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The evaluation of System Performance of UWB (ultra-wide band) jointing in MC (multi-carrier) signaling in correlated environments is presented in the report. The correlated Nakagami-m statistical distribution for the multipath fading model is assumed in this scenario. In fact to establish the model for analyzing in this article is using MC-CDMA (multi-carrier code-division multiple-access) system characterization combined with a UWB scheme. The average BER (bit error rate) is calculated and compared to a special case of previously published results. Studied results from this paper can be implied to approve the system performance for a UWB system combined with a MC-CDMA wireless communication system. It is worth noting that the Nakagami-m distributed fading parameter significantly dominates UWB system performance when it cooperates with MC signaling under a fading environment. Finally, it is worthy of noting that when the SNR (signal-to-noise ratio) at system’s receiver reaches a preset high threshold value, the parameter of power decay ratio effect could be not included.

The UWB (ultra-wide band) system is explained as a system which the impulse wireless radio system transmits data without a sinusoidal modulating signal, it is instead of sub-nanosecond pulses, recently raised growing interest. The UWB system, which has joint bandwidth in excess 1 GHz with very low PSDs (power spectral densities) but without inducing significant extra interference to incumbent subscribers, has emerged as a solution for the TG3a (IEEE 802.15a) standard to provide low complexity, low power consumption, low cost and high data rate among WPAN (Wireless Personal Area Networking) devices. The modulating sub-nanosecond pulses obtain a 10 dB bandwidth benefit which exceeds 500 MHz or on the order of one to several Gigahertz and is traditionally 20% of their center frequency. It is known that UWB signals have large bandwidth when appropriately modulated with spreading spectrum techniques and provide low error intercept and detection probability as well as robustness to jamming. UWB signals occupy large bandwidth and account mainly for both the drawback and advantages of UWB radio systems. For the purpose of minimizing interference to these systems, UWB system operation must strictly follow certain necessary limitations, e.g., transmission range, power control implementation and achievable data rate. In order to break through these limitations, UWB systems could be developed to coexist with wideband and narrowband systems already allocated for a dedicated frequency spectrum, such as multi-carrier modulated schemes, SS-CDMA (spreadspectrum code-division multiple-access), DS-CDMA (direct-sequence CDMA) or MB-OFDM (multi-band orthogonal frequency division multiplexing) systems. UWB channels are characterized as frequency selective while the waveform of UWB system propagates over a fading channel, because of their extremely high bandwidth, as reported in [

The aforementioned research motivated this paper’s contribution in analyzing MC (multi-carrier) techniques combined with the UWB system performance over the correlated-Nakagami-m fading channel. The correlation phenomenon caused by the distance between subchannels is not sufficient, a fact validated in [

It is known that the MC (multi-carrier) system can be incorporated in the UWB system, e.g. the MC-CDMA system overlay for the UWB communication system [

The binary source data sequence, , where indicates the integer part of , of the k-th user is first spread by a set of random time-hopping sequences, , with processing gain , where and stand for the bit duration and the hopping time, respectively.

Based on TH-PPM scheme utilization for the UWB system, the BER of the coherent BPSK (binary phase shift keying) modulation with AWGN (additive white Gaussian noise) channel is calculated in [

where, in which represents the time shift of the comparison between the pulse in a data bit “1” to a data bit “0”, is defined as the auto-correlation function of the received pulse signal and represents the SNR (signal-to-noise ratio) of the frame. The BER system for evaluating the communication system with BPSK modulation can be expressed as

where is the guard interval time, indicates the bit symbol time without guard interval time 1 (1 Since PPM (pulse position modulation) is adopted for UWB system, guard interval exists between each pulse position is necessary). Consider a channel that has frequencyselective fading, the channel impulse response is then modeled as a linear filter and can be written as

where are mutually uncorrelated, denotes the Dirac delta function, is the time delay of the 1-th diversity path, the phase , are uniformly distributed over , and the amplitude is assumed characterized as Nagakami-m distributed, which is denoted by [

where is the Gamma function, denotes the average power of the received signal and are the fading parameters, with representing the fading severity. The smaller the values are, the greater the fading in the channel environment. Assuming that the fading parameters of all taps are identical, i.e., for and exponential MIP (multipath intensity profile), , where is the average power of the first channel path, indicates the average power decay rate, however, corresponds to the constant MIP condition. To claim that the pulsed-UWB channel is modeled as correlated Nakagami-m fading in this report is necessary. Generally the relevant UWB techniques are applied in a short distance and the standard channel model with IEEE 802.15.3a is always the one model suitable for the UWB system. It is reasonable to adopt the correlated-Nakagami-m distributed system as the channel model in the indoor communications case [

The approximate expression is adopted to analyze the system performance showing that the fading intensity follows the Nagakami-m distribution with fading parameter , and mean power , i.e., , where and are given by [

respectively, where , while δ = 0, and then , and . It is worth noting that the accuracy of the approximate expression becomes better when the number of multipaths, L, increase. After the channel fading model is modeled and established, we describe the receiver model, as shown in

where represents the sub-channel fading intensity with attenuation defined in (5), and is the phase of the nth subcarrier of the kth user. The channel impulse response taps, , are assumed i.i.d (identically independent distributed) for different users with the same index, thus, and , is the time misalignment of user with respect to the referenced user at the receiver which is i.i.d for different users and uniformly distributed in [qT_{b}, (q + 1)T_{b}]. n(t) represents the AWGN (additive white Gaussian noise) having a two-side power spectral density of .

Without loss of generality, the 0th user is considered the referenced user operating on each subcarrier suffering from a flat fading channel. Assuming that the channel fading, , and phase shift, , are constant over one bit duration. The decision variable, , of the 0th data bit of the referenced user after the coherent demodulation is given by

where the weighting factors, , of the 0th user for an

MRC (maximal ratio combining) diversity is employed in detection scheme, thus, let

By multiplying each subcarrier with the weighting factor , and the decision variable, , after despreading the signal of the 0th user can be expressed as

where the first term, , denotes the desired signal of the referenced user can be calculated as (assuming that the case of )

the second term, , in (11) indicates the MAI (multiple-access interference) which is including all the disturbed terms from the other located in the same BS (base station), and the last term, , is the background noise component. Assuming that the number of multi-carriers is large enough, the chips for spreading code and the input data symbol are modeled as random variables, conditioned on channel state . Hence, the MAI is able to be well approximated by zero mean Gaussian statistics and with variance obtained as [

where is given in (7). By utilizing the CLT (central limit theorem) definition, the random variable of can also be approximated using a zero mean Gaussian distribution with variance yields as

Once the desired value and all statistical characteristic variances for the interference of the referenced user are computed, the instantaneous SINR (signal-to-interferencenoise ratio) can then be determined as

where is the SNR per transmission bit. Now, the phenomena of the correlation characteristic between transmission branches should be taken into account. Different delay times could be caused for each path, when they arrive at the receiver via different propagation routes. Each path with the same delay may meet correlated fading among the separated paths. All of the possibilities may occur in the path of each subcarrier or the multi-path channel. In this article, the correlation will be ignored for the multi-path channel, since the spatial technique is not considered, thus, , are mutually uncorrelated for different. It is believed that the correlation between the propagation channel cannot be canceled while the MC scheme is applied. However, the CFO (carrier frequency offset) is negligible when the channel estimation is considered. Hence, the correlation of and is expressed as

It can be shown that in the case of, in order to satisfy the case avoiding correlation, , the relationship between paths of

must be satisfied, where nonzero integer . Among all sub-carriers, for a fixed sub-carrier n, only other subcarriers are uncorrelated to sub-carrier n. It is well known that when a random variable is modeled with Nakagami-m distribution, then the square of , is a random variable characterized as Gamma distribution. Thus, the pdf of SINR, , shown in (15) can be denoted as

where and have been described in (4). By using of the symbol of express the summation of all the fading components at the system output, i.e.,

, since are mutually uncorrelated, the statistical results of can be shown as, where

and

where the first path arrival average power at the receiver output, and are the same as shown in (7). By following the average system BER procedure for a coherent BPSK signaling system, the BER evaluation, , for UWB system adoption in a MCCDMA system equipped with a MRC receiver can be calculated using the formula shown in [

where is the well known Gaussian Q-function, alternatively expressed as

and by substituting (15) and (18) into (21), the average system BER result, , of MC-CDMA system with MRC diversity scheme combined with UWB signaling can be obtained as (see the Appendix )

where, with conditions of and.

In this section the UWB system performance combined with a MC-CDMA system is illustrated by implementation with a computer software package. Some of the parameters will be taken into account for accuracy validation and performance comparison, e.g., the number of received resolvable multi-paths, , the user number, , the fading parameter, , and the sub-carrier number, , of the MC-CDMA system. The role of correlation, which is always fixed at , is implicitly applied all through the numerical operation for evaluating the system performance. The bit duration is considered equal to that of a frame time, i.e.,. First the BER versus SNR, , of the transmitted bits are illustrated in

parameter value, the better the system performance. This fact can be explained as the system performance is dominated by the fading parameter. On the other hand, the system performance of an UWB system combined with MC techniques is dominated mainly by the fading environment. In contrast to the results shown in

We proposed a system which adopted the UWB system in the MC system, i.e., the MC-CDMA system, in this paper. The system performance of the proposed configuration is also evaluated in the article. The fading channel is characterized as the correlated-Nakagami-m distribution. Some of the usual parameters were taken into account in the numerical analysis for accuracy validation and comparison. The most important thing worth noting is the fading parameter always dominates the system performance when considering the Nakagami-m distribution as the fading model.

In this appendix the Formula (23) is derived in detail. From the SINR description shown in (15), it is known that the distribution has Gamma statistics. After put the SINR into (18), the Gamma distribution expression can be obtained. By substituting the Gamma distribution formula into (21) the result can be calculated as

where the Q-function can be alternatively replaced with the formula shown in (22), and the integral expression now becomes

By applying the integral equivalent of the formula shown as [

and using of the variable changed with, the other variable is denoted as

By following the results illustrated in the previous equation, the indefinite integral in (A.2) becomes

The formula shown in (23) can be obtained by substituting (A.5) back into (A.2) and yielded as