Frequency diversity occurs in OFDMA systems by sending multiple replicas of the transmitted signal at different carrier frequencies. The idea behind diversity is to provide independent replicas of the same transmitted signal at the receiver and appropriately process them to make the detection more reliable. Different copies of the signal should be transmitted in different frequency bands, a condition which guarantees their independence. The subchannels given in (5) are correlated and no more than a L_{
h
}order frequency diversity gain can be possible since there are only L_{
h
} independent channel coefficients, i.e., {h}_{0}^{\left(u\right)},\dots ,{h}_{{L}_{h}1}^{\left(u\right)}, [14]. Intuitively, we can imagine that there are L_{
h
} groups of \frac{{N}_{p}}{{L}_{h}} frequencies which are ‘identical’. Each user retrieves the maximal diversity if it has at least L_{
h
} blocks of size b which implies b\le \frac{{N}_{p}}{{L}_{h}{N}_{u}}. Therefore, users can achieve full diversity when the blocksize is within the coherent bandwidth \frac{{N}_{p}}{{L}_{h}{N}_{u}}.
In this work, we propose an original approach to find the blocksize that guarantees the maximum diversity. This approach is based on the minimization of the subchannels/subcarrier sumcorrelation function.
In the sequel, we define a measure of correlation between subcarriers, that we call sumcorrelation function, and we derive its properties. Moreover, we find the set of optimal blocksizes b^{∗}∈β which minimizes this correlation in order to minimize the effects that it produces.
3.1 Properties of the sumcorrelation function
Assuming that the channel impulse responses are independent but distributed accordingly to the complex Gaussian distribution \mathcal{C}\mathcal{N}(0,{\sigma}_{h}^{\left(u\right)2})[14], we define for each user the sum of correlations of each subcarrier with respect to the others in the same allocation scheme as follows:
\begin{array}{c}{\Gamma}_{u,m}\left(b\right)=\sum _{{c}_{u}\in {\mathcal{C}}_{u}}\mathbb{E}\left[{H}_{m}^{\left(u\right)}{H}_{{c}_{u}}^{\left(u\right)\ast}\right]\hfill \\ \phantom{\rule{4.3em}{0ex}}=\mathbb{E}\left[{H}_{m}^{\left(u\right)}{}^{2}\right]+\sum _{\begin{array}{c}{c}_{u}\in {\mathcal{C}}_{u}\\ {c}_{u}\ne m\end{array}}\mathbb{E}\left[{H}_{m}^{\left(u\right)}{H}_{{c}_{u}}^{\left(u\right)\ast}\right]\hfill \\ \phantom{\rule{4.3em}{0ex}}=\frac{{\sigma}_{h}^{\left(u\right)2}}{{N}_{p}}{L}_{h}+\frac{{\sigma}_{h}^{\left(u\right)2}}{{N}_{p}}\sum _{{c}_{u}\in {\mathcal{C}}_{u}}\frac{\text{sin}\left[\frac{\pi {L}_{h}}{{N}_{p}}(m{c}_{u})\right]}{\text{sin}\left[\frac{\pi}{{N}_{p}}(m{c}_{u})\right]}\hfill \\ \phantom{\rule{5.5em}{0ex}}\times \phantom{\rule{2.77626pt}{0ex}}{e}^{\mathrm{\pi j}\frac{({L}_{h}1)}{{N}_{p}}(m{c}_{u})}\hfill \end{array}
(6)
where m\in {\mathcal{C}}_{u} is the reference subcarrier and
\begin{array}{c}{\mathcal{C}}_{u}=\bigcup _{k\in \left\{1,2,3,\dots ,\frac{{N}_{p}}{{\mathit{\text{bN}}}_{u}}\right\}}\left\{(k1){\mathit{\text{bN}}}_{u}\phantom{\rule{0.3em}{0ex}}+\phantom{\rule{0.3em}{0ex}}(u1)b+i,\forall i\in \phantom{\rule{0.3em}{0ex}}\left\{1,\dots ,b\right\}\right\}\phantom{\rule{0.3em}{0ex}}.\end{array}
(7)
The set {\mathcal{C}}_{u} is composed of all indices of subcarriers allocated to user u given a blocksize b allocation scheme. The ratio \frac{{N}_{p}}{{\mathit{\text{bN}}}_{u}} represents the number of blocks that can be allocated to each user given a blocksize b. We consider, therefore, that the total N_{
p
} subcarriers are divided into \frac{{N}_{p}}{{\mathit{\text{bN}}}_{u}} largeblocks that contain b N_{
u
} subcarriers corresponding to the N_{
u
} blocks, one for each user, of size b. The set {\mathcal{C}}_{u} is the union of indices of subcarriers allocated to user u in all these largeblocks. Inside of the largeblock of index k, the indices of the b subcarriers allocated to user u are (k1)b N_{
u
} + (u1)b + i, ∀i∈{1,…,b}, where (k1)b N_{
u
} corresponds to the previous k1 largeblocks and (u  1)b corresponds to the previous allocated users (1,2,…,u  1), see Figure 4. The function Γ_{u,m}(b) is the sum of the correlations between subcarriers that are in the same allocation scheme.
Considering the subcarriers m and c_{
u
} in {\mathcal{C}}_{u}, we denote the distance between them by
\begin{array}{l}d:=m{c}_{u}\hfill \\ \phantom{\rule{1em}{0ex}}=({k}^{\prime}{k}^{\mathrm{\prime \prime}}){\mathit{\text{bN}}}_{u}+{i}^{\prime}{i}^{\mathrm{\prime \prime}}\hfill \end{array}
(8)
with {k}^{\prime},{k}^{\mathrm{\prime \prime}}\in \left\{1,2,3,\dots ,\frac{{N}_{p}}{{\mathit{\text{bN}}}_{u}}\right\} and i^{′},i^{′′}∈{1,…,b}. We define the function
f\left(d\right)\triangleq \left\{\begin{array}{cc}{L}_{h}\hfill & \phantom{\rule{1em}{0ex}}\text{if}\phantom{\rule{1em}{0ex}}d=0\\ \frac{\text{sin}\left[\pi \frac{{L}_{h}}{{N}_{p}}d\right]}{\text{sin}\left[\frac{\pi}{{N}_{p}}d\right]}{e}^{\mathrm{\pi j}\frac{({L}_{h}1)}{{N}_{p}}d}& \phantom{\rule{1em}{0ex}}\text{otherwise}.\end{array}\right.
(9)
The next result guarantees that the function f(d) is independent of the user index.
Proposition
1
Given the parameters N_{
p
}, N_{
u
}, and L_{
h
}, the value f(d) of the function in (9) for any d = mc_{
u
} in (8) is independent of the user index u.
Proof: The dependence of f on the user index u is expressed by the term (mc_{
u
}), representing the distance between two subcarriers in the same allocation scheme, where m,{c}_{u}\in {\mathcal{C}}_{u}. If m and c_{
u
} are in two different blocks there exist two indices k^{′} and k^{′′} in \left\{1,2,3,\dots ,\frac{{N}_{p}}{{\mathit{\text{bN}}}_{u}}\right\} such that
\begin{array}{c}m=({k}^{\prime}1){\mathit{\text{bN}}}_{u}+(u1)b+{i}^{\prime}\hfill \end{array}
(10)
\begin{array}{c}{c}_{u}=({k}^{\mathrm{\prime \prime}}1){\mathit{\text{bN}}}_{u}+(u1)b+{i}^{\mathrm{\prime \prime}}\hfill \end{array}
(11)
with i^{′} and i^{′′} in {1,…,b}. Therefore, the distance
\begin{array}{c}m{c}_{u}=\left({k}^{\prime}{k}^{\mathrm{\prime \prime}}\right){\mathit{\text{bN}}}_{u}+{i}^{\prime}{i}^{\mathrm{\prime \prime}}\hfill \end{array}
(12)
does not depend on the user index u. If m and c_{
u
} are in the same block, k^{′} = k^{′′} and the same reasoning holds. This guarantees the independence of the function f(d) on the particular user u. ■
The independence between f(d) and user index u comes from the fact that given a blocksize b, the set of distances between subcarriers are the same for all users.
We observe that the function f(d) has the following circularity property.
Proposition 2(Circularity of f(d)).
Given N_{
p
} the number of subcarriers and for any distance d between subcarriers given in (8), we have
f(d+{N}_{p})=f\left(d\right).
(13)
Proof: From (9),
\begin{array}{c}f\left(d+{N}_{p}\right)=\frac{\text{sin}\left[\pi \frac{{L}_{h}}{{N}_{p}}(d+{N}_{p})\right]}{\text{sin}\left[\frac{\pi}{{N}_{p}}(d+{N}_{p})\right]}{e}^{\mathrm{\pi j}\frac{({L}_{h}1)}{{N}_{p}}(d+{N}_{p})}\hfill \\ \phantom{\rule{5em}{0ex}}=\frac{{e}^{\mathrm{j\pi}\frac{{L}_{h}}{{N}_{p}}d}{e}^{\mathrm{j\pi}{L}_{h}}{e}^{\mathrm{j\pi}\frac{{L}_{h}}{{N}_{p}}d}{e}^{j\pi {L}_{h}}}{{e}^{j\pi \frac{1}{{N}_{p}}d}{e}^{j\pi}{e}^{j\pi \frac{1}{{N}_{p}}d}{e}^{j\pi}}\frac{{e}^{j\pi \frac{{L}_{h}}{{N}_{p}}d}{e}^{j\pi {L}_{h}}}{{e}^{j\pi \frac{1}{{N}_{p}}d}{e}^{j\pi}}\hfill \\ \phantom{\rule{5em}{0ex}}=\frac{1{e}^{j2\pi \frac{{L}_{h}}{{N}_{p}}d}{e}^{j2\pi {L}_{h}}}{1{e}^{j2\pi \frac{1}{{N}_{p}}d}{e}^{j2\pi}}\hfill \\ \phantom{\rule{5em}{0ex}}=\frac{{e}^{j\pi \frac{{L}_{h}}{{N}_{p}}d}{e}^{j\pi \frac{{L}_{h}}{{N}_{p}}d}}{{e}^{j\pi \frac{1}{{N}_{p}}d}{e}^{j\pi \frac{1}{{N}_{p}}d}}{e}^{j\pi \frac{({L}_{h}1)}{{N}_{p}}d}\hfill \\ \phantom{\rule{5em}{0ex}}=\frac{\text{sin}\left[\pi \frac{{L}_{h}}{{N}_{p}}d\right]}{\text{sin}\left[\frac{\pi}{{N}_{p}}d\right]}{e}^{\pi j\frac{({L}_{h}1)}{{N}_{p}}d}\hfill \\ \phantom{\rule{5em}{0ex}}=f\left(d\right).\hfill \end{array}
■
We consider the following sumcorrelation metric:
\begin{array}{l}\Gamma \left(b\right)=\sum _{u=1}^{{N}_{u}}\sum _{\begin{array}{c}m\in {\mathcal{C}}_{u}\end{array}}{\Gamma}_{u,m}\left(b\right).\end{array}
(14)
Thanks to Proposition 1 and Proposition 2, it can be expressed as follows:
\begin{array}{lcr}\Gamma \left(b\right)& =& \sum _{u=1}^{{N}_{u}}\left{\mathcal{C}}_{u}\right\frac{{\sigma}_{h}^{\left(u\right)2}}{{N}_{p}}\left(\sum _{\begin{array}{c}d\in {\mathcal{D}}_{b}\end{array}}f\left(d\right)\right)\end{array}
(15)
\begin{array}{lc}=& \sum _{u=1}^{{N}_{u}}\frac{{N}_{p}}{{N}_{u}}\frac{{\sigma}_{h}^{\left(u\right)2}}{{N}_{p}}\left({L}_{h}+\sum _{\begin{array}{c}d\in {\mathcal{D}}_{b}\\ d\ne 0\end{array}}f\left(d\right)\right)\end{array}
(16)
where
\begin{array}{l}{\mathcal{D}}_{b}=\left\{\begin{array}{c}d={\mathit{\text{kbN}}}_{u}+i,\phantom{\rule{1em}{0ex}}i\in \left\{0,\dots ,b1\right\}\\ k\in \left\{0,\dots ,\left(\frac{{N}_{p}}{{\mathit{\text{bN}}}_{u}}1\right)\right\}\hfill \end{array}\right\}.\end{array}
(17)
Given a subcarrier of reference, without loss of generality, the set {\mathcal{D}}_{b} represents the set of the distances^{b} between the subcarrier of reference and all the other subcarriers in the same allocation scheme (the k factor represents here the distance between the largeblocks). This definition is consistent since the function f(d) has the circularity property with respect to N_{
p
}. This means that it does not matter which subcarrier of reference we consider. Therefore, the sumcorrelation function defined in (15) is independent on the reference subcarrier. This guarantees that the next results hold for each user in the system.
We observe that there is no correlation between the subcarrier of reference and other carriers which are spaced from it at a distance equal to a multiple of \frac{{N}_{p}}{{L}_{h}}. It is obvious from the definition of the function f(d) that it is equal to zero when the distance d is a multiple of the ratio \frac{{N}_{p}}{{L}_{h}}:
f\left(r\frac{{N}_{p}}{{L}_{h}}\right)=0,\phantom{\rule{1em}{0ex}}\forall r\in {\mathbb{N}}^{\ast}.
(18)
This is what we observe in Figure 5, in which we plot the absolute value of the function f(d), with m = 1, N_{
p
} = 32, N_{
u
} = 2 and L_{
h
} = 4. We observe that this function is equal to zero for all the multiples of \frac{{N}_{p}}{{L}_{h}}=8.
We have seen that the function f(d) equals zero for all multiples of \frac{{N}_{p}}{{L}_{h}} and that the distance d can be expressed in function of the blocksize b (see (8) and (17)). In the following, we provide the expression of the blocksize b such that d=\frac{{N}_{p}}{{L}_{h}}.
Proposition 3.
Given a fixed distance d=\frac{{N}_{p}}{{L}_{h}} between subcarriers, the corresponding blocksize is equal to b=\frac{{N}_{p}}{{L}_{h}{N}_{u}}.
Proof: Notice that, given our allocation policy in Figure 4, not all the distances can be achieved for any possible blocksize. We consider an arbitrary distance d in {\mathcal{D}}_{b} and we are interested to find the blocksize that ensures d=\frac{{N}_{p}}{{L}_{h}}:
\begin{array}{c}d={\mathit{\text{kbN}}}_{u}+i\phantom{\rule{1em}{0ex}}\text{with}\phantom{\rule{1em}{0ex}}i\in \left\{0,\dots ,b1\right\},\hfill \\ \phantom{\rule{1.5em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}k\in \left\{0,\dots ,\frac{{N}_{p}}{{\mathit{\text{bN}}}_{u}}1\right\}\hfill \\ \phantom{\rule{.5em}{0ex}}=r\frac{{N}_{p}}{{L}_{h}}\phantom{\rule{1em}{0ex}}\text{with}\phantom{\rule{1em}{0ex}}r\le 1.\hfill \end{array}
First, we analyze the case when b\ge \frac{{N}_{p}}{{L}_{h}}. In this case, we observe that the condition d=\frac{{N}_{p}}{{L}_{h}} is satisfied if k = 0 and i=\frac{{N}_{p}}{{L}_{h}}. This means that for all b\ge \frac{{N}_{p}}{{L}_{h}} there are two subcarriers in the same block such that their distance is \frac{{N}_{p}}{{L}_{h}}. More interesting is the case when b<\frac{{N}_{p}}{{L}_{h}}. From the fact that i∈{0,…,b1} and b<\frac{{N}_{p}}{{L}_{h}}, in order to have the distance d=\frac{{N}_{p}}{{L}_{h}}, we have just to consider the case i = 0. Then, it is obvious that we have d equal to \frac{{N}_{p}}{{L}_{h}} when {\mathit{\text{bN}}}_{u}=\frac{{N}_{p}}{{L}_{h}}. This means that the blocksize is b=\frac{{N}_{p}}{{L}_{h}{N}_{u}}. ■
3.2 Novel blockwise allocation scheme
In the next Theorem, we give the set of optimal blocksizes that minimize the sumcorrelation function Γ(b). This set is given by {\beta}^{\ast}=\left\{1,\mathrm{...},\frac{{N}_{p}}{{N}_{u}L},\frac{{N}_{p}}{{N}_{u}{L}_{h}}\right\}\subseteq \beta
Theorem 1.
We consider our uplink system with N_{
p
} subcarriers, N_{
u
} users and a channel impulse response length L_{
h
}. Given {\beta}^{\ast}\triangleq \left\{1,\dots ,\frac{{N}_{p}}{{N}_{u}L},\frac{{N}_{p}}{{N}_{u}{L}_{h}}\right\}, we have

1.
The elements in the set β ^{∗} minimize the sumcorrelation function Γ(b):
{\beta}^{\ast}=\underset{b\in \beta}{\text{arg min}}\phantom{\rule{1em}{0ex}}\Gamma \left(b\right)
(19)

2.
The optimal value of the sumcorrelation function depends only on the system parameters.
\Gamma \left({b}^{\ast}\right)=\sum _{u=1}^{{N}_{u}}{\sigma}_{h}^{\left(u\right)2}\frac{{N}_{p}}{{N}_{u}^{2}},\phantom{\rule{1em}{0ex}}\forall {b}^{\ast}\in {\beta}^{\ast}.
(20)
Proof: The proof is given in the Appendix 5. ■
Proposition 4.
In the case without CSI, assuming that L_{
h
} is not known at the transmitter side, we propose to use β^{∗} restricted to \left\{1,\dots ,\frac{{N}_{p}}{{N}_{u}L}\right\}.
Proof: We observe that \left\{1,\dots ,\frac{{N}_{p}}{{N}_{u}L}\right\} is included in \left\{1,\dots ,\frac{{N}_{p}}{{N}_{u}L},\dots ,\frac{{N}_{p}}{{N}_{u}{L}_{h}}\right\}. Intuitively, this means that, in a more realistic scenario in which only the knowledge of L and not of the channel length L_{
h
} is available, we can still provide the subset of optimal blocksizes. ■
We observe that, in the case without CSI, the minimum value of the sumcorrelation function depends only on the number of subcarriers, number of users, and the variance of the channel impulse response and that the largest optimal blocksize is given by
{b}_{\text{max}}^{\ast}=\frac{{N}_{p}}{{N}_{u}{}_{L}},
(21)
which is a function of system’s parameters: number of subcarriers, number of users, and cyclic prefix length.
In a more general scenario in which the channel length is different for each user, i.e., {L}_{h}^{\left(u\right)}\ne {L}_{h}, the optimal blocksize maximizing the sumcorrelation function is a difficult problem and an open issue. However, in a realistic scenario in which these parameters {L}_{h}^{\left(u\right)} are not known, the system planner would assume the worse case scenario and approximate them with the length of the cyclic prefix L. Since the length of the cyclic prefix L is bigger, the chosen blocklength is suboptimal and given by (21).
3.3 Numerical results: diversity and sumcorrelation
In this section, the aim is to highlight the close relationship between diversity gain, outage probability and the sumcorrelation function. We define the outage probability of the system under consideration as the maximum of the outage probabilities of the users:
{P}_{\text{out}}=\underset{\begin{array}{c}1\le u\le {N}_{u}\end{array}}{max}{P}_{\text{out}}^{\left(u\right)}
(22)
where {P}_{\text{out}}^{\left(u\right)}=\mathit{\text{Pr}}\left\{{C}_{b}^{\left(u\right)}<R\right\} with R is a fixed target transmission rate and {C}_{b}^{\left(u\right)} is the instantaneous mutual information of the user u defined in the following. We consider the transmitted symbols in (2) distributed accordingly to the Gaussian distribution such that \mathbb{E}\left[{\mathbf{\text{x}}}^{\left(u\right)}{\mathbf{\text{x}}}^{\left(u\right)H}\right]=\mathbf{\text{I}}. The user instantaneous achievable spectral efficiency assuming singleuser decoding at the BS [15] in the case without CFO is as follows:
\begin{array}{c}{C}_{b}^{\left(u\right)}=\frac{{N}_{u}}{B}\underset{2}{log}det\left[\mathbf{\text{I}}+{\mathbf{\text{H}}}^{\left(u\right)}{\mathit{\Pi}}_{b}^{\left(u\right)}{\mathbf{\text{H}}}^{\left(u\right)\u2020}\right.\hfill \\ \phantom{\rule{3em}{0ex}}\times \left(\right)close="]">{\left(\mathbf{\text{I}}{\sigma}_{n}^{2}+\sum _{\begin{array}{c}v=1\\ v\ne u\end{array}}^{{N}_{u}}{\mathbf{\text{H}}}^{\left(v\right)}{\mathbf{\Pi}}_{b}^{\left(v\right)}{\mathbf{\text{H}}}^{\left(v\right)\u2020}\right)}^{1}\hfill \end{array}\n \n \n \n =\n \n \n \n \n N\n \n \n u\n \n \n \n \n B\n \n \n \n \n \u2211\n \n \n m\n \u2208\n \n \n C\n \n \n u\n \n \n \n \n \n \n log\n \n \n 2\n \n \n \n \n 1\n +\n \n \n 1\n \n \n \n \n \sigma \n \n \n n\n \n \n 2\n \n \n \n \n \n \n \n H\n \n \n m\n \n \n (\n u\n )\n \n \n \n \n \n \n \n 2\n \n \n \n \n .\n \n \n
(23)
An explicit analytical relation between the sumcorrelation function and the outage probability is still an open problem. The major issue is the fact that the distribution of the mutual information is very complex and closedform expressions for the outage probability are not available in general. For example, Emre Telatar’s conjecture on the optimal covariance matrix minimizing the outage probability in the singleuser MIMO channels [13] is yet to be proven. We propose a new metric, the sumcorrelation function, and show by simulations that there is an underlying relation between the sumcorrelation function and the outage probability. Indeed, it is intuitive that, in SCFDMA systems, correlation among the subcarriers decreases the diversity gain and, thus, the transmission reliability decreases [16, 17]. This explains that the outage probability increases when the correlation among subcarriers is increasing. This connection has been validated via extensive numerical simulations. The interest behind this connection is that the sumcorrelation function has a closedform expression allowing us to perform a rigorous analysis and to find the blockwise subcarrier allocation minimizing the sumcorrelation which is consistent with the optimal blockwise subcarrier allocation minimizing the outage probability. The following results illustrate numerically this connection.
3.3.1 Uncorrelated subcarriers
We consider the case of a SCFDMA system with independent subcarriers. Since the subcarriers are independent the correlation between them is zero, which means that the sumcorrelation Γ(b) is equal to zero for any blocksize b. In the next simulation, we observe that we obtain the same performance in terms of the outage probability regardless of the particular allocation scheme and the blocksize, see Figure 6. Although this scenario is unrealistic from a practical standpoint, it is important to notice that, in this case, there are no privileged blocksizes to achieve better diversity gain.
In Figure 6, we plot the outage probability in the SCFDMA system with independent subcarriers (subchannels) generated by complex Gaussian distribution with respect to SNR for the scenario N_{
p
} = 64, N_{
u
} = 2, and fixed rate R = 1 bits/s/Hz. In particular, in this case with independent subcarriers, we consider the matrix H^{(u)} in (4) to be diagonal with entries {H}_{k}^{\left(u\right)} i.i.d \sim \phantom{\rule{1em}{0ex}}\mathcal{C}\mathcal{N}(0,{\sigma}^{2}). It is clear that for any blocksize (hence, for any subcarrier allocation scheme) we obtain the same performance in terms of outage probability. Therefore, there are not any privileged blocksize allocations to achieve better diversity gain. This motivates and strengthens our observation that the subcarrier correlation has a direct impact on the outage probability.
3.3.2 Correlated subcarriers
In this section, we consider a more interesting and realistic SCFDMA system given in (4). For simplicity and lack of spacerelated reasons, the simulations presented here have been done for the particular case L = L_{
h
}. Numerous other simulations were performed in the general case L_{
h
}≤L, which confirm the theoretical result of Theorem 1.
In Figure 7, we have plotted with respect to the blocksize b, the sumcorrelation function Γ(b) in the SCFDMA system without CFO for the scenario N_{
u
} = 4, L = 4 and σ^{2}(1) = 0.25,σ^{2}(2) = 0.5,σ^{2}(3) = 0.125,σ^{2}(4) = 0.3. The illustrated markers represent the values of the function Γ(b) for the given choice of the parameters of the system. We observe that the minimal values of Γ(b) are obtained for the blocksizes b^{∗}∈β^{∗} = {1,2,4}. In particular, ∀ b^{∗}∈β^{∗} = {1,2,4} we have \Gamma \left({b}^{\ast}\right)={\sum}_{u=1}^{{N}_{u}}\frac{{\sigma}_{h}^{\left(u\right)2}{N}_{p}}{{N}_{u}^{2}}=4.7.
In Figure 8, we use Binary Phase Shift Keying (BPSK) modulation in the following scenario: N_{
p
} = 64, N_{
u
} = 2, L = 8, and {\sigma}_{h}^{\left(1\right)2}={\sigma}_{h}^{\left(2\right)2}. We observe that the optimal blocksizes are in β^{∗} = {1,2,4} (but here, we just plot the smallest and the biggest values) for the BER which confirms that these blocksizes optimize also the sumcorrelation function we have proposed.
In Figure 9, we use BPSK modulation, N_{
p
} = 64, N_{
u
} = 2, L = 4, and {\sigma}_{h}^{\left(2\right)2}=2{\sigma}_{h}^{\left(1\right)2}. The optimal blocksizes are given in the set β^{∗} = {1,2,4,8}.
In Figure 10, we use BPSK modulation in the following scenario: N_{
p
} = 128, N_{
u
} = 2, and L = 8. In this case, we consider an exponential power delay profile which means that {\sigma}_{h}^{\left(u\right)2}=\frac{{e}^{\tau /L}}{{\sum}_{\tau =0}^{L1}{e}^{\tau /L}} with τ∈{0,1,…,L1}. The theoretical results are confirmed since the optimal blocksizes are in β^{∗} = {1,2,4,8}.
In Figure 11, we evaluate the outage probability P_{out} in the SCFDMA system without CFO for the scenario N_{
u
} = 4, L = 4, and R = 1 bits/s/Hz. We observe that the optimal blocksizes are the ones that correspond to the outage probabilities which have a higher decreasing rate as a function of the SNR. We see that the curves with b^{∗}∈β^{∗} = {1,2,4} (in this case {b}_{\text{max}}^{\ast}=\frac{64}{4\times 4}=4) are overlapped and they represent the lower outage probability. These blocksizes are the same that minimize the sumcorrelation function (see Figure 7).
In Figure 12, we plot the outage probability for the SCFDMA system without CFO for the scenario N_{
p
} = 64, N_{
u
} = 2, L = 4, and R = 1 bits/s/Hz so that {b}_{\text{max}}^{\ast}=\frac{64}{2\times 4}=8. In this case in which the subcarriers are correlated, we observe that the curves with b^{∗}∈β^{∗} = {1,2,4,8} have a higher diversity.
Many others simulations, changing the values of the parameters (in particular, N_{
p
} and L), have been performed, and similar observations were made. Moreover, we have done simulations choosing the following as a performance metric:
{\stackrel{~}{P}}_{\text{out},b}=1\prod _{u=1}^{{N}_{u}}\left(1{P}_{\text{out},b}^{\left(u\right)}\right).
(24)
The same observation can be made with this outage metric.