono-bit A/D conversion and receive Eb /N0 will be considered in the following BER evaluation. In addition, we assume one pulse per data symbol and W = 2 GHz. A. Practical Aspects and Impacts on Performance Measurements are necessary for dealing with location sensitive issues. A time-domain channel sounding system was used for measurements in a typical office with wooden and metallic furniture [12]. Linear antenna arrays with 20 cm spacing were used and all individual antenna elements have identical omnidirectional pattern. Distances between the transmitter and the receiver are over six meters. In this research the single-pulse waveform measured at 1 meter distance serves for both the pilot pulse ptx (t) and the interpolating pulse ytx (t). The impacts of mono-bit A/D conversion, the pre-filter length (truncation length), and the integration window size TI
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 12, DECEMBER 2007
10
0.15
Zoom-in 0.1
10
Without time reversal
−1
Main lobe 0.05
10
Simplified
−2
BER
A mplitude (V olt)
0
Ideal Simplified
0
10
-0.05
Waveform of simplified time reversal
10
−3
no time reversal no time reversal (CM4) Ep/N‘o = 20 dB Ep/N‘o = 30 dB Ep/N‘o = infinity Ep/N‘o = infinity (CM4) ideal ideal (CM4) AWGN
−4
-0.1 0
20
40
60
80
100
120
140
10
Time (ns)
Fig. 2.
Ideal AWGN
Fig. 3.
Time reversal’s temporal focusing effect (single-antenna).
10
11
12
13
14 Eb/No
15
16
17
18
BER comparison of various system settings (Tb = 30 ns).
0
0.15
10
0.1
−1
Main Lobe
10
−2
BER
Amplitude (Volt)
AWGN 0.05
0
10
−3
−0.05
10
element 1 element 2 element 3 element 4
−4
MISO
−0.1 0
20
40
60
80
100
120
140
Time (ns)
Fig. 4.
Received waveforms of MISO time reversal.
are studied. The pre-filter length is truncated to 45 ns, which is almost 2.5 times of the received RMS pulse width (typically 18 ns). The integration window size TI is chosen such that most of the main-lobe energy is captured, and in this paper TI = 6 ns is chosen as default. Ideally the received symbol waveform can be represented by [h(−t) ⊗ h(t)] ⊗ [ptx (−t) ⊗ ptx (t)], implying the main lobe in the received waveform almost has the same profile as the autocorrelation waveform ptx (−t) ⊗ ptx (t). Temporal focusing is visually demonstrated in Fig.2 based on the result measured in the office, where “Ideal” refers to using a perfect waveform prx (−t) to set the pre-filter, and “Simplified” corresponds to a practical time reversal scheme with mono-bit A/D conversion, Nyquist rate sampling (4 Gs/s), truncation in length, and Ep /N0 = 30 dB. Note that for either “Ideal” and ”Simplified” cases the main-lobe waveforms are shaped almost the same. A semi-analytical approach is adopted to evaluate BER. The pre-filter’s coefficients are generated based on measured pilot waveform and simulated noise. To obtain average performance, J( 1) realizations of the coefficients are used to generate J receive symbol waveforms based on measured
10
−5
10
Fig. 5.
11
12
13
14 Eb/No
15
16
17
18
BER comparison of MISO and SISO (Tb = 30 ns).
CIR. Then, for a given value of Eb /N0 , each receive symbol waveform yields a BER from calculation described in last section. Finally, the overall BER is obtained by averaging over these J individual BERs. A BER performance comparison based on ACD for Tb = 30 ns under various channel conditions is presented in Fig.3. For comparison purpose, results generated from 100 realizations of IEEE802.15.3a model CM4 are included. The reference curve labeled “AWGN” refers to ordinary ACD under no-ISI condition, and its integration window is simply set to Tb . The results suggest that time reversal is a simple and effective means to combat ISI, and simplifying from the “Ideal” time reversal only leads to a performance penalty of less than one dB at Ep /N0 ≥ 30 dB and BER=10−3 . Also, the impact of noise embedded in the received pilot waveform can be seen: performance improves as Ep /N0 increases, but with the system configuration considered in this paper, very little improvement can be expected when Ep /N0 is over 30 dB. To isolate the central issues, in the following sufficiently large Ep /N0 is set.
IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 12, DECEMBER 2007
C. Time-Reversal Multi-user System Time reversal can enable SDMA, although its temporalspatial focusing mechanism is not able to remove ISI and IUI completely. According to our indoor trials, an SDMA multiuser system solely based on time reversal can suffer from strong inter-user interferences because of cross-correlations between the radio links, and even increasing the number of transmit antennas cannot fully solve the problem. It is difficult to find an optimal multi-user waveform precoding solution and it is worth modifying the time-reversal based SDMA system. Simply sampling the signal peaks at the receiver can reduce IUI impact since the SINR (Signal to Interference plus Noise Ratio) at the instant of a signal peak is higher. To reduce the IUI impact completely, a stage of pre-decorrelating transformation can be added to the transmitter to remove the IUI at the instants of signal peaks. Recall the original downlink multi-user system described in equation (12) where the diagonal matrix A plays power control function. In the modified scheme the matrix A plays decorrelating and is given by −1 (T0 ), A = Wrx
(20)
10
10
BER
B. MISO Time Reversal The use of antenna array at the transmitter is a simple way to achieve performance gain. Following each transmit antenna element is a pre-filter with corresponding setting. Shown in Fig.4 is the received waveforms of the MISO time reversal, where a 4-element linear array with 20-cm separation is employed at the transmitter. Comparing to the SISO case (see Fig.2 ), MISO not only leads to a stronger main lobe, but also strengthens the signal focusing, which is especially meaningful when ISI exists. With the same total transmit power, it is shown that the main-lobe energy is roughly in proportion to the number of transmit antenna elements, which is because the individual main lobes coherently combine. As data rate goes higher, ISI would have more impact on the performance. A BER performance comparison for Tb = 30 ns is presented in Fig.5, where the label “element m” refers to a SISO configuration using antenna element m. For fair comparison, in the plot Eb /N0 is the receive per-bit SNR for MISO time reversal and the AWGN case, and it has been assumed that all time reversal schemes have the same transmit power. Note that the SISO time reversal schemes may have different receive Eb /N0 s, given the same level of transmit Eb /N0 . In our office environment, at a bit rate 1/Tb = 33.3 bps (implying some ISI suffering), the use of MISO time reversal results in more than 5 dB gain at BER = 10−3 over any SISO time reversal schemes. Furthermore, with ACD receiver, MISO time reversal is able to beat the no-ISI AWGN scheme. This is because, for MISO time reversal, as the number of antenna elements continuously increases, the receive signal energy will eventually condense in the main lobe, leading to a reduction of ISI and an increased SNR in the decision variable. It can be concluded that, compared to SISO, the benefit of using MISO time reversal is three-fold: (1) increased SNR because of strengthened main lobe, (2) reduced ISI impact due to increase of focusing, and (3) enhanced link robustness gained from spatial diversity.
10
10
10
Fig. 6.
5
0
3 users 3 users, ideal 3 users, peak 3 users, pre−decorr 3 users, ideal, pre−decorr 4 users, pre−decorr
−1
−2
−3
−4
12
13
14
15
16
17 Eb/No
18
19
20
21
22
Multi-user performance (Tb = 50 ns). (i,i)
where T0 is the time of the signal peak at which wrx (t)’s, 1 ≤ i ≤ I, reach their maximums. Note that this decorrelating matrix only inverts the equivalent channel matrix Wrx (t) at the instants of signal peaks. Shown in Fig.6 are a set of simulation results for downlink multi-user schemes with different settings, where “pre-decorr” refers to the proposed time-reversal SDMA system with predecorrelating at the transmitter and peak sampling at the receivers. The BER performance is mainly affected by prefiltering accuracy, ISI, IUI as well as integration window size TI . Two integration window sizes are considered here: TI = 6 ns (default) and TI = 0.1 ns (for peak-sampling method). Based on these trials we have the following observations: (1) the mono-bit time-reversal scheme performs worst as expected; (2) the peak-sampling scheme yields improved performance; (3) pre-decorrelating combined with peak sampling improves the performance further and leads to more than 1.5 dB gain over the the mono-bit time-reversal scheme; and (4) in the ideal time-reversal scenario, applying pre-decorrelating plus peak sampling does not provide additional gain. V. C ONCLUSION This paper proposes a simplified time reversal technique combined with the ACD scheme, and provides the baseline performances based on experimental results and practical pulse waveform. The equivalent discrete channel models and the BER formulas for a single-user scenario are derived. Major issues associated with implementation of time reversal, such as sampling, quantization, truncation length and interpolation, are discussed. Thanks to multipath scattering mechanism, SDMA (also location-based information leaking resistance) can be effectively provided in the time reversal system without consuming additional radio resources. It has been shown that by adding a stage of pre-decorrelating transformation, the downlink multi-user system with simplified time reversal and simple ADC receivers performs remarkably. Our results suggest that the proposed reduced-complexity time reversal works satisfactorily, thus it is very promising for applications where receiver complexity/cost, performance and security are highly concerned.
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R EFERENCES [1] R. A. Scholtz, “Multiple access with time-hopping impulse modulator (invited paper),” in Proc. IEEE MILCOM’93, Oct. 1993, pp. 447-450. [2] 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. 679-689, April 2000. [3] J. D. Choi and W. S. Stark, “Performance of ultra-wideband communications with suboptimal receivers in multipath channels,” IEEE J. Sel. Areas Commun., vol. 20, pp. 1754-1766, Dec. 2002. [4] R. C. Qiu, H. P. Liu, X. Shen, and M. Guizani, “Ultra-wideband for Multiple Access,” IEEE Commun. Mag., vol. 43, pp. 80-87, Feb. 2005. [5] A. F. Molisch, J. R. Foerster, and M. Pendergrass, “Channel models for ultrawideband personal area networks,” IEEE Wireless Commun. Mag., [see also IEEE Personal Commun. Mag.], pp. 14-21, Dec. 2003. [6] Y. Chao and R. A. Scholtz, “Optimal and suboptimal receivers for Ultrawideband transmitted reference systems,” in Proc. IEEE Globecom’03, Dec. 2003, pp. 759-763. [7] S. Hoyos, B. M. Sadler, and G. R. Arce, “Monobit digital receivers for ultrawideband communications,” IEEE Trans. Wireless Commun., vol. 4, pp. 1337-1344, July 2005. [8] N. Guo and R. C. Qiu, “Improved autocorrelation demodulation receivers
based on multiple-symbol detection for UWB communications,” IEEE Trans. Wireless Commun., vol. 5, pp. 2026-2031, Aug. 2006. [9] Y. Souilmi and R. Knopp, “On the achievable rates of ultra-wideband PPM with non-coherent detection in multipath environments,” in Proc. IEEE ICC’03, vol. 5, May 2003, pp. 3530-3534. [10] M. Weisenhorn and W. Hirt, “Robust noncoherent receiver exploiting UWB channel properties,” in Proc. IEEE UWBST’04, May 2004, pp. 156-160. [11] T. Strohmer, M. Emami, J. Hansen, G. Pananicolaou, and A. J. Paulraj, “Application of time-reversal with MMSE equalizer to UWB communications,” in Proc. Globecom’04, Dec. 2004, pp. 3123-3127. [12] R. C. Qiu, C. Zhou, N. Guo, and J. Q. Zhang, “Time reversal with MISO for ultra-wideband communications: experimental results,” IEEE Antennas Wireless Propag. Lett., vol. 5, pp. 269-273, Dec. 2006. [13] M. Fink, “Time reversal of ultrasonic fields–part I: basic principles,” IEEE Trans. Ultrason., Ferroelectr. Freq. Control, vol. 39, no. 5, pp. 555-566, Sept. 1992. [14] M. K. Simon, S. M. Hinedi, and W. C. Lindsey, Digital Communication Techniques - Signal Design and Detection. Englewood Cliffs, NJ: Prentice Hall, 1994.
