A channel and delay estimation algorithm for both positive and negative delay, based on the distributed Alamouti scheme, has been recently discussed for basestation–based asynchronous cooperative
systems in frequencyflat fading channels. This paper extends the algorithm, the maximum likelihood estimator, to work in frequencyselective fading channels. The minimum mean square error (MMSE)
performance of channel estimation for both packet schemes and normal schemes is discussed in this paper. The symbol error rate (SER) performance of equalisation and detection for both timereversal
spacetime block code (STBC) and singlecarrier STBC is also discussed in this paper. The MMSE simulation results demonstrated the superior performance of the packet scheme over the normal scheme
with an improvement in performance of up to 6 dB when feedback was used in the frequencyselective channel at a MSE of 3 x 10^{–2}. The SER simulation results showed that, although
both the normal and packet schemes achieved similar diversity orders, the packet scheme demonstrated a 1 dB coding gain over the normal scheme at a SER of 10^{–5}. Finally,
the SER simulations showed that the frequencyselective fading system outperformed the frequencyflat fading system.
Spacetime block coding is a technique used to achieve spatial diversity in both synchronous multipleinput multipleoutput and synchronous cooperative communication systems in frequencyflat fading
channels.^{1} However, cooperative communication is generally asynchronous, not synchronous. This is because different relays have different locations, and the transmitted signals from
different relays may arrive at different times. In asynchronous cooperative communication systems, when the relays use orthogonal spacetime block codes (STBC) to forward the received data from
source to destination, the code structure at the receiver is not orthogonal.^{2} The system can only achieve a diversity order of 1. Therefore, new transmission schemes based on STBC are
required, and the estimation of the relative delays between different paths at the destination is needed.^{2} New transmission schemes based on STBC have been studied, but these studies
did not take into account delay estimation.^{2,3,4,5} In this paper, we focus on channel and delay estimation for asynchronous cooperative communication systems.A typical example of asynchronous cooperative communication systems is a basestation–based cooperative communication in macrocell downlink networks, proposed by Skjevling et al.^{6}
Tourki and Deneire^{7} proposed a channel and delay estimation algorithm that achieved a lower Cramér–Rao bound (CRB) in Skjevling et al.’s system. But Tourki and Deneire’s
system was derived only for positive delays, that is, when Transmitter one’s data always arrives at the receiver before Transmitter two’s data.^{7} Recently, Xu and
Padayachee^{8} extended Tourki and Deneire’s scheme to accommodate negative delays (i.e. when Transmitter two’s data arrives at the receiver before Transmitter
one’s data). However, Xu and Padayachee’s scheme^{8} works only in frequencyflat fading channels. The main motivation of this paper was to extend the scheme described in Xu and Padayachee^{8} to work in frequencyselective fading channels. Extending the scheme to accommodate
transmission in frequencyselective channels is pivotal because current and future broadband wireless communication systems aim to have high data rates, which gives rise to frequencyselective
propagation effects. A typical example is that of a mobile communicating with a base station. As a result of the reflections of buildings, hills, cars and other obstacles, there are multiple
delayed receptions of the transmitted signals at the receiver. This causes frequencyselective propagation effects. Mheidat et al.^{9} have already discussed cooperative communication
over frequencyselective channels, but their scheme assumed perfect channel knowledge at the receiver, as well as perfect synchronisation. Simeone and Spagnolini^{10} discussed channel
estimation in frequencyselective fading channels but still maintained the assumption of synchronicity. Sirbu^{11} addressed channel and delay estimation in wireless communication systems,
but assumed that the delay was fixed. Very recently Li et al.^{12} proposed a simple orthogonal spacetime coding scheme for asynchronous cooperative systems for frequencyselective fading
channels. Their proposed system used an orthogonal frequency division multiplexing technique to convert the delay in the time domain into phase in the frequency domain, but they assumed that full
channel state information was known at the receiver.
The basestation–based cooperative diversity system shown in Figure 1 is a macrocell consisting of two base stations transmitting wirelessly to a single mobile receiver.^{6,7} The base stations
(BS1, BS2) and mobile node have only one transceiver each. It is assumed that both base stations have the same set of data to transfer to the mobile receiver. This can be achieved in reality via either a
wired highspeed connection such as Ethernet, or a wireless transmission of the data between the base stations preceding the transmission scheme.To begin with, the data set to be transmitted is parsed into two blocks of N symbols each, d_{}n and d_{}n_{+1}, where n is the block
number. Training symbols d_{}1 and d_{}2, each of length L, are then added at the end of d_{}n and d_{}n_{+1},
respectively, to form two (N + L) x 1 vectors. To insert a cyclic prefix of training symbols between any two successive blocks these vectors are premultiplied by a precoding matrix
F_{p}. Inserting the pilot sequences in clusters in the middle of the data frame minimises the CRB.^{13} Multiplying the vectors by F_{p} results in two
(N + 2L) x 1 blocks s_{}n and s_{}n_{+1}. The blocks are then transmitted according to the timereversed block form of Alamouti’s
scheme shown in Figure 2. In Figure 2, T is a timereversal matrix which is described in [Eqn 3]. The channel impulse responses of the frequencyselective channels from BS1 and BS2 to the mobile node are given by h_{}1 = [h_{1}(0), h_{1}(1),...,h_{1}(L_{1})]_{ } and h_{}2 = [h_{2}(0),
h_{2}(1),...,h_{2}(L_{2})], where L_{i}, i = 1,2 is the channel memory length which is assumed to be known at the
receiver. These channel impulse responses are also assumed to be Rayleigh block flatfading, that is, the amplitude of the fading envelope follows a Rayleigh distribution which stays constant
for the duration of each frame but varies independently from frame to frame. Each h_{i}(l_{i}), where i = 1,2, and l_{i}
= 0,1,..., L_{i}, is assumed to be an independently and identically distributed zeromean complex Gaussian with a variance of 1/(L_{i} + 1).^{14}
For simplicity it is also assumed that L_{1} = L_{2} = L_{c} in this paper. When designing the system model, the aim was to follow the following technique for linear equalisation^{7,15}: Step 1 – model the received signals as
and
where r_{}n and r_{}n_{+1} are (N + 2L) x 1 received symbol vectors. b_{}n and
b_{}n_{+1} represent (N + 2L) x 1 zeromean complex Gaussian noise with each entry having a variance of N_{0}/2 per dimension,
T is an (N + 2L) x (N + 2L) timereversal matrix as described by
and H_{}1 and H_{}2 are (N + 2L) x (N + 2L) circulant channel matrices given by
Step 2 – conjugate and timereverse r_{}n_{+1} and allow the received signals r_{}n and r_{}n_{+1}
to be grouped in matrix form by using the identity TH_{}i * T = H_{}i ^{H},^{9} which is shown in [Eqn 5]
where H_{}i^{H} , i = 1,2, denotes the Hermitian matrix.Step 3 – define a unitary matrix K_{TR} as^{16}
where represents the Kronecker product and I_{}A denotes a square identity matrix of size A.Step 4 – because multiplying by unitary matrices does not result in any loss of decoding optimality,^{14} if [Eqn 5] is multiplied by K_{TR}
the output streams can be decoupled according to [Eqn 7]
where b_{}n^{out} represents the filtered noise, which is still zeromean complex Gaussian noise with each entry having a variance of N_{0}/2
per dimension.^{16} The decoupled data streams can then be sent to a standard equaliser for detection.In order to extend the scheme in Xu and Padayachee^{8 }to follow the above technique, it is required that the received signal is broken up into two parts, namely r_{}n
and r_{}n_{+1}, which are then treated separately. A new (N + 2L) x (N + 2L) delay matrix P(τ) is then introduced to account for
the asynchronicity in the spacetime block coded system. P(τ) is given by
where. denotes the absolute operator and 0_{A}_{x}_{B} denotes a matrix of zeros of dimensions (A x B). P(τ) is basically a circularly shifted identity matrix used to circularly shift a symbol vector by τ symbols, τ is the difference between the arrival times of the two signals,
τ_{1} and τ_{2,} where τ_{1} and τ_{2} are the arrival times of the first (BS1) and second (BS2) signals, respectively. With the inclusion of
P(τ), the asynchronous received signal can be modelled as follows: When τ_{ }_{} ≥_{ }0,
and
And when τ_{ }_{} <_{ }0,
and
[Eqn 9] to [Eqn 12] are derived simply by noting that for positive delays the symbols from BS2 are shifted with respect to BS1, whilst for negative delays the symbols from BS1 are shifted with
respect to BS2.The asynchronicity introduced by cooperative communication poses a threat to the system’s orthogonality. One method proposed here to ensure orthogonality is to impose certain conditions
on the choice of pilot symbols, which will enable the receiver to employ linear equalisation. This is explained in Figure 3 and the subsequent discussion. The insertion of P(τ) into
the system model will only accurately describe the system if and only if d_{}2 = ts_{}1*_{ } when τ ≥ 0 and if d_{}1
= – ts_{}2*_{ } when τ < 0. ts_{}1 and ts_{}2 are given by
and T_{}s is an L x L timereversal matrix.Figure 3 shows the difference between the actual system and system model for τ ≥ 0. In Figure 3, T_{}N is an (N + L) x (N + L) timereversal
matrix. It can be seen that when there is a positive delay in the actual system, the last τ symbols of ts_{}1*_{ } are received in
the beginning of the current frame interval of BS2’s transmission, which destroys the orthogonality of the STBC. In the system model, however, P(τ) performs a cyclic shift
on s_{}n_{+1}, which results in the last τ symbols of d_{}2 being placed at the beginning of BS2’s frame. To overcome this discrepancy
between the actual system and system model, setting d_{}2 = ts_{}1*_{ }when designing the pilot symbols will yield no difference between
the system model and the actual system for positive delays, thus allowing the insertion of P(τ) to restore the orthogonality of the system. Using similar reasoning, one can see
the need for d_{}1 =–ts_{}2* for negative delays.

FIGURE 1: An illustration of the system model of the basestation–based cooperative
diversity system, showing the macrocell consisting of two base stations transmitting
wirelessly to a single mobile receiver.



FIGURE 2: A block transmission of the timereversed block form of Alamouti’s
scheme.



FIGURE 3: An illustration showing the differences between the actual system
and the system model for Base station 1 (BS1) and Base station 2 (BS2) when
the difference between the arrival times of the two signals from BS1 and BS2 is
greater than or equal to zero (τ > 0).


Channel and delay estimation algorithm


A maximum likelihood estimation algorithm in this paper essentially follows the same steps as those used by Tourki and Deneire^{7} and Sirbu^{11}:Step 1 – define ss1, ss2 and S In order to account for the effects of the frequencyselective channels, ss_{}1, ss_{}2 and S have to be defined differently to their definitions used by Tourki and Deneire^{7}.
Before one endeavours to understand how ss_{}1 and ss_{}2 are defined by equations, one should first understand how the pilot symbols are used to create these matrices. For a clearer understanding, the effect of P(τ), an L x L delay matrix, is shown in Figure 4 for τ = 1. The composition of S is illustrated in Figure 5 for L = 4, L_{c} = 2 and τ = 1. Note that d_{}2 = ts_{}1* and d_{}1 = –ts_{}2* are used in Figure 5. In Figure 5, the symbols denoted by ‘τ’ and ‘x’ are unknown at the receiver and hence cannot be used in the estimation process because S is the region of the received signal where only pilot symbols overlap. When τ ≥ 0
When τ < 0
Figure 6 illustrates how the indices of S(τ) are obtained.Step 2 – define z and find minimum error Define z as S(τ)h = r(N + L + τ + L_{c} + 1 : N + 3L), where h is a column vector containing the channel
coefficients of h_{}1 followed by the coefficients of h_{}2. It then follows that
where (.)^{#} denotes the pseudoinverse operator and h^{}~ is the linear least squares estimate of h for a given value of τ.The maximum likelihood estimator, which is modified to accommodate frequencyselective channels and negative delays, is given by
and
where hˆ is the final estimate of h based on the delay estimate τˆ.The mean square error (MSE) is a widely used evaluation criteria for channel estimation. It is given by

FIGURE 4: An illustration showing the effect of the L x L delay matrix, P(τ), when
the difference between the arrival times of the two signals from BS1 and BS2
equals 1 (τ = 1).



FIGURE 5: An illustration showing the composition of the S matrices for Base
station 1 (BS1) and Base station 2 (BS2) when the channel memory length equals
2 (L_{c} = 2), the length of the training symbol equals 4 (L = 4) and the difference
between the arrival times of the two signals from BS1 and BS2 equals 1 (τ = 1).



FIGURE 6: An illustration showing how the indices of S(τ), the region of the
received signal where only pilot symbols overlap, were obtained.


Equalisation and detection


Once the channel parameters and delay have been estimated, the next step is to equalise the received signal
before extracting the data. In this paper, two equalisation techniques will be presented: timereversal STBC^{16} and singlecarrier STBC.^{14}
Timereversal STBC
The first step in the timereversal equalisation process is to conjugate and time reverse r_{}n_{+1} as defined in [Eqn 10] for τ ≥ 0, then multiply the result by the delay matrix P(τˆ) . This will produce the following equation:
The identity, [Eqn 35], is proven in Appendix 1:
Using [Eqn 35] and the identity TH_{}i* T = H_{}i^{H},^{9} [Eqn 34] can be reduced to
Combining [Eqn 9] and [Eqn 36] in matrix form yields
Next we define the unitary matrix K_{TR} as^{16}
[Eqn 38] is identical to the definition in [Eqn 6] and is just repeated here for convenience. Multiplying [Eqn 37] by K_{TR} allows for the decoupling of the output streams according to
and
where b_{}n^{out} and b_{}n+_{1} still represent zeromean complex Gaussian noise with each entry having a variance of N_{0}/2 per dimension. Because the output streams are decoupled, standard equalisation techniques such as minimum mean square error (MMSE) or maximum likelihood sequence estimation (MLSE) may be employed.The analysis that yielded [Eqn 39] and [Eqn 40] was carried out assuming that τ ≥ 0. A similar analysis for τ < 0 will yield the following decoupled output equations:
and
Singlecarrier STBC
The first step in the singlecarrier equalisation process is to once again conjugate r_{}n_{+1} and then multiply it by P(τ)T to obtain [Eqn 34]. Next, transfer r_{}n and [Eqn 34] to the frequency domain by multiplying them by the discrete Fourier transform (DFT) matrix
Q where
and x and y are the row and column numbers of Q, respectively. In this paper it is assumed that the size of the DFT matrix, N + 2L, is a power of 2 and thus
the terms FFT (fast Fourier transform) and DFT are interchangeable.^{14}The frequency domain equations are given by
and
Once again using the identities TH_{}i* T = H_{}i^{H }^{(}^{14)} and P(τ) TH_{}i* P(τ) T = H_{}i^{H}, [Eqn 45] reduces to
Using the circulant property of the channel matrices one can easily prove that^{16}
where Λ_{}i is a diagonal matrix whose (k,k) element is the kth DFT coefficient h_{}i.Using [Eqn 47], [Eqn 44] and [Eqn 46] reduce to
and
Combining [Eqn 48] and [Eqn 49] in matrix form yields
Next, we define the unitary matrix K_{SC} as^{16}
Multiplying [Eqn 50] by K_{SC} results in the decoupling of the output streams according to
and
where b_{}n^{out} and b_{}n+_{1} represent filtered noise vectors, which are still zeromean complex Gaussian noise with variances of N_{0}/2 per dimension. Because the output streams are now decoupled, standard equalisation techniques such as MMSE or MLSE may also be employed.The analysis that yielded [Eqn 52] and [Eqn 53] was carried out assuming that τ ≥ 0. A similar analysis for τ < 0 will yield the following decoupled output equations
and
Pilot symbol design for frequencyselective fading channels
The CramérRao bound (CRB) of the channel estimation for a given τ is^{17}
where tr(.) is the trace operator.If optimal sequences are used, then the trace in the CRB definition in [Eqn 56] will be minimised, and the lowest achievable MSE of the channel estimation will thus also be reduced. It was shown that the
use of optimal sequences in a similar synchronous system will result in the following condition^{18}:
where (2L – L_{c}) is the number of rows of S. The number of rows of S is basically related to the number of pilot symbols available for use in the
estimation process. As expected, the estimation performance improves when more pilot symbols are available. Using a similar reasoning, the condition for optimality for an asynchronous system
can be defined as:
where (2L – L_{c} – τ) is the number of rows of S(τ).Once again, because the CRB is inversely proportional to [Eqn 58], it can be seen that increased pilot sequence lengths and decreased channel memory lengths and delay values will decrease the CRB and
improve the channel estimation performance. [Eqn 58] also demonstrates that the orthogonality of S(τ) hinges upon the channel memory lengths and the delay value, hence one would need to
know what the channel memory lengths and delay value are, before attempting to design optimal pilot sequences. This implies that the base stations would require knowledge of the channel and delay.
Generally, however, the base stations do not have access to this information, and hence it is not possible to satisfy [Eqn 58]. In these scenarios, using an exhaustive search to identify a pilot
sequence that has an impulselike autocorrelation is suggested. The impulselike autocorrelation criteria will not minimise the MSE of the channel estimation, but it will optimise the delay
estimation and hence improve the channel estimation and overall performance of the system. It was mentioned that the number of rows of S(τ), in this case (2L – L_{c} – τ ), must be even,^{8} because it would otherwise be
impossible to obtain zeros in the offdiagonal terms of (S(τ))^{H}S(τ) and hence [Eqn 58] would not be satisfied. This implies that optimal sequences only exist
if the channel memory lengths and delay are both even or both odd. When this condition is not satisfied, suboptimal sequences for neighbouring delay values are used. An additional condition specific to the system model described in this paper is that d_{}2 =_{ }T_{}sd_{}1* for τ ≥ 0 and
d_{}1 = –T_{}sd_{}2* for τ < 0. While these conditions were stipulated in order to maintain the orthogonality of the system
as a whole, they minimise the possibility of finding sequences that will ensure the orthogonality of S(τ). Although all the above conditions generally result in only suboptimal sequences being found, they tremendously reduce the size of the search space and hence decrease the computational
complexity required to find these suboptimal sequences.
In our simulation, a 4PSK modulation was used with N = 100, L = 14 and L_{c} = 3. The channel and noise parameters used in this paper were described from [Eqn 1] and [Eqn 2].
The channels were assumed to be unknown and hence needed to be estimated. Unless otherwise stated, the delay was assumed to remain constant over each frame but was allowed to vary randomly from frame
to frame. The delays were uniformly distributed between –(L–1) and (L–1). The signaltonoise ratio (SNR) values used refer to the ratio between symbol and noise energy.
The singleinput singleoutput (SISO) equaliser used was a MMSE equaliser.Two schemes, a normal scheme and a packet scheme, were used for data transmission in this paper. Normal schemes transmit pilot symbols, followed by data, whilst packet schemes transmit x pilot
frames after every y normal frames. The packet scheme can be divided into three phases. In the first phase, the base stations transmit x pilot frames consisting of pilot symbols only. The mobile receiver then uses these frames
to obtain an average estimate of the channel delay, and then transmits the estimated delay back to the base stations via the feedback channel in phase two. In phase three, the base stations use this
delay estimate to select the corresponding optimal pilot sequences for channel estimation to be used in the next y normal frames consisting of data and pilot symbols. It is assumed that the
delay remains constant over each round of the above three phases, but is allowed to vary from round to round. The first simulation was done for a packet transmission scheme, which shows how the delay error probability varies in the presence of a frequencyselective channel as a function of the number of
pilot frames, x, and the SNR at which these frames are transmitted. Figure 7 shows the delay estimation performance for a varying number of pilot frames. As can be seen, the delay estimation
performance improves as the number of pilot frames increases. Depending on the bandwidth and power requirements of the system, as well as its error tolerances, the user may choose the number of pilot
frames and transmission power that is best suited for their system. Also, it is worthy to note that in a system with a relatively high SNR, users may transmit the pilot frames at different and lower
power levels than the data frames, depending on the desired error performance. The next simulation was done for both a normal transmission scheme and a packet transmission scheme. The simulation investigated the performance of channel estimation in a frequencyselective fading
channel. The CRB given by [Eqn 56] was derived assuming that the delay was a known parameter, therefore it was once again only used here in order to obtain a tractable comparison. Figure 8 shows the MSE of
the channel estimation for varying scenarios. It has the following plots: • MSE (normal) – MSE of the channel estimation in the normal system where the packet scheme was not used. • MSE (no feedback) – MSE of the channel estimation when the packet scheme was used but no feedback was employed. • CRB (no feedback) – CRB for the channel estimation when the packet scheme was used but no feedback was employed, that is, a fixed pilot sequence was used for channel estimation. • MSE (with feedback) – MSE of the channel estimation when the packet scheme was used and feedback was employed. • CRB (with feedback) – CRB for the channel estimation when the packet scheme was used and feedback was employed, that is, the base stations used specific pilot sequences for channel
estimation based on the delay estimates that they received. From Figure 8 it can immediately be seen that the use of the packet scheme provided a substantial decrease in the MSE of the channel estimation when compared to the normal scheme – an
approximate 5dB decrease without feedback, and a further 2dB decrease when feedback was employed. Also important to note is that the packet scheme overlapped with the respective CRB, whether
feedback was employed or not. This highlights the fact that the use of the packet scheme resulted in a near zero delay error probability, as the CRB used was derived assuming the delay was a
known parameter. Figure 9 shows the symbol error rate (SER) performance of the normal scheme compared to that of the packet scheme. Figure 9 also shows the performance curves of the system under ideal conditions,
that is, when the system is assumed to be synchronous and the channel parameters are assumed to be known at the receiver. Figure 9 contains the following plots: • SER TR (normal) – SER performance when the timereversal equalisation technique was used and the packet scheme was not employed. • SER SC (normal) – SER performance when the singlecarrier equalisation technique was used and the packet scheme was not employed. • SER TR (packet) – SER performance when the timereversal equalisation technique was used and the packet scheme was employed. After every 200 normal frames, 5 pilot frames at an
SNR of 10 dB were transmitted. • SER SC (packet) – SER performance when the singlecarrier equalisation technique was used and the packet scheme was employed. After every 200 normal frames, 5 pilot frames at an
SNR of 10 dB were transmitted. • SER TR (ideal) – SER performance of the system under ideal conditions with timereversal equalisation used at the receiver. • SER TR (ideal) – SER performance of the system under ideal conditions with singlecarrier equalisation used at the receiver. • SER SC (ref) – SER performance of a similar system as presented by Mheidat and Uysal^{16}, in which ideal conditions were assumed. The first notable observation from Figure 9 is that the performances of the two equalisation techniques overlap regardless of which scheme was used. Mheidat and Uysal^{16} also
reported the same results. It can also be seen that the performance of the system presented here when ideal conditions were assumed, overlapped with the performance of the system presented
by Mheidat and Uysal^{16}, as expected. The packet scheme shows a 1 dB improvement in performance over the normal scheme and is only between 1 dB and 1.5 dB ‘off ideal’ performance. This reiterates the packet
scheme’s credibility and further justifies the minimal increase in overhead that the system may incur as a result of its employment. Finally, to demonstrate the benefits of the mulitpath diversity, Figure 10 shows the SER performance of the system when transmitting over frequencyflat fading channels (SER FF)
versus the performance of the system when transmitting over frequencyselective fading channels (SER FS). As can be seen, even though there were more parameters to estimate, such
are the benefits of multipath diversity that it enabled the frequencyselective fading system to outperform the frequencyflat fading system by 6 dB at an SER of 8 x 10^{4},
even when ideal conditions were assumed in the frequencyflat fading system.

FIGURE 7: The delay error probability with an increase in the signaltonoise
ratio (SNR, dB) for a varying number of pilot frames (x = 1 ,…, 5).



FIGURE 8: The mean square error (MSE) of the channel estimation with an
increasing signaltonoise ratio (SNR, dB) under different scenarios: Cramér–Rao
bound (CRB)(no feedback) – CRB for the channel estimation when the packet
scheme was used but no feedback was employed; CRB with feedback – CRB
for the channel estimation when the packet scheme was used and feedback
was employed; MSE (normal) – MSE of the channel estimation in the normal
system where the packet scheme was not used; MSE (no feedback) – MSE of
the channel estimation when the packet scheme was used but no feedback was
employed; and MSE (with feedback) – MSE of the channel estimation when the
packet scheme was used and feedback was employed.



FIGURE 9: The symbol error rate (SER) performance with an increasing signalto
noise ratio (SNR, dB) of the normal scheme compared to the packet scheme
under different scenarios: SER TR (normal) – SER performance when the
timereversal equalisation technique was used and the packet scheme was
not employed; SER SC (normal) – SER performance when the singlecarrier
equalisation technique was used and the packet scheme was not employed;
SER TR (packet) – SER performance when the timereversal equalisation
technique was used and the packet scheme was employed; SER SC (packet) –
SER performance when the singlecarrier equalisation technique was used and
the packet scheme was employed; SER TR (ideal) – SER performance of the
system under ideal conditions when timereversal equalisation was used at the
receiver; SER TR (ideal) – SER performance of the system under ideal conditions
when singlecarrier equalisation was used at the receiver; and SER SC (ref) –
SER performance of a system such as that presented by Mheidat and Uysal16, in
which ideal conditions were assumed.



FIGURE 10: The symbol error rate (SER) performances of the frequencyflat
fading channels (SER FF and Ideal FF) compared to that of the frequencyselective
fading channels (SER FS and Ideal FS) with an increasing signaltonoise
ratio (SNR, dB), demonstrating the benefits of multipath diversity.


The channel and delay estimation algorithm proposed by Xu and Padayachee^{8} was extended to accommodate frequencyselective channels. New timereversal and singlecarrier equalisation
techniques proposed were capable of accommodating the asynchronicity of the system. The required criteria to design optimal pilot sequences for frequencyselective channels were presented, with the
imposed conditions suggesting that suboptimal sequences will have to be used in most cases. The MSE simulation results demonstrated the superior performance of the packet scheme over the normal
scheme with an improvement in performance of up to 5 dB when feedback was used. The SER simulations showed that, although both the normal and packet schemes achieved similar diversity orders,
the packet scheme demonstrated a 1 dB gain over the normal scheme. Also, it was important to note that, although there were more fading coefficients to estimate, the frequencyselective fading
channel model still outperformed the frequencyflat fading channel model in terms of SER performance because of the diversity benefits provided by the multipath nature of the frequencyselective
fading channel.
Appendix 1: Proof of P(τ)TH_{}i*^{ }P(τ) T = H_{}i^{H}


The proof of P(τ)TH_{}i*^{ }P(τ) T = H_{}i^{H }translates into proving that
This is because H_{}i is a (N + 2L) x (N + 2L) circulant matrix defined by
The step by step proof is outlined as follows:• Using [Eqn 3] the matrix TH_{}i corresponds to swapping the (N + 2L – k – 1)st rows with the (k + 1)st rows of
H_{}i, which results in
• The matrix P(τ)TH_{}i corresponds to performing τ downward cyclic shifts on the rows of [Eqn A3], resulting in
• The matrix P(τ)TH_{}iP(τ) corresponds to performing τ left cyclic shifts on the columns of [Eqn A4], resulting in
• Finally, using [Eqn 3], P(τ)TH_{}iP(τ)T corresponds to swapping the (N + 2L – (l – τ) – 1)st columns with the ((l – τ) + 1)st columns of [Eqn A5], resulting in
• Conjugating both sides of [Eqn A6] results in P(τ) T H_{}i* P (τ)T = H^{}H_{i}
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