In CI/MC-CDMA system described in Section 2, *N* length CI codes support *N* orthogonal users and additional *N* users are added by pseudo-orthogonal CI codes [21, 22]. To support more users, a high-capacity CI/MC-CDMA system is proposed in [29], where the capacity is increased up to 3*N* users through the splitting of pseudo-orthogonal CI (PO-CI) codes. As defined earlier, the CI code for *k* th user (1 <= *k* <= *K*) is given by \left[1,{e}^{j\mathrm{\Delta}{\theta}_{k}},{e}^{2j\mathrm{\Delta}{\theta}_{k}},\dots ,{e}^{\left(N-1\right)j\mathrm{\Delta}{\theta}_{k}}\right]. This code is divided into odd and even parts. Further, orthogonal subcarriers are also divided into odd and even parts. The odd/even partitioning of PO-CI and odd/even separation of available subcarriers are useful in adding extra users and hence the system capacity.

In multimedia communication, users transmit at variable data rate. In this paper, different data rate users are broadly grouped into high data rate users (HDR) and low data rate users (LoDR). HDR users are assigned by *N* contiguous subcarriers. Non-orthogonal odd/even subcarriers with odd/even CI code are allocated to LoDR users. In multipath fading channel, if some of the subcarriers are passed through deep fade, then other subcarriers are used to ensure low BER. The non-contiguous odd-even subcarrier allocation ensures better performance in deep fade as compared to contiguous subcarrier allocation. Proper user allocation algorithm [29] is maintained to minimize the cross-correlation between different user sets. In multirate high-capacity system model, there are five user sets.

*U*_{1} : assigned normal CI; transmit through all subcarriers

*U*_{2}: assigned odd CI codes; transmit through odd subcarriers

*U*_{3}: assigned even CI codes; transmit through odd subcarriers

*U*_{4}: assigned odd CI codes; transmit through even subcarriers

*U*_{5}: assigned even CI codes; transmit through even subcarriers

The transmitted signal for multirate high-capacity system can be expressed as

\begin{array}{cc}\hfill S\left(t\right)=& \sum _{k=0}^{N-1}\sum _{i=0}^{N-1}{a}_{k}\left[n\right].{e}^{j\left(2\pi {f}_{i}t+\frac{2\pi}{N}.i.k\right)}.p\left(t-n{T}_{b}\right)\hfill \\ +\sum _{k=N}^{\left(3N\u22152\right)-1}\sum _{i=0\forall i=\mathsf{\text{odd}}}^{N-1}{a}_{k}\left[n\right].{e}^{j\left(2\pi {f}_{i}t+\frac{2\pi}{N}.i.k+i\mathrm{\Delta}{\Phi}_{1}\right)}.p\left(t-q.n{T}_{b}\right)\hfill \\ +\sum _{k=3N\u22152}^{2N-1}\phantom{\rule{1em}{0ex}}\sum _{i=0\forall i=\mathsf{\text{odd}}}^{N-1}{a}_{k}\left[n\right]{e}^{j\left(2\pi {f}_{i}t+\frac{2\pi}{N}.\left(i+1\right).k+i\mathrm{\Delta}{\Phi}_{2}\right)}.p\left(t-q.n{T}_{b}\right)\hfill \\ +\sum _{k=2N}^{\left(5N\u22152\right)-1}\sum _{i=0\forall i=\mathsf{\text{even}}}^{N-1}{a}_{k}\left[n\right]{e}^{j\left(2\pi {f}_{i}t+\frac{2\pi}{N}.i.k+i\mathrm{\Delta}{\Phi}_{3}\right)}.p\left(t-q.n{T}_{b}\right)\hfill \\ +\sum _{k=5N\u22152}^{3N-1}\phantom{\rule{1em}{0ex}}\sum _{i=0\forall i=\mathsf{\text{even}}}^{N-1}{a}_{k}\left[n\right].{e}^{j\left(2\pi {f}_{i}t+\frac{2\pi}{N}.\left(i+1\right).k+i\mathrm{\Delta}{\Phi}_{4}\right)}.p\left(t-q.n{T}_{b}\right)\hfill \end{array}

(31)

It is assumed that HDR users transmit data at 'q' times higher than LoDR users. The angles ΔΦ_{1}, ΔΦ_{2}, ΔΦ_{3} and ΔΦ_{4} are phase shift for the different LoDR sets (*U*_{
i
} , *i* = 2, 3, 4, 5) with respect to HDR users assigned by normal CI codes. Different angles are shown in Figure 1.

\begin{array}{c}\mathrm{\Delta}{\Phi}_{1}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}=\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\pi \u22152\\ \mathrm{\Delta}{\Phi}_{2}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}=\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}-\pi \u22152\\ \mathrm{\Delta}{\Phi}_{3}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}=\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}-\left(\pi +\pi \u2215N\right)\\ \mathrm{\Delta}{\Phi}_{4}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}=\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}-\pi \u2215N\end{array}

(32)

These phase angles are chosen such that the interferences between different sets is reduced. Let us assume that *R*_{1,2}(*j*, *k*) represents the cross-correlation between *j* th user in group 1 and *k* th user in group 2.

{R}_{1,2}\left(j,k\right)=\frac{1}{2\mathrm{\Delta}f}\sum _{i=0}^{N-1}\mathsf{\text{cos}}\left[i\left(\mathrm{\Delta}{\theta}_{j}-\mathrm{\Delta}{\theta}_{k}\right)\right]

(33)

Here, the cross-correlation between *j* th user in orthogonal group 1 and all the users in group 2 is identical to the cross-correlation between (*j* + 1)th user in orthogonal group 1 and all the users in group 2. The total numbers of users in group 1 and group 2 are *K*_{1} and *K*_{2}, respectively.

Let *R*_{1,2}(*j*) is the total cross-correlation between *j* th user and all the users in group 2.

{R}_{1,2}\left(j\right)=\frac{1}{{K}_{2}}\sum _{k=1}^{{K}_{2}}{R}_{1,2}\left(j,k\right),\dots \phantom{\rule{1em}{0ex}}\mathsf{\text{for}}j\mathsf{\text{thuser}}

(34)

{R}_{1,2}\left(j+1\right)=\frac{1}{{K}_{2}}\sum _{k=1}^{{K}_{2}}{R}_{1,2}\left(j+1,k\right),\dots \phantom{\rule{1em}{0ex}}\mathsf{\text{for}}\left(j+1\right)\mathsf{\text{thuser}}

(35)

In CI-based system, *R*_{1,2}(*j*) = *R*_{1,2}(*j* + 1), i.e., every user in one set has same total cross-correlation from users of the other set. If both sets have same number of users, i.e., *K*_{1} = *K*_{2}, then the total cross-correlation between *j* th user in orthogonal group 1 and all the users in group 2 is identical to the cross-correlation between *k'* th user in orthogonal group 2 and all the users in group 1. Total cross-correlation between group 1 and group 2 can be written as

{R}_{1,2}={\left[\frac{1}{{K}_{1}\times {K}_{2}}\sum _{j=1}^{{K}_{1}}\sum _{k=1}^{{K}_{2}}{\left({R}_{1,2}\left(j,k\right)\right)}^{2}\right]}^{\frac{1}{2}}

(36)

If *K*_{1} = *K*_{2} = *N*, then *R*_{1,2} becomes

{R}_{1,2}=\frac{1}{N}{\left[\sum _{j=1}^{N}{\left({R}_{1,2}\left(j,0\right)\right)}^{2}\right]}^{\frac{1}{2}}

(37)

Let {R}_{{U}_{x},{U}_{y}}\left(j,k\right) refers to cross-correlation between *j* th spreading sequence in *U*_{
x
} user set and *k* th spreading sequence in *U*_{
y
} user set. For real signal, the expression is

\begin{array}{cc}\hfill {{R}_{{U}_{x}}}_{,{U}_{y}}\left(j,k\right)\phantom{\rule{1em}{0ex}}& =\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\frac{1}{2\mathrm{\Delta}f}\sum _{i=0}^{N-1}\mathsf{\text{cos}}\left[i\left(\mathrm{\Delta}{\theta}_{j}-\mathrm{\Delta}{\theta}_{k}\right)\right]\hfill \\ =\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\frac{1}{2\mathrm{\Delta}f}\sum _{i=0}^{N-1}\mathsf{\text{cos}}\left[i\left(\frac{2\pi}{N}j-\frac{2\pi}{N}k\right)\right]\hfill \end{array}

(38)

{R}_{{U}_{1},{U}_{2}}\left(j,k\right)\phantom{\rule{2.77695pt}{0ex}}=\frac{1}{2\mathrm{\Delta}f}\sum _{i=0\forall i=\mathsf{\text{odd}}}^{N-1}\mathsf{\text{cos}}\left[i\left(\mathrm{\Delta}{\theta}_{j}-\mathrm{\Delta}{\theta}_{k}\right)\right]

(39)

Total cross-correlation between *j* th user and all the user of *U*_{2} set becomes

{R}_{{U}_{1},{U}_{2}}\left(j\right)=\frac{1}{{K}_{{U}_{2}}}\sum _{k=1}^{{K}_{{U}_{2}}}{R}_{{U}_{1},{U}_{2}}\left(j,k\right)

(40)

where {K}_{{U}_{x}} represents total number of users in *U*_{
x
} set. In general,

{R}_{{U}_{1},{U}_{m}}\left(j\right)=\frac{1}{{K}_{{U}_{m}}}\sum _{k=1}^{{K}_{{U}_{m}}}{R}_{{U}_{1},{U}_{2}}\left(j,k\right),\phantom{\rule{1em}{0ex}}m\in 2,3,4,5

(41)

{{R}_{{U}_{1}}}_{,{U}_{m}}\left(j,k\right)=\frac{1}{2\mathrm{\Delta}f}\sum _{i=0\forall i=\mathsf{\text{odd}}}^{N-1}\mathsf{\text{cos}}\left[i\left(\mathrm{\Delta}{\theta}_{j}-\mathrm{\Delta}{\theta}_{k}\right)\right],\phantom{\rule{1em}{0ex}}m\in 2,3

(42)

and

{{R}_{{U}_{1}}}_{,{U}_{m}}\left(j,k\right)=\frac{1}{2\mathrm{\Delta}f}\sum _{i=0\forall i=\mathsf{\text{even}}}^{N-1}\mathsf{\text{cos}}\left[i\left(\mathrm{\Delta}{\theta}_{j}-\mathrm{\Delta}{\theta}_{k}\right)\right],\phantom{\rule{1em}{0ex}}m\in 4,5

(43)

So, total cross-correlation between *j* th user in *U*_{1} set and all the users in other set is given by

{R}_{{U}_{1},\left({U}_{2},{U}_{3},{U}_{4},{U}_{5}\right)}\left(j\right)=\sqrt{{R}_{{U}_{1},{U}_{2}}^{2}\left(j\right)+{R}_{{U}_{1},{U}_{3}}^{2}\left(j\right)+{R}_{{U}_{1},{U}_{4}}^{2}\left(j\right)+{R}_{{U}_{1},{U}_{5}}^{2}\left(j\right)}

(44)

From Equation (44), it is clear that the users of the same set of subcarrier used by *U*_{1} user set create interference to the *j* th user of *U*_{1}. set. Assuming orthogonality is maintained in subcarrier, there is no cross-correlation between [*U*_{2}, *U*_{4}] set and [*U*_{2}, *U*_{5}] set. *U*_{2} and *U*_{3} user sets are using different set of subcarriers that is utilized by *U*_{4} and/or *U*_{5} sets. In same subcarriers, the cross-correlation between two different user set is minimized by proper phase separation described in Equation (32). For *U*_{2} user set, all users from *U*_{1} set and *U*_{3} user create interference on odd subcarrier. Then, total interference for *j* th user in *U*_{2} user is obtained by

{R}_{{U}_{2},\left({U}_{1},{U}_{3}\right)}\left(j\right)=\sqrt{{R}_{{U}_{2},{U}_{1}}^{2}\left(j\right)+{R}_{{U}_{2},{U}_{3}}^{2}\left(j\right)}

(45)

In multipath channel, intercarrier interference (ICI) occurs due to non-orthogonality between subcarrier. So, MAI in multipath fading channel is more than AWGN channel due to ICI.