Time reversal (TR) is an effective solution in both single user and multiuser communications for moving complexity from the receiver to the transmitter, in comparison to traditional postfiltering based on Rake receivers. Imperfect channel estimation may, however, affect pre- versus postfiltering schemes in a different way; this paper analyzes the robustness of time reversal versus All-Rake (AR) transceivers, in multiple access communications, with respect to channel estimation errors. Two performance indicators are adopted in the analysis: symbol error probability and spectral efficiency. Analytic expressions for both indicators are derived and used as the basis for simulation-based performance evaluation. Results show that while TR leads to slight performance advantage over AR when channel estimation is accurate, its performance is severely degraded by large channel estimation errors, indicating a clear advantage for AR receivers in this case, in particular when extremely short impulsive waveforms are adopted. Results however also show a stronger non-Gaussianity of interference in the TR case suggesting that the adoption of a receiver structure adapted to non-Gaussian interference might tilt the balance towards TR.

1. Introduction

Time reversal (TR) is a signal processing technique that takes advantage of the field equivalence principle [14] and was originally proposed in acoustics [57]. By prefiltering transmissions with a scaled version of the channel impulse response, reversed in time, TR allows simplification of receiver design, since the channel is compensated by precoding at the transmitter and can simplify the task of synchronization at the transmitter.

In a legacy network, e.g., 3G or WiFi, where Base Stations (BS) or Access Points (APs) were typically characterized by higher computational power, TR would be typically implemented in downlink, in order to concentrate most of the complexity on the network side. However, the evolution of networks is leading more and more to use cases where devices with identical characteristics can play different roles at different times, as for example in Device-2-Device (D2D) connections that are already planned in LTE and are expected to become common in 5G networks [8]. This is particularly true in light of the extremely high density of devices and, in turn, of massive amount of data expected in 5G networks, that will call for efficient data fusion and data concentration techniques [9], where in turn one device will take the responsibility of collecting and merging the data generated by a large number of other devices. In such a use case, moving the complexity on the device that is playing the role of sink/data collector is not necessarily the optimal choice and in some cases might not even be feasible. In this context, this work focuses on the analysis of what happens at the sink when comparing two approaches: TR, where the complexity is distributed among devices, versus AR, where the complexity is concentrated at the BS, taking care of compensating the impact of the channels for all links. In agreement with such use case scenario identified above, the network model in this work, shown in Figure 1, considers multiple access by user terminals (UTs) communicating to one BS, where UTs and BS differ in their functional roles (sources versus sink), but not in general in their hardware characteristics.

Pioneering work on single-antenna TR spread-spectrum communications dates back to the nineties, where the TR prefilter was named pre-Rake [10, 11]. The basic idea was to prefilter the transmitted pulse with the channel impulse response reversed in time, therefore matching the transmitted signal with the subsequent channel.

Precoding techniques for multiuser spread-spectrum systems were developed along similar lines of receive filters: transmit Zero-Forcing (ZF) [12], that attempts to preequalize the channel by flattening the effective channel formed by the cascade of the prefilter and the actual channel, is optimum in the high- regime; transmit matched filter (MF), that has been recognized to be equivalent to the pre-Rake filter in [13], conversely is optimum in the low- regime; and finally, transmit MMSE (Wiener) filter minimizing the was derived in [14] following previous attempts [15, 16].

In recent years, along with the fast developing of narrowband MIMO systems, precoding techniques using multiple antennas at the transmitter were thoroughly studied (for a complete overview on MIMO precoding; see [17]). Since the mathematical formulation of multiuser spread-spectrum is very close to that of MIMO communications (see [18] for an overview of this analogy), MIMO linear precoders can be derived along similar techniques. Significant research efforts were then devoted to the analysis of the impact of imperfect channel estimation on such systems, with several authors proposing closed-formulas for BER evaluation in both SISO and MIMO systems affected by channel estimation errors [1922].

TR was proposed in connection with UWB communications in [23] that also addressed equalization through an MMSE receiver. In [24], early experimental data, showing the feasibility of TR, were collected. Shortly after, experimental investigations on multiple-antenna systems with TR [2527] and performance analyses [28] were also pursued. In [29, 30], compensation for pulse distortion in connection with TR was investigated. In [31], the trade-off between the complexity of transmitter versus receiver in terms of number of paths was analyzed. In [32], the robustness of TR to the lack of correlation between channels in a MISO system was investigated. Finally, in [33], the effect of TR on statistical properties of multiuser interference in communication versus positioning was explored.

The above investigations were all carried out based on the hypothesis of perfect channel estimation. Only recently, the impact of imperfect channel state information (CSI) on the performance of TR was experimentally investigated in the case of UWB signals in [34], and analytic expressions for BER evaluation in a TR-UWB system in presence of channel estimation errors were provided in [35], where an iterative algorithm to compensate for estimation errors was also proposed. The impact of a channel estimation error in a TR-UWB system was analyzed in terms of Bit Error Probability also in [36], focusing however only on BPSK modulation, and in [37], where an uncoordinated network of IR-UWB devices was considered. The impact of nonstationarity in the channel on TR-UWB communications was addressed in [38], comparing a TR and a conventional system in terms of BER and mutual information assuming that the error is determined by temporal variation in the channel coefficients and carrying thus out the analysis as a function of the correlation coefficient between real and estimated channel. The impact of channel nonstationarity was also addressed in [39], and the analysis was extended in [40] by considering the combined impact of nonstationarity and channel coefficients quantization in terms of BER.

The above works highlighted that channel estimation errors, in particular due to nonstationarity, may affect the performance of TR systems, although the actual impact mainly depends on the degree of stationarity of the channel. Several aspects are however still open: in particular, an exhaustive comparison of how channel estimation error affects TR in comparison to other techniques; e.g., receiver-based schemes is still missing, and the impact of imperfect CSI in presence of Multiple User Interference (MUI) was only partially addressed.

This paper investigates such issues, focusing on the specific case of single-antenna UWB systems using TR versus as an exemplary case of a receiver-based scheme, that is an AR scheme. The contributions of this paper can be thus identified as follows:(i)It carries out an extensive comparison of TR versus AR under imperfect CSI, not carried out to this extent in the previous works on this topic, and adopts an estimation error model more general than the use of correlation coefficient adopted in most of such works.(ii)It derives mutual information for TR versus AR schemes in presence of both CSI estimation error and MUI.(iii)It analyzes the statistical properties of MUI in order to provide hints on how such properties might be used to improve the performance of a TR system under imperfect CSI.

Comparison of performance of TR versus AR transceivers will be carried out in terms of effect of imperfect channel state information (CSI) on symbol error probability of a generic information-bearing symbol for a given UT (see [41] for a work adopting a similar approach in other systems, in particular CDMA). The analysis will further explore robustness of TR versus AR, by finding the maximum achievable rate for the uplink channel. Finally, the maximum information rate, that takes into account channel estimation overhead, will be explored.

The paper is organized as follows: Section 2 contains the reference model; Section 3 is devoted to the performance analysis in terms of symbol error probability, while Section 4 presents the comparison between TR and AR in terms of uplink rate as measured by the mutual information. Finally, Section 5 contains the conclusions.

2. Reference Model

This section is organized as follows: Section 2.1. System model; Section 2.2. Single User Channel; Section 2.3. Multiuser Channel; Section 2.4. Channel Estimation and Data Transmission.

2.1. System Model

A generic transmits data that is encoded into a sequence of information-bearing symbols . This set of symbols is partitioned into blocks of symbols each, where , where the length of the block is typically determined based on the coherence time of the channel. Each block is transmitted by using the following signal:where (sec) is the symbol period and is the unit energy waveform associated with the -th symbol of user . In general, is a spread-spectrum signal at user prefilter output and has band ; that is, its spectrum is nonzero for . Assuming that are orthonormal, or very mildly cross-correlated, the energy of in (1) is ; since the block has duration , the average power is .

In the adopted model, demodulation at BS is performed on a block-by-block basis. Index , that specifies the block number, can be thus dropped. Consider for the sake of simplicity in (1):Figure 2 shows the system model, including modulators producing transmitted signals, , , affected by propagation within different channels and corrupted at the receiver by white Gaussian noise . The receiver consists in one demodulator.

Transmitted signal of each user propagates over a multipath channel with impulse response leading to the signal :where and are amplitudes and delays of the paths of , respectively. The received signal is as follows:where is a white Gaussian noise with flat power spectral density (W/Hz).

Throughout the paper it is assumed that the receiver estimates transmitted symbols of user , , on a symbol-by-symbol basis, by considering users as unknown interference over user ; for example, Figure 2 shows the demodulation of user . As detailed below, transmissions are symbol-synchronous but not necessarily chip-synchronous; therefore the symbol-by-symbol demodulation does not imply any performance loss. In the adopted model, in agreement with the scenario defined in Section 1, the receiver is a single user detector and as such suboptimal, since it does not take into account the possibility of joint multiuser detection, which is left for future work. In addition, throughout the paper it is assumed that the transmission data rate is such that Intersymbol Interference (ISI) is negligible in agreement with other works operating under the same assumption in the literature; see, for example, [38, 39].

How channel is estimated and how errors in channel estimation play a role in the model will be explained later in this section, in association with the different modulation and demodulation structures. Considering signals for the symbol at time epoch , notation can be simplified by setting , and (2), (3), and (4) become the following:

2.2. Single User Channel

Since the system is symbol-synchronous, analysis may refer to transmission of one generic symbol, that is chosen as symbol ; the index can be dropped in order to simplify the notation, leading to , .

If transmission does not foresee prefiltering, that is, a zero-excess bandwidth pulse with bandwidth and unit energy is transmitted to modulate , the received signal is as follows:where the spreading sequence and the chip period are made explicit. In the following, time-hopping is considered, for which all are zero, but one.

Since , and therefore also , are bandlimited to , these signals can be represented by their samples spaced by ; in particular, will be represented by its samples , . One has thus the following:In general, one will have , where the case corresponds to a pulse duration shorter than chip duration, as common in UWB communications. This is taken into account in the model by defining the impulsiveness index , a positive integer defined such that ; note that can be equivalently interpreted as the number of samples per chip time . Furthermore it can be safely assumed that the channel impulse response is causal and has finite delay spread . By introducing the approximation , where is an integer, one has for and for . As a result, one can write the following:By projecting (10) onto , the following discrete model is obtained:where the symbol indicates the Kronecker product, is the first vector of the canonical basis of , is , and is a banded Toeplitz matrix with dimensions and elements .

In general, for a system with prefiltering, with prefiltering impulse response , (11) generalizes to the following (see, e.g., [18]):where is a Toeplitz matrix with dimensions and elements .

In this paper, prefiltering is introduced in order to compensate channel effects; in particular, prefiltering is based on an estimated version of the channel impulse response. In other words, imperfect prefiltering may be matched to channel estimation error patterns. If prefiltering is imperfect, as will be justified in Section 2.4, the error due to the estimation process can be modeled as a white Gaussian process , that is, added to as follows [42]:where , where accounts for estimation accuracy, and is such that , so as to ensure an equal energy comparison.

No Prefiltering, All-Rake Receiver. The traditional (or conventional) receiver is a matched filter, i.e., an AR receiver in the case of a multipath channel. Knowing the time-hopping spreading sequence and the resolved channel , a sufficient statistic for is obtained by projecting the received signal onto , or, equivalently, onto :where .

As occurs in the prefiltering, also the AR receiver is affected by possible channel estimation errors. If the AR receiver is provided with imperfect CSI, that is, operates using an estimation of channel that is impaired by an error , then it combines paths through instead of , and inference of is based on the following:where , being the nonzero dimension of .

TR Prefiltering, 1Rake Receiver. The TR prefilter is represented by , where guarantees that prefiltered and non-prefiltered transmitted waveforms have same energy. Time-hopping implies , and . A 1Rake receiver is given by . Denoting by the TR prefilter matrix, one has the following:being and having defined .

If the transmitter is provided with imperfect CSI, then model of (13) holds, and (16) becomes the following:

AR versus TR. As well-known [33], TR is equivalent to a system without prefiltering and AR in terms of the signal-to-noise ratio. From a single user perspective, there is no performance difference in both uncoded (symbol error probability) and coded (channel capacity) regimes between the two transceiver structures. Moreover, previous work [31] suggested that sets of equivalent systems can be obtained with partial Rakes compensating for partial TR transmitter structures. This paper extends the comparison of AR versus TR to the case where imperfect CSI is available.

2.3. Multiuser Channel

A straightforward extension of (11) to users is as follows:where and models the chip-asynchronism by making i.i.d. according to a uniform distribution. This extension holds based on the hypothesis that all UTs are symbol-synchronous. This hypothesis is reasonable since, as further discussed in Section 2.4, the BS broadcasts in a link setup phase a known sequence to the UTs.

Denoting by the spreading sequence after transition in the multipath channel, and bythe spreading matrix, (11) can also be rewritten as follows:where .

For systems with prefiltering, (18) generalizes towhere matrices and have same dimensions as and of (12), respectively, and (20) holds with .

In the presence of imperfect CSI, in (21) is substituted by , as defined in (13), where estimation errors are independent of .

No Prefiltering, All-Rake Receiver. The decision variable following the matched filter of user iswhere , represents the MUI and .

If the AR is provided with imperfect CSI, then signal is projected onto instead of , hencewhere .

TR Prefiltering, 1Rake Receiver. With TR, the decision variable for user becomes the following:where is the delay (in samples) to which the 1Rake is synchronized. In the presence of imperfect CSI, the decision variable is

AR versus TR. As well-known [33, 43], TR usually increases the kurtosis of the interference at the output of Rake receivers. This follows from the fact that the effective channel impulse response formed by the combination of prefilter and multipath channel has a peaked behavior, whereas without TR the behavior is nonpeaked. While in the single user case the two schemes are equivalent, this equivalence does not hold in the multiuser case. The impact of estimation errors will be investigated in Sections 3 and 4.

2.4. Channel Estimation and Data Transmission

For both AR and TR, the channel impulse response estimation requires cooperation between the transmitter and in the receiver. Actual transmission of the set of information-bearing symbols requires, therefore, additional symbols to be sent either in a preamble or in a postamble of the block [44]. Note that in the scenario considered in this work precoding of does not depend on channels experienced by users , and thus feedback from the BS to each UT is not required (see [42] for a thorough discussion).

Assuming Time-Division Duplexing (TDD) is adopted, transmission can follow the scheme shown in Figure 3, summarizing the organization of the different links (downlink, i.e., broadcast, versus uplinks) over time, where durations of data, preamble, and postamble are indicated in terms of number of chips (, , , ). The information exchange between the BS and each UT consists in four phases:(1)Phase 1, Downlink Channel Training: the BS broadcasts a training sequence of length samples (corresponding to seconds) that is received by each UT starting at time . Each multipath channel spreads the sequence for samples; hence each UT listens from time to time samples. This training sequence may be also used for network synchronization at symbol level.(2)Phase 2, Uplink Channel Training: each UT transmits its own training sequence of length samples to the BS (preamble). By reciprocity, channels spread these sequences for samples; therefore each UT remains idle for samples.(3)Phase 3, Data Transmission: each UT transmits a sequence of information-bearing symbols for samples.(4)Phase 4, Idle: each UT transmits a sequence of null symbols of duration samples denoted with (postamble).

During the downlink training, the BS broadcasts its training sequence to the UTs. With reference to model of Section 2, and in particular to (11) and impulsiveness index , the received signal at iswhere is vector of received samples, is Toeplitz channel matrix, is training sequence with energy , is training sequence accounting for impulsiveness with energy , and is white Gaussian noise vector.

Equation (26) can be rewritten as follows:where now is a Toeplitz matrix and is the channel vector.

In order to minimize the signal-plus-interference-to-noise ratio, may use an MMSE estimation of , where the interference is caused by multipath. However, the use of PN sequences as training sequences is very common, due to their good autocorrelation properties. In fact, PN sequences have periodic ACF of the following form [45, 46]:that is asymptotically impulse-like, as .

Asymptotically then, and dropping the superscript to unclutter notation, , and MMSE reduces to a matched filter, and estimation is as follows:Dividing by the previous expression yields the following:where . Note that, as well-known (e.g., [47]), the estimation can be made as accurately as desired by increasing . For antipodal sequences, say with , the energy of the training sequence is ; therefore, can be increased either by increasing power spent on training, that is, by increasing , or by increasing time spent for training, that is, by increasing , or both.

In the uplink training, the BS receives the superposition of the sequences of users each filtered by the corresponding channel, that is,having definedAs previously mentioned, the superscript is dropped to unclutter notation.

The goal of the BS is to linearly estimate by observing , knowing :where is the vector of channel estimations, being the vector representing estimate, and is the matrix representing the estimator. All common linear estimators, that is, ZF (Zero-Forcing), RZF (Regularized Zero-Forcing), MMSE (minimum mean square error), and MF (matched filter), can be described by the following expression, parametrized by and :Indeed, MMSE is obtained with ; ZF with ; MF with ; RZF with .

In the simple case of ZF, the form assumed by (33) is as follows:and, therefore, the -th tap of the channel of generic user isHere, is a correlated Gaussian random variable with variance coinciding with the -th diagonal element of .

Assuming all UTs are transmitting the same power, i.e., is the same for each , the approximation allows assuming uncorrelated estimation errors, since , and thus

2.5. Performance Indicators

In both system structures, the statistic for inferring the transmitted symbol of user can be written in the following form:where is a r.v. representing noise, and are r.vs. depending on multipath channels, random time-hopping codes, random delays, and estimation errors.

Two Performance Indicators Are Considered.

In the uncoded regime, the probability of error as defined byis considered.

In the coded regime, mutual information with Gaussian inputs and a bank of matched-filters followed by independent decoders is considered; for the generic user , this is given bywhere is the mutual information between the transmitted symbol and the decision variable . Since a channel use corresponds to seconds, the sum-rate achieved by the set of users ishaving indicated with the mutual information (40) for a generic user. Finally, a spectral efficiency equal tois obtained.

3. Probability of Error

3.1. Single User

The main contribution of this subsection is to show that imperfect TR and AR achieve the same probability of error, and, therefore, that the same accuracy is needed for channel estimation at transmitter and receiver in order to achieve a given error probability.

With reference to decision variables of (17) and of (15), the probability of error, in both cases, iswhere the first equality follows from belonging to with equal probability, and the second equality follows from the distribution of being an even function. For the power constraint, results. Equivalence of for the two cases is derived by showing that and have the same distribution.

To this end, rewrite the decision variable conditioned on . Without loss of generality, and for the sake of simplicity, consider . ThenSimilarly, the decision variable conditioned on iswhere . By comparing the two expressions, is equivalent to , the equivalence being defined as producing the same , iff term in is distributed as in . This is, indeed, the case; by choosing an orthogonal matrix such that , one haswhere ; hence the equivalence in terms of distributions, and, therefore, probability of error is verified.

3.2. Multiuser

In the multiuser setting, although the expression for the probability of error remains as in (43), there are three sources of errors: thermal noise, imperfect CSI, and multiuser interference (MUI). In particular, as increases, the last two factors both lead to a probability error floor, i.e., as , being the load of the system.

Figures 4 and 5 show the probability of error versus (dB) for systems with versus , respectively, and for different values of , with , and , with . All results were obtained by Monte-Carlo simulations of finite-dimensional systems with chips. In both figures, the left-hand side plot (Figures 4(a) and 5(a)) refers to TR, while the right-hand side plot (Figures 4(b) and 5(b)) refers to AR.

For low-SNR, i.e., , is not very sensitive to estimation errors: the performance of the systems thus mainly depends on the considered load. The impact of estimation error becomes however evident for high SNR, i.e., , with significantly different values depending on the combination of estimation error variance and system load (and thus MUI).

Figures 4(a) and 4(b), presenting the results for both systems with , show that AR outperforms TR when imperfect CSI is the main cause of error (see, for example, the case , for which the floor in the case of TR is more than an order of magnitude larger), while the opposite is true when the load is the main cause of error, with a slight advantage of TR with respect to AR for high traffic loads; see, for example, the case where TR guarantees a gain of about 3 dBs in terms of for a .

Figure 5 presents results for impulsive systems, characterized by . Focusing first on the case of perfect CSI estimation (plots with ), it can be observed that, in presence of MUI, that is for medium () and high () system loads, the performance of both TR and AR is significantly worse than in the case with . An increase of impulsiveness leads thus to a lower robustness with respect to MUI for both AR and TR schemes, although this phenomenon is arguably more evident in the TR case, since this scheme shows a slight performance advantage in the case with , while for TR and AR have exactly the same performance for all system loads. When one considers the impact of CSI estimation error (plots with and ), the introduction of a more impulsive waveform has the opposite effect: both AR and TR prove to be more robust to CSI estimation errors, with lower error probability floors. The extent to which the phenomenon is visible in the two schemes is however very different: while a moderate improvement can be seen for TR, the AR scheme shows a dramatic performance increase, in particular for a low network load (), where the error probability floors are not visible. The stronger positive impact of impulsiveness on the AR is a rather surprising result, which might be explained by the role of impulsiveness in the number of resolvable paths at the receiver: a shorter pulse makes in fact more paths resolvable, and TR, which allows resolving more paths than AR by approximately a factor of 2, appears to be more sensitive to estimation errors, thus reducing the advantage of increasing the impulsiveness of the system; further research is however required in order to properly explain this result.

4. Mutual Information, Sum-Rate, and Spectral Efficiency

In this section, mutual information defined in (40) is derived for AR and TR. The other merit figures defined in (41) and (42) follow directly, although all the elements for a comparison are already included in (40).

4.1. Derivation of Mutual Information

The decision variable for both the imperfect TR (cf. (25)) and AR (cf. (23)) can be cast in the following form:where .

Let us specify and give an interpretation of the terms , , for both TR and AR.

TR Coupling Coefficients. For TR, the term is given by where denotes a vector with same components of vector in reversed order, i.e., , . The term , , iswhere being the autocorrelation sequence of , and, similarly, with a Gaussian random vector with non-identity correlation.

In order to provide an interpretation of the above expressions, it is useful to start with the case of no estimation error. In general, the decision variable at the output of the matched filter of user is given by the -th sample of the sum of both intended and interference signals, plus noise. In the special case of no estimation errors, is the square-root energy of channel , i.e., the maximum tap of the effective channel, while is either equal to zero if the effective channel of user , that occupies out of degrees of freedom in a symbol period, is not present at delay , or to a random resolved path of the effective channel of user , the randomness owing to random hopping and asynchronism. In presence of estimation errors, is smaller than, although in general in the neighbourhood of, the square-root energy of channel due to the mismatch between and , and is either equal to zero if the perturbed effective channel of user , that occupies out of degrees of freedom in a symbol period, is not present at delay , or equal to a random path of the perturbed effective channel of user , the perturbation owing to the imperfect channel estimation of user .

AR Coupling Coefficients. For AR, the set of coupling coefficients arewhere , , and . We can think of as the perturbed direction along which the received signal is projected in order to decode user ; represents, therefore, the “mismatch” between the perturbed and unperturbed channels of user ; represents the coupling between user , that is perturbed, and another user . As in the TR case, a channel impulse response occupies a fraction, that is approximately equal to , of the available degrees of freedom in a symbol period; opposite of the TR case, where the perturbation affects user in , user is perturbed in the AR case (through ) and user appears with the true channel impulse response .

Derivation. Being each term in the r.h.s. of (47) independent of the other terms, mutual information can be derived once the distributions of and are known. The former depends on both the random channel impulse response and estimation errors of user , and the latter on the random channel impulse responses and estimation errors, and the random delays with respect to user . Hence, the final form assumed by strongly depends on the channel model; however, in the following, the effect of the time-hopping and random asynchronism will be enucleated, without entering in the computation of a mutual information when a particular channel model is adopted; this last task was addressed by simulations, where Channel Model 1 (CM1) specified in the IEEE 802.15.3a standard [48], that is valid for bandwidths up to several gigahertz, was selected. Details on the settings and results of these simulations are provided in Section 4.2.

In the case of TR, since the effective channel of user occupies a fraction , and user , due to the assumptions on independence and uniformity of hopping codes and asynchronism, selects uniformly at random one of the samples available per symbol period, then , , is equal to zero with probability , and is distributed as the generic path of the effective channel with probability , that iswhere indicates the distribution of the generic path of the effective channel of user (that is independent of ). In presence of estimation errors, the above argument holds, that is, , , is equal to zero with probability , and is distributed as the generic path of the perturbed effective channel with probability :where indicates the distribution of the generic path of the perturbed effective channel of user (that is independent of ).

In the case of AR, let us start by finding the distribution of , , i.e., the coupling coefficient between two users in absence of estimation error. Each channel spans a subspace of dimension in a space with dimensions (the number of degrees of freedom in a symbol period is ; in the large system limit, as , the difference between and due to the convolution is negligible.); in other words, just entries of are nonzero. From the hypotheses of independence and uniformity of delays due to asynchronism between users and time-hopping codes, there exists a probability such that the inner product is nonzero, and the remaining probability that the inner product is zero. We may think of the “nonzero event” as the partial overlapping between two channels. As , results, where the assumption allows neglecting border effects. Indicating with the distribution of conditioned on the nonzero event, one has the following:In presence of estimation errors, the above discussion remains valid, since an error changes, in general, the direction of vector with respect to , i.e., and , is, in general, not colinear, but it does not change the subspace spanned by the two channels, i.e., the subspace spanned by the true channel is equal to the subspace spanned by the perturbed channel. Indicating with the distribution of conditioned on the nonzero event, one has the following: reduces to when the estimation error is nil.

In terms of c.fs., (55) and (53) become the following:wherebeing equal to either or in (55) and (53), respectively. In general, given two independent r.vs. and and their product , results; therefore, the r.v. has c.f.:where the expectation is taken with respect to , and is independent of . Since with are independent, has c.f.:that, in the large system limit, where , , , converges to where is the effective load; without multipath () and one pulse per chip , reduces to the usual load as given by . The interference-plus-noise variable has thus c.f. given by the following:

Knowing the distribution of conditioned on , or equivalently its c.f., mutual information follows directly. Hence, the c.f. of given is the following:hence, the c.f. of is as follows:where the expectation is over . Explicitly, one has the following:whereandwhere denotes a Gaussian distribution with zero mean and variance , and

The above derivation allows one to find as a function of the distribution of , , and the distribution accounting for the loss of correlation incurred by the user to be decoded because of the estimation error. Both and account for the channel model and the estimation error, in particular its variance.

As baseline comparison, we also provide the following lower bound for , that is achieved when the interference is Gaussian:where the expectation is over . The corresponding spectral efficiency lower bound is (cf. (42)).

Results are shown on Figure 6, where (solid curve) and (dashed curve) are presented as a function of , for different values of and . The receiver structure shows a mutual information floor at high SNR. By comparing Figures 6(a) and 6(b), one observes that increases sublinearly as increases, while by comparing Figures 6(a) and 6(c), or Figures 6(b) and 6(d), a reduction in spectral efficiency due to the presence of an estimation error is observed. scales with as shown on Figures 6(e) and 6(f).

For all results presented in Figure 6, AR outperforms TR, again indicating that, under the assumptions considered in this work, performance of TR is significantly degraded by channel estimation errors. It is interesting to note, however, that the gap can be interpreted as a measure of the non-Gaussianity of and , and is indeed higher in the TR case with respect to the AR case, because of the different distribution of the interference term. A detailed discussion on the characteristics of interference is provided in the next subsection.

4.2. Impact of Channel Model and Interference Distribution

As previously said, the distributions of the channel paths and in turn of the coupling coefficients used in (52)-(55) depend on the channel model and were obtained by simulation. All simulations were carried out for a system with fixed chip duration ns and bandwidth . Power control was assumed; in particular, for all users, . The delay spread of each channel impulse response was fixed at a value ns that includes most of the energy of typical CM1 channels. For a given bandwidth , the length of the channel expressed in number of samples per channel is ; i.e., is a vector.

Figure 7 shows the distributions , , , and in (52), (53), (54), and (55), respectively, and the distribution of the term .

Figure 7(a) versus Figure 7(b), in particular, shows the distributions of the coupling coefficient , , in case of no estimation errors, for AR and TR, respectively. As may be expected, the variance of the latter is larger than the variance of the former, as follows from the property of TR to increase the total energy of the effective channel; in the specific case, one has , while . In Figure 7(b) the presence of a strong interference () is highlighted, not present in the AR case; this is coherent with the fact that, in the TR case, there is, indeed, among the paths of the effective channel, one path with amplitude equal to the square-root energy of the channel, , that is, therefore, selected with probability . is also more leptokurtic than , showing a kurtosis approximately equal to