In this section, the transceiver structure of a proposed SIMO OTR-UWB system is described. As discussed in previous section, OTR-UWB data rate is limited due to ISI effect; therefore, its performance and capacity are degraded. We would like the ISI to be as low as possible and the system capacity to be as high as possible. The system capacity can be increased by using a SIMO structure. A SIMO-OTR system configuration is illustrated in Figure 3. Let *h*_{
i
}(*t*) denotes the CIR between antennas at the transmitter and the *i* th antenna at the receiver, *p*(*t*) is pulse shaping function, and \mathsf{\text{sign}}\left({h}_{i}^{*}\left(-t\right)\right) is the corresponding prefilter code employed in the *i* th antenna branch at the transmitter. In this method, data are transmitted by a transmitter comprising of a single antenna and received by multiple antennas. The transceiver structure is based on spatial focusing property of OTR-UWB system. One-bit TR retains the spatial property of classic time reversal [7]. In office environment, this directivity dropping (20 dB) can be achieved by setting the spacing between any two adjacent antenna elements greater than 20 cm, which corresponds to the half wavelength of the lowest frequency [2]. In rectangular metal cavity environment, directivity of time reversal drops by 20 dB when the antenna is 5 cm away from the intended receiver [8]. Low spatial focusing gain of OTR-UWB system at distance *r* away from the intended receiver indicates that a nearby receiver at that location would not be able to detect the signal. The receiving antenna should not be placed in spatial focusing depth of each other, therefore the channels \left({h}_{1}\left(t\right),{h}_{2}\left(t\right),...,{h}_{{N}_{r}}\left(t\right)\right) are almost uncorrelated.

Data rate and power in each branch of transmitter decrease to *R*/*N*_{
r
} and *P*_{0}/*N*_{
r
}, respectively, where *N*_{
r
} is the number of received antenna. Therefore, the ISI power in the proposed scheme reduces due to lower rate (greater symbol interval) and the symbols can successfully be decoded by simple sampling the received signal at the appropriate instance.

The transmitted signal in proposed SIMO OTR-UWB is

\begin{array}{cc}\hfill s\left(t\right)& ={x}_{1}\left(t\right)\otimes \mathsf{\text{sign}}\left({h}_{{}^{1}}^{*}\left(-t\right)\right)+{x}_{2}\left(t\right)\otimes \mathsf{\text{sign}}\left({h}_{2}^{*}\left(-t\right)\right)+\cdots +{x}_{{N}_{r}}\left(t\right)\otimes \mathsf{\text{sign}}\left({h}_{{N}_{r}}^{*}\left(-t\right)\right)\hfill \\ =\sum _{i=1}^{{N}_{r}}{x}_{i}\left(t\right)\otimes \mathsf{\text{sign}}\left({h}_{{}^{i}}^{*}\left(-t\right)\right)\hfill \end{array}

(15)

And the output of *l* th antenna at the receiver is denoted by

\begin{array}{cc}\hfill {y}_{l}\left(t\right)& ={h}_{l}\left(t\right)\otimes \sum _{i=1}^{{N}_{r}}{x}_{i}\left(t\right)\otimes \mathsf{\text{sign}}\left({h}_{{}^{i}}^{*}\left(-t\right)\right)+n\left(t\right)\hfill \\ =\underset{\mathsf{\text{Signal}}}{\underset{\u23df}{x\left(t\right)\otimes {h}_{l}\left(t\right)\otimes \mathsf{\text{sign}}\left({h}_{{}^{1}}^{*}\left(-t\right)\right)}}+\underset{\mathsf{\text{Cochannelinterference}}}{\underset{\u23df}{{h}_{l}\left(t\right)\otimes \sum _{i=1,i\ne l}^{{N}_{r}}{x}_{i}\left(t\right)\otimes \mathsf{\text{sign}}\left({h}_{{}^{i}}^{*}\left(-t\right)\right)}}+\underset{\mathsf{\text{Noise}}}{\underset{\u23df}{n\left(t\right)}}\hfill \end{array}

(16)

{y}_{l}\left(t\right)=x\left(t\right)\otimes {h}_{l}\left(t\right)\otimes \mathsf{\text{sign}}\left({h}_{{}^{1}}^{*}\left(-t\right)\right)+{i}_{s}\left(t\right)+n\left(t\right)=x\left(t\right)\otimes {h}_{\mathsf{\text{eq}}}\left(t\right)+{i}_{s}\left(t\right)+n\left(t\right)

(17)

where *i*_{
s
}(*t*) denotes cochannel interference part in (16). As it is seen the output of each branch in SIMO OTR-UWB transceiver is similar to a SISO OTR-UWB system, but the power and rate of each branch at transmitter is decreased and an interfering term is added to each of them. With the constant transmitted power *P*_{0}, the associated power for each transmitter branch of SIMO OTR-UWB reduces to *P*_{0}/*N*_{
r
}. There is usually a power scaling factor included in the OTR code for SIMO to make sure that the transmit power remain the same, after OTR precoding. Without loss of generality, we suppose that this scaling factor is equal to 1.

If the rate of each branch in transmitter is less than coherence bandwidth (*R*/*N*_{
r
} < *B*_{
c
}), the ISI does not occur. Therefore, the received SNR of proposed SIMO-OTR is

\mathsf{\text{SNR}}=\frac{{P}_{0}\pi}{4{N}_{r}\times \left({P}_{N}+{I}_{s}\right)}\left[\frac{\left(4-\pi \right)\left(1-{\gamma}^{L}\right)}{\left(1-\gamma \right)}+{\left(\frac{1-{\gamma}^{L/2}}{1-{\gamma}^{1/2}}\right)}^{2}\right]

(18)

Comparing (18) and (14) in rate higher than coherence bandwidth shows that the SIMO OTR performance is better than SISO OTR if ISI_{pr} + ISI_{po} ≥ (*N*_{
r
}-1)*P*_{
N
}. In the other world, in low transmitted power, the ISI power is very less than noise power and the noise power is the dominant term (ISI_{pr} + ISI_{po} ≪ (*N*_{
r
}-1)*P*_{
N
}). Consequently, the proposed SIMO technique performs better than conventional SISO at higher values of *P*_{0}/*N*_{0} and data rate.

It is possible to compensate the reduced power using a receiver with more sensitivity. But, more sensitive receiver cannot compensate for the ISI effect and in presence of ISI, equalizer or MMSE receiver should be used. The optimal receiver is the maximum likelihood sequence estimator (MLSE), since the computational complexity grows exponentially with channel length. Most channels of practical interest require too much computation for MLSE to be feasible. Therefore, suboptimal schemes like equalizers should be used to compensate for the ISI effects. Traditional equalizers use training sequences to adjust the tap weights. However, using training sequences decrease bandwidth efficiency.

We define a quality factor (*ϑ*) for rates greater than coherence bandwidth (*R*/*N*_{
r
} < *B*_{
c
}, *R* > *B*_{
c
}) ISI condition, which is the ratio of SIMO SNR to SISO SINR:

\begin{array}{cc}\hfill \vartheta & =\frac{\mathsf{\text{SN}}{\mathsf{\text{R}}}_{\mathsf{\text{SIMO}}}}{\mathsf{\text{SIN}}{\mathsf{\text{R}}}_{\mathsf{\text{SISO}}}}\approx \frac{\frac{{P}_{0}\pi}{4{N}_{r}\times \left({P}_{N}+{I}_{s}\right)}\left[\frac{\left(4-\pi \right)\left(1-{\gamma}^{L}\right)}{\left(1-\gamma \right)}+{\left(\frac{1-{\gamma}^{L/2}}{1-{\gamma}^{1/2}}\right)}^{2}\right]}{\frac{\frac{\pi {P}_{0}\left(1-\gamma \right)}{4}\left[\frac{\left(4-\pi \right)\left(1-{\gamma}^{L}\right)}{\left(1-\gamma \right)}+{\left(\frac{1-{\gamma}^{L/2}}{1-{\gamma}^{1/2}}\right)}^{2}\right]}{{P}_{N}\left(1-\gamma \right)+2{P}_{0}{L}_{0}-{P}_{0}\frac{\left(1-{\gamma}^{\left({L}_{0}+1\right)M}\right){\gamma}^{L}}{1-{\gamma}^{M}}-{P}_{0}\frac{\left(1-{\gamma}^{-\left({L}_{0}+1\right)M}\right){\gamma}^{L}}{1-{\gamma}^{-M}}}}\hfill \\ =\frac{1/\left({N}_{r}\times \left({P}_{N}+{I}_{s}\right)\right)}{\frac{\left(1-\gamma \right)}{{P}_{N}\left(1-\gamma \right)+2{P}_{0}{L}_{0}-{P}_{0}\frac{\left(1-{\gamma}^{\left({L}_{0}+1\right)M}\right){\gamma}^{L}}{1-{\gamma}^{M}}-{P}_{0}\frac{\left(1-{\gamma}^{-\left({L}_{0}+1\right)M}\right){\gamma}^{L}}{1-{\gamma}^{-M}}}}\hfill \\ =\frac{{P}_{N}\left(1-\gamma \right)+2{P}_{0}{L}_{0}-{P}_{0}\frac{\left(1-{\gamma}^{\left({L}_{0}+1\right)M}\right){\gamma}^{L}}{1-{\gamma}^{M}}-{P}_{0}\frac{\left(1-{\gamma}^{-\left({L}_{0}+1\right)M}\right){\gamma}^{L}}{1-{\gamma}^{-M}}}{{N}_{r}\times \left({P}_{N}+{I}_{s}\right)\left(1-\gamma \right)}\hfill \\ =\frac{{P}_{N}}{{N}_{r}\left({P}_{N}+{I}_{s}\right)}+\frac{{P}_{0}}{{N}_{r}\times \left(1-\gamma \right)\left({P}_{N}+{I}_{s}\right)}\left[2{L}_{0}-\frac{\left(1-{\gamma}^{\left({L}_{0}+1\right)M}\right){\gamma}^{L}}{1-{\gamma}^{M}}-\frac{\left(1-{\gamma}^{-\left({L}_{0}+1\right)M}\right){\gamma}^{L}}{1-{\gamma}^{-M}}\right]\hfill \end{array}

(19)

From (19) it is seen that for *γ* < 1 and *L*, *M* ≫ 1, *L* > *M*, the quality factor is

\vartheta \approx \frac{{P}_{N}}{{N}_{r}\left({P}_{N}+{I}_{s}\right)}+\frac{{P}_{0}}{{N}_{r}\times \left(1-\gamma \right)\left({P}_{N}+{I}_{s}\right)}\left[2{L}_{0}-\frac{{\gamma}^{-\left({L}_{0}+1\right)M+L}}{{\gamma}^{-M}-1}\right]

(20)

For high *P*_{0}/(*P*_{
N
} + *I*_{
s
}), the quality factor is

\vartheta \approx \frac{{P}_{0}\left[2{L}_{0}-\frac{1}{{\gamma}^{-M}-1}\right]}{{N}_{r}\times \left(1-\gamma \right)\left({P}_{N}+{I}_{s}\right)}

(21)

As it is seen from (20) and (21), *ϑ* increases by increment of *P*_{
0
}/(*P*_{
N
} + *I*_{
s
}) and *L*_{0}. But increment of the received antenna number decreases the quality factor because of power reduction in each branch of SIMO transmitter. The reduced power can be compensated using a receiver with more sensitivity.

For distance larger than 3 m, *I*_{
s
} is negligible (Appendix A), if *γ* < 1 and *L* ≫ 1, the SNR, SINR and *ϑ* of SIMO-OTR is

\mathsf{\text{SNR}}=\frac{\pi {P}_{0}}{4{N}_{r}{P}_{N}}\left[\frac{\left(4-\pi \right)}{\left(1-\gamma \right)}+\frac{1}{{\left(1-{\gamma}^{1/2}\right)}^{2}}\right]

(22)

\vartheta \approx \frac{1}{{N}_{r}}+\frac{{P}_{0}\left(2{L}_{0}-1/\left({\gamma}^{-M}-1\right)\right)}{{N}_{r}\times \left(1-\gamma \right){P}_{N}}

(23)

If the received antennas do not placed in focusing depth, the interfering term in (13) is considerable. Similar to previous section, the power of this interfering part with consideration of directivity factor (*d*_{
f
}) is

\begin{array}{cc}\hfill {I}_{s}& =\frac{{P}_{0}}{{N}_{r}}{\left\{E{\left(\sum _{\begin{array}{c}i=1\\ i\ne l\end{array}}^{{N}_{r}}{h}_{l}\left(t,{r}_{i}\right)\otimes sign\left({h}_{i}^{*}\left(-t,{r}_{1}\right)\right)\right)}^{2}\right\}}_{t=\left(L-1\right)\Delta}<\frac{{P}_{0}}{{N}_{r}}{d}_{f}{\left\{E{\left(\sum _{\begin{array}{c}i=1\\ i\ne l\end{array}}^{{N}_{r}}{h}_{l}\left(t,{r}_{l}\right)\otimes \mathsf{\text{sign}}\left({h}_{i}^{*}\left(-t,{r}_{1}\right)\right)\right)}^{2}\right\}}_{t=\left(L-1\right)\Delta}\hfill \\ <\frac{\left({N}_{r}-1\right){d}_{f}}{{N}_{r}}{P}_{0}E{\left(\sum _{j=0}^{L-1}{\alpha}_{j}{p}_{j}{}_{p}^{j}\right)}^{2}<\left({N}_{r}-1\right)S{d}_{f}=\frac{\pi {P}_{0}\left({N}_{r}-1\right){d}_{f}}{4}\left[\frac{\left(4-\pi \right)\left(1-{\gamma}^{L}\right)}{\left(1-\gamma \right)}+{\left(\frac{1-{\gamma}^{L/2}}{1-{\gamma}^{1/2}}\right)}^{2}\right]\hfill \end{array}

(24)

where *h*_{
i
}(*t*, *r*_{
i
}) is the CIR between transmitter and receiver located in *r*_{1}.

It is possible to accept a limited level of ISI, data rate of each branch greater than coherence bandwidth, and design the system with lower complexity (smaller *N*_{
r
}). In this case, the received SINR of the proposed SIMO-OTR scheme is

\begin{array}{cc}\hfill \mathsf{\text{SINR}}& =\frac{\frac{\pi {P}_{0}\left(1-\gamma \right)}{4{N}_{r}}\left[\frac{\left(4-\pi \right)\left(1-{\gamma}^{L}\right)}{\left(1-\gamma \right)}+{\left(\frac{1-{\gamma}^{L/2}}{1-{\gamma}^{1/2}}\right)}^{2}\right]}{\left({P}_{N}+{I}_{s}\right)\left(1-\gamma \right)+\left({P}_{0}/{N}_{r}\right)\sum _{j=1}^{{L}_{0I}}\left\{2-{\gamma}^{L-jM}-{\gamma}^{L+jM}\right\}}=\frac{\frac{\pi {P}_{0}\left(1-\gamma \right)}{4}\left[\frac{\left(4-\pi \right)\left(1-{\gamma}^{L}\right)}{\left(1-\gamma \right)}+{\left(\frac{1-{\gamma}^{L/2}}{1-{\gamma}^{1/2}}\right)}^{2}\right]}{\left({P}_{N}+{I}_{s}\right)\left(1-\gamma \right){N}_{r}+{P}_{0}\sum _{j=1}^{{L}_{0I}}\left\{2-{\gamma}^{L-jM}-{\gamma}^{L+jM}\right\}}\hfill \\ \approx \frac{\frac{\pi {P}_{0}\left(1+{\gamma}^{1/2}\right)}{4\left(1-{\gamma}^{1/2}\right)}{\left(1-{\gamma}^{L/2}\right)}^{2}}{\left({P}_{N}+{I}_{s}\right)\left(1-\gamma \right){N}_{r}+2{P}_{0}{L}_{0I}-\frac{{P}_{0}\left(1-{\gamma}^{\left({L}_{0I}+1\right)M}\right){\gamma}^{L}}{1-{\gamma}^{M}}-\frac{{P}_{0}\left(1-{\gamma}^{-\left({L}_{0I}+1\right)M}\right){\gamma}^{L}}{1-{\gamma}^{-M}}}\hfill \end{array}

(25)

where *L*_{0I}is the same as *L*_{0} in limited ISI scenario.

In this case, it is possible to use of optimal one-bit time reversal UWB. In optimal OTR scheme, optimum number of taps is used to design the prefilter to obtain the best system performance. It is shown that in the temporal domain the performance of the one-bit TR system does not necessary improve when the number of prefilter coefficients increases [7, 9]. Since the prefilter length selection criteria is based on the output SIR maximization, which is a highly nonlinear function of code rate (or symbol interval). It was observed that the optimal code length is the same as the symbol interval in the low rate scenario. However, in the high rate scenario, the optimal code length is just equal to multiple symbol intervals [6]. Therefore, due to lower rate of each branch in proposed SIMO-OTR scheme, its required prefilter length is smaller than SISO-OTR scheme. This means simple prefilter and smaller rat in feedback link for tap coefficients.

### 4.1. Analytical and simulation results

In this section, performance of SISO and proposed SIMO OTR-UWB systems are evaluated. The most widely adopted UWB multipath channel model has been proposed by the IEEE 802.15.3a Task Group [10]. Therefore, the CIR is simulated (generated) according to IEEE 802.15.3a channel model. This channel model is designed for SISO scenario; the extension to a SIMO scheme is achieved by assuming that the SIMO channel parameters are independent and identically distributed from the same statistical model. CM3 channel model is used in all simulation.

The performance of SISO OTR-UWB communication system in CM3 channel is shown in Figure 4, indicating a satisfying agreement between analytical and simulation results. Figure 5 shows the performance of SIMO OTR-UWB communication system in bit rate of 50 Mbps. As it is seen in SISO OTR the maximum received SINR is about 15 dB and for high *P*_{0}/*N*_{0} quality factor is very high. By increasing *P*_{0}/*N*_{0} in Equation 17, *P*_{0}/*N*_{0} ≫ 1 *or P*_{0}/*N*_{0} → ∞, it could be seen that the SINR is not increased and limited to SIR. Therefore, the performance is not improved by increasing *P*_{0}/*N*_{0} and so the system capacity is limited. But, for proposed SIMO, the performance depends on *P*_{0}/(*N*_{0}*N*_{
r
}) and more increasing *P*_{0}/*N*_{0} results in more performance improvement. Figure 6 shows the bit error rate (BER) of SIMO and SISO OTR-UWB system in a CM3 UWB channel at data rate of 50 and 25 Mbps. As it is seen for a certain *P*_{0}/*N*_{0}, SIMO OTR performance is better than SISO OTR. As it is observed in BER of 10^{-4}, SIMO-OTR with *N*_{
r
} = 3 rate and of 25 Mbps is 2 dB better than SISO; also the performance of SIMO-OTR with *N*_{
r
} = 6 and rate of 50 Mbps is 1.8 dB better than SISO-OTR. Figure 7 shows that in higher SNR, the SIMO-OTR performance is better than SISO-OTR and in BER of 10^{-4}, SIMO-OTR with *N*_{
r
} = 2 and rate of 50 Mbps is 0.8 dB better than SISO-OTR with two transmitted antennas. Performance comparison of SIMO-OTR in BER of BER of 10^{-4} shows that the performance of SIMO-OTR with six received antennas is only 1 dB better than SIMO-OTR with two received antennas.