3.1 Methodology
In this research, we demonstrate how subcarrier selection can be exploited to achieve reduction in average transmit optical power in MCCDMAbased indoor optical wireless communications with IM/DD. We propose subcarrier selection based on linear equalization, which is considered to be the simplest and least expensive technique to be implemented [26] to mitigate various impairments in conventional MCCDMA systems. Along with taking advantage of these positive features of linear equalization, we derive upper bounds for obtaining fixed DC bias values for both and pre and postequalization in MCCDMAbased indoor optical wireless communications with IM/DD. Based on these upper bounds, we devise subcarrier selection criteria for both pre and postequalization implementation.The subcarrier selection algorithms obtained analytically are used in the simulation to observe the average transmit optical power reduction separately for both cases.
Optical wireless channels exhibit frequencyselective nature at high data rates for multicarrier signals. To use the transmission bandwidth efficiently, subcarriers with better channel gains are used through an equalizationbased criterion. For this, we investigate downlink optical wireless transmissions with the data rates of 10 Mbps, which is considered moderate for optical wirelessbased industrial applications [1, 2]. The mentioned data transmission rate is used, keeping in view the two important factors in considering IRbased indoor optical wireless systems with diffused configuration. Firstly, the standards framed by IrDA and IEEE for typical room sizes in indoor optical wireless systems are of the order of this range [9, 29]. Secondly, the operating speed of currently available commercial devices in the market is typically in the same order as for our system. According to [30], there are theoretical limits and practical constraints like suitable optical sources and drive electronics in achieving high data rates for such systems. Data transmissions are simulated separately using MATLAB (MathWorks Inc., Natick, MA, USA), for pre and postequalization cases. The key parameters used for the selection of subcarriers are the channel gains of individual subcarriers. Since channel variations are slow in indoor environments, the channel state information (CSI) derived at a particular instant can be used for some subsequent time duration in indoor optical wireless systems. Subcarrier selection for MCCDMA can be further advocated by the fact that different subcarriers contain information on the same data symbol; therefore, noisedominated subcarriers can be discarded and signal energy can be reallocated to better subcarriers.
3.2 System model
The system model is specified by the following notations. For analysis, it is sufficient to focus on the transmission of a single symbol from each user

L, number of active users

N, number of subcarriers

N_{cp}=γ N, length of cyclic prefix where 0 ≤ γ ≤ 1

{\mathcal{N}}_{\mathrm{a}}, set of active (selected) subcarrier indices

N_{a}, number of active subcarriers indices, i.e.,
{N}_{\mathrm{a}}=\left{\mathcal{N}}_{\mathrm{a}}\right

d_{
l
}, data symbol from user l

c_{l,n}, CDMA codeword component for user l on subcarrier
n\in {\mathcal{N}}_{\mathrm{a}}

T, MCCDMA symbol period

f_{c}, electronic carrier (passband) frequency

p(t), transmit pulse shape

B, fixed bias of nonnegative passband signal for IM

A_{max}, maximum amplitude of QPSK symbol

s_{b}(t), baseband MCCDMA signal

s_{p}(t), passband MCCDMA signal

s_{opt}(t), transmit optical signal

E_{s}, QPSK symbol energy

H_{
n
}, FFT of the discretetime channel impulse response with length N symbol periods ( n∈{0,…,N−1})

W_{
n
}, complex AWGN values, which are iid circularly symmetric complex Gaussian rv with variance σ^{2} (the variance of a complex random variable is the sum of the variance of the real part and that of the imaginary part).
Each user’s binary data from the source is mapped onto complexvalued symbols d_{
l
} and spread in the frequency domain using a userspecific WalshHadamard spreading code. Each CDMA codeword is assumed to be bipolar with its components belonging to the set {1,−1}. However, for convenience, we assume that a codeword component is equal to 0 on each inactive subcarrier. For example, out of eight subcarriers, if the first two and the last two subcarriers are active for user 0 and if the CDMA codeword for user 0 is (1,−1,1,−1), then we write
\begin{array}{l}({c}_{0}^{0},{c}_{1}^{0},{c}_{2}^{0},{c}_{3}^{0},{c}_{4}^{0},{c}_{5}^{0},{c}_{6}^{0},{c}_{7}^{0})=(1,1,0,0,0,0,1,1).\end{array}
Hence, users are distinguished by their respective code sequences. Every chip of the spreading code representing a fraction of the information symbol is transmitted through an active subcarrier. As described in Section 3.1, these active subcarriers are selected based on the CSI.
Figure 1 illustrates the transmission system model. An IFFT block modulates all QPSK data symbols corresponding to the total number of subcarriers. A cyclic prefix is inserted between each pair of successive symbols after converting them into a serial stream. Finally, the data signal is converted from digital to analog after which a fixed DC bias B is added to the signal to facilitate intensity modulation. When this signal travels through the medium, different subcarriers suffer different degradation and lose their mutual orthogonality. At the receiver, the DC bias is removed. After demodulation, each cyclic prefix is removed to obtain Nsubcarrier components. The N−N_{a} components are discarded after FFT and the remaining N_{a} components are despreaded to get each respective userspecific data symbol.
The above model is used to derive upper bounds for a fixed DC bias by employing preequalization and postequalization schemes. For each case under investigation, CSI is assumed to be available at the transmitter or at the receiver as per post or preequalization requirements, respectively. Then these fixed biases, denoted by B_{post} and B_{pre}, are employed in finding the optimal criteria for subcarrier selection in both cases. The following two subsections show the analytical models for both cases along with the derivation of conservative bounds for fixed DC biases. In both cases, subcarrier selection is based on maximizing the argument of the Q function in the BER expression, where the Q function is the complementary cumulative distribution function of zeromean unitvariance Gaussian random variable.
3.2.1 Postequalization case
At the transmitter, regardless of the number of active subcarriers N_{a}, the number of distinct symbols to be transmitted will always be N. The data symbols after the IFFT operation are represented as
\begin{array}{l}{D}_{k}=\frac{1}{\sqrt{N}}\sum _{l=0}^{L1}\sum _{n=0}^{N1}{d}_{l}\xb7{c}_{l,n}{e}^{j2\mathrm{\pi kn}/N},k\in \phantom{\rule{0.3em}{0ex}}\left\{0,\dots ,N1\right\}.\end{array}
(3)
The complex baseband transmitted signal is
\begin{array}{l}{s}_{\mathrm{b}}\left(t\right)=\sum _{k=0}^{N1}{D}_{k}p(t\mathrm{kT}/(1+\gamma \left)N\right)\phantom{\rule{2em}{0ex}}\end{array}
(4)
where p(t) is the unit norm rectangular transmit pulse shape of width T/(1+γ)N
\begin{array}{l}p\left(t\right)=\left\{\begin{array}{ll}\sqrt{\frac{(1+\gamma )N}{T}}& t\in (0,T/((1+\gamma )N\left)\right]\\ 0,& \text{otherwise.}\end{array}\right.\end{array}
(5)
Hence, the real passband MCCDMA signal to be used for IM is
\begin{array}{l}s\left(t\right)=\text{Re}\left[{e}^{j2\pi {f}_{\mathrm{c}}t}\sum _{k=0}^{N1}{D}_{k}p(t\mathrm{kt}/(1+\gamma \left)N\right)\right],\phantom{\rule{2em}{0ex}}\end{array}
(6)
while the transmitted optical signal will be
\begin{array}{l}{s}_{\text{opt}}\left(t\right)=s\left(t\right)+{B}_{\text{post}}.\end{array}
(7)
We focus on the transmission period of the k th pulse, i.e., p(t−k T/(1+γ)N). The real signal amplitude from (5) and (6) is given by
\begin{array}{l}s\left(t\right)=\sqrt{\frac{(1+\gamma )N}{T}}\text{Re}\left\{{D}_{k}{e}^{j2\pi {f}_{\mathrm{c}}t}\right\},\phantom{\rule{2em}{0ex}}\\ t\in \left[k\frac{T}{(1+\gamma )N},(k+1)\frac{T}{(1+\gamma )N}\right].\phantom{\rule{2em}{0ex}}\end{array}
For f_{c} > > (1+γ)N/T (i.e., several periods of carrier waveform during the pulse interval), which is typically the case, the minimum amplitude of s(t) in \left[k\frac{T}{(1+\gamma )N},(k+1)\frac{T}{(1+\gamma )N}\right] can be taken as
\begin{array}{l}\sqrt{\frac{(1+\gamma )N}{T}}\left{D}_{k}{e}^{j2\pi {f}_{\mathrm{c}}t}\right\phantom{\rule{1pt}{0ex}}=\sqrt{\frac{(1+\gamma )N}{T}}\left{D}_{k}\right.\phantom{\rule{2em}{0ex}}\end{array}
It follows that the required bias during this interval is \sqrt{\frac{(1+\gamma )N}{T}}\left{D}_{k}\right, which can be bounded as
\begin{array}{ll}{B}_{\text{post}}& =\sqrt{\frac{(1+\gamma )N}{T}}\left{D}_{k}\right\phantom{\rule{2em}{0ex}}\\ =\sqrt{\frac{(1+\gamma )N}{T}}\left\frac{1}{\sqrt{N}}\sum _{l=0}^{L1}\sum _{n=0}^{N1}{d}_{l}\xb7{c}_{l,n}{e}^{j2\mathrm{\pi kn}/N}\right\phantom{\rule{2em}{0ex}}\\ =\sqrt{\frac{1+\gamma}{T}}\left\sum _{l=0}^{L1}\sum _{n=0}^{N1}{d}_{l}\xb7{c}_{l,n}{e}^{j2\mathrm{\pi kn}/N}\right\phantom{\rule{2em}{0ex}}\\ \le \sqrt{\frac{1+\gamma}{T}}\sum _{l=0}^{L1}\left{d}_{l}\sum _{n=0}^{N1}{c}_{l,n}{e}^{j2\mathrm{\pi kn}/N}\right\phantom{\rule{2em}{0ex}}\\ \le \sqrt{\frac{1+\gamma}{T}}\sum _{l=0}^{L1}\left{d}_{l}\right\left\sum _{n=0}^{N1}{c}_{l,n}{e}^{j2\mathrm{\pi kn}/N}\right.\phantom{\rule{2em}{0ex}}\end{array}
(8)
We are interested in the maximum negative value of the signal. Hence, we set d_{
l
}≤A_{max}, which, in turn, depends upon the constellation configuration and the allocated subcarrier powers. Moreover, we set {C}_{l,k}=\sum _{n=0}^{N1}{c}_{l,n}{e}^{j2\mathrm{\pi kn}/N}. So inequality (8) becomes
\begin{array}{l}{B}_{\text{post}}\le \sqrt{\frac{1+\gamma}{T}}{A}_{\text{max}}\sum _{l=0}^{L1}{C}_{l,k}.\end{array}
To give an upper bound that is independent of k, i.e., fixed bias, the final expression is
\begin{array}{l}{B}_{\text{post}}\le \underset{k\in \{0,1,\dots ,N1\}}{\text{max}}\sqrt{\frac{1+\gamma}{T}}{A}_{\text{max}}\sum _{l=0}^{L1}{C}_{l,k}.\end{array}
(9)
Since only the data on the active subcarriers will be processed at the receiver
\begin{array}{ll}{C}_{l,k}& =\left\sum _{n\in {\mathcal{N}}_{\mathrm{a}}}{c}_{l,n}{e}^{j2\mathrm{\pi kn}/N}\right\phantom{\rule{2em}{0ex}}\\ \le \sum _{n\in {\mathcal{N}}_{\mathrm{a}}}\left{c}_{l,n}{e}^{j2\mathrm{\pi kn}/N}\right={N}_{\mathrm{a}},\phantom{\rule{2em}{0ex}}\end{array}
(10)
yielding an upper bound
\begin{array}{ll}\sum _{l=0}^{L1}{C}_{l,k}& \le L{N}_{\mathrm{a}}.\phantom{\rule{2em}{0ex}}\end{array}
(11)
The expression in (10) imposes an upper bound on the value of C_{l,k}, thus yielding a conservative value of the DC bias. Thus, from (9) and (11), we get
\begin{array}{l}{B}_{\text{post}}\le \sqrt{\frac{1+\gamma}{T}}{A}_{\text{max}}L{N}_{\mathrm{a}}.\phantom{\rule{2em}{0ex}}\end{array}
(12)
For the BER analysis, we shall focus on the equivalent baseband complex discretetime system model. The received signals at the OFDM receiver can be expressed as vector r=(r_{0},…,r_{N−1}) such that
{r}_{n}={H}_{n}{s}_{n}+{W}_{n},
(13)
where (H_{0},…,H_{N−1}) is the FFT of the discretetime channel impulse response (whose length is N symbol periods), and W_{
n
}s are complex AWGN values. In particular, we assume that W_{
n
}s are iid circularly symmetric Gaussian random variables with variance σ^{2} (when the noise PSD is given by N_{0}/2, we have σ^{2}=N_{0}/N). For equalization as well as restoring the CDMA code orthogonality, we apply onetap equalization to obtain {\mathbf{\text{r}}}^{{}^{\prime}}=\left({r}_{0}^{{}^{\prime}},\dots ,{r}_{N1}^{{}^{\prime}}\right) such that
{r}_{n}^{{}^{\prime}}={s}_{n}+{W}_{n}/{H}_{n}.
(14)
From the equalized received signals, the QPSK symbol d_{
l
} can be obtained using the codeword c_{
l
}, i.e., despreading. Let {\widehat{d}}_{l} be the received signal after despreading. In particular,
{\widehat{d}}_{l}={\mathbf{\text{c}}}_{l}^{\mathsf{T}}{\mathbf{\text{r}}}^{{}^{\prime}}={N}_{\mathrm{a}}{d}_{l}+\sum _{n=1}^{N}{c}_{l,n}{W}_{n}/{H}_{n}.
(15)
For analyzing a userspecific performance, without loss of generality, we consider the received signal of user 0 ({\widehat{d}}_{0}).
\begin{array}{l}{\widehat{d}}_{0}={N}_{\mathrm{a}}{d}_{0}+\sum _{n=1}^{N}{c}_{0,n}{W}_{n}/{H}_{n}.\end{array}
(16)
For QPSK, the BER can be found from the received symbol energy and the noise variance as
\begin{array}{ll}\phantom{\rule{12.0pt}{0ex}}\text{Received symbol energy}& ={N}_{\mathrm{a}}^{2}{E}_{\mathrm{s}}\phantom{\rule{2em}{0ex}}\\ \text{Noise variance}& ={\sigma}^{2}\left(\sum _{n\in {\mathcal{N}}_{\mathrm{a}}}1/{\left{H}_{n}\right}^{2}\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{2em}{0ex}}{\text{BER}}_{\text{posteq}}& =\mathcal{Q}\left(\frac{{N}_{\mathrm{a}}}{\sigma}\sqrt{\frac{{E}_{\mathrm{s}}}{\left(\sum _{n\in {\mathcal{N}}_{\mathrm{a}}}1/{\left{H}_{n}\right}^{2}\right)}}\right).\phantom{\rule{2em}{0ex}}\end{array}
(17)
For QPSK, using {A}_{\text{max}}=\sqrt{{E}_{\mathrm{s}}} in (12), the BER in terms of the DC bias is found to be
\begin{array}{ll}{\text{BER}}_{\text{posteq}}& =\mathcal{Q}\left(\frac{{B}_{\text{post}}}{\mathrm{L\sigma}}\sqrt{\frac{T}{1+\gamma}}\sqrt{\frac{1}{\sum _{n\in {\mathcal{N}}_{\mathrm{a}}}1/{\left{H}_{n}\right}^{2}}}\right)\phantom{\rule{2em}{0ex}}\end{array}
(18)
Subcarrier selection: postequalization case. The BER expression in (18) yields the following theorem that provides a method to select active subcarriers. Let N_{c}(L) denote the length of CDMA codewords required to accommodate L users. Note that the value N_{c}(L) depends on the type of codewords used.
Theorem 1
Given that we use the DC bias in (12), the set of active subcarriers that yields the minimum BER is selected as follows:

1.
The number of active subcarriers is N _{c}(L).

2.
The selected active subcarriers are the N _{c}(L) subcarriers with the highest gain magnitudes, i.e., highest H _{
n
}s.
Proof
Minimize the BER expression in (18) over N_{c}(L)≤N_{a}≤N is equivalent to solving the following optimization problem:
\begin{array}{l}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\text{maximize}\phantom{\rule{2em}{0ex}}\xi =\sqrt{\frac{1}{\sum _{n\in {\mathcal{N}}_{\mathrm{a}}}1/{H}_{n}{}^{2}}}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\text{subject to}\phantom{\rule{1em}{0ex}}{N}_{\mathrm{c}}\left(L\right)\le {N}_{\mathrm{a}}\le N\phantom{\rule{2em}{0ex}}\end{array}
The value of ξ is decreasing with N_{a} since H_{
n
}’s are always positive. Consequently, the smallest possible value of N_{a} is optimal. Since we need to use at least N_{c}(L) subcarriers, it follows that N_{a}=N_{c}(L). This proves part 1.
Now, given that we must use N_{c}(L) subcarriers, the best choice is to select the subcarriers with the highest magnitude gains to maximize ξ. This proves part 2. □
3.2.2 Preequalization case
When equalization is employed at the transmitter, every data symbol is weighted by 1/H_{
n
}. Accordingly, the data symbols after the IFFT operation are represented as
\begin{array}{l}{D}_{k}=\frac{1}{\sqrt{N}}\sum _{l=0}^{L1}\sum _{n=0}^{N1}\frac{{d}_{l}\xb7{c}_{l,n}}{{H}_{n}}{e}^{j2\mathrm{\pi kn}/N}.\end{array}
(19)
After a similar treatment as in the case of postequalization, we get the fixed DC bias B_{pre} as follows
\begin{array}{ll}{B}_{\text{pre}}& =\sqrt{\frac{(1+\gamma )N}{T}}\left{D}_{k}\right\phantom{\rule{2em}{0ex}}\\ =\sqrt{\frac{(1+\gamma )N}{T}}\left\frac{1}{\sqrt{N}}\sum _{l=0}^{L1}\sum _{n=0}^{N1}\frac{{d}_{l}\xb7{c}_{l,n}}{{H}_{n}}{e}^{j2\mathrm{\pi kn}/N}\right\phantom{\rule{2em}{0ex}}\\ =\sqrt{\frac{1+\gamma}{T}}\left\sum _{l=0}^{L1}\sum _{n=0}^{N1}\frac{{d}_{l}\xb7{c}_{l,n}}{{H}_{n}}{e}^{j2\mathrm{\pi kn}/N}\right\phantom{\rule{2em}{0ex}}\end{array}
(20)
\begin{array}{l}\le \sqrt{\frac{1+\gamma}{T}}\sum _{l=0}^{L1}\left{d}_{l}\sum _{n=0}^{N1}\frac{{c}_{l,n}}{{H}_{n}}{e}^{j2\mathrm{\pi kn}/N}\right\phantom{\rule{2em}{0ex}}\\ \le \sqrt{\frac{1+\gamma}{T}}\sum _{l=0}^{L1}\left{d}_{l}\right\left\sum _{n=0}^{N1}\frac{{c}_{l,n}}{{H}_{n}}{e}^{j2\mathrm{\pi kn}/N}\right.\phantom{\rule{2em}{0ex}}\end{array}
(21)
We are interested in the maximum negative value of the signal. Hence, we set d_{
l
}≤A_{max}, which, in turn, depends upon the constellation configuration and the allocated subcarrier power. Moreover, we set {C}_{l,k}^{{}^{\prime}}=\left\sum _{n=0}^{N1}\frac{{c}_{l,n}}{{H}_{n}}{e}^{j2\mathrm{\pi kn}/N}\right. So inequality (21) becomes
\begin{array}{l}{B}_{\text{pre}}\le \sqrt{\frac{1+\gamma}{T}}{A}_{\text{max}}\sum _{l=0}^{L1}{C}_{l,k}^{{}^{\prime}}.\end{array}
To give an upper bound that is independent of k, i.e., fixed bias, the final expression is
\begin{array}{l}{B}_{\text{pre}}\le \underset{k\in \{0,1,\dots ,N1\}}{\text{max}}\sqrt{\frac{1+\gamma}{T}}{A}_{\text{max}}\sum _{l=0}^{L1}{C}_{l,k}^{{}^{\prime}}.\end{array}
(22)
Since only the data on the active subcarriers will be processed at the receiver, we can generalize in a similar fashion as in the case of postequalization,
\begin{array}{ll}{C}_{l,k}^{{}^{\prime}}& =\left\sum _{n\in {\mathcal{N}}_{\mathrm{a}}}\frac{{c}_{l,n}}{{H}_{n}}{e}^{j2\mathrm{\pi kn}/N}\right\phantom{\rule{2em}{0ex}}\\ \le \sum _{n\in {\mathcal{N}}_{\mathrm{a}}}\left\frac{{c}_{l,n}}{{H}_{n}}{e}^{j2\mathrm{\pi kn}/N}\right=\sum _{n\in {\mathcal{N}}_{\mathrm{a}}}\frac{1}{\left{H}_{n}\right},\phantom{\rule{2em}{0ex}}\end{array}
(23)
yielding an upper bound
\begin{array}{ll}\sum _{l=0}^{L1}{C}_{l,k}^{{}^{\prime}}& \le L\sum _{n\in {\mathcal{N}}_{\mathrm{a}}}\frac{1}{\left{H}_{n}\right}.\phantom{\rule{2em}{0ex}}\end{array}
(24)
The expression in (23) imposes an upper bound on the value of {C}_{l,k}^{{}^{\prime}}, thus yielding a conservative value of the DC bias. Thus, from (22) and (24), we get
\begin{array}{l}{B}_{\text{pre}}\le \sqrt{\frac{1+\gamma}{T}}{A}_{\text{max}}L\sum _{n\in {\mathcal{N}}_{\mathrm{a}}}\frac{1}{\left{H}_{n}\right}.\phantom{\rule{2em}{0ex}}\end{array}
(25)
By proceeding in the similar fashion as in the case of postequalization and incorporating noise variance=σ^{2}N_{a}, we get the BER expression as
\begin{array}{l}{\text{BER}}_{\text{preeq}}=\mathcal{Q}\left(\frac{1}{\sigma}\sqrt{{E}_{\mathrm{s}}{N}_{\mathrm{a}}}\right).\phantom{\rule{2em}{0ex}}\end{array}
(26)
Using (25) and incorporating {A}_{\text{max}}=\sqrt{{E}_{\mathrm{s}}} for QPSK, the BER expressing in terms of the DC bias is
\begin{array}{l}{\text{BER}}_{\text{preeq}}=Q\left(\sqrt{\frac{T{N}_{\mathrm{a}}}{1+\gamma}}\frac{{B}_{\text{pre}}}{\mathrm{L\sigma}\sum _{n\in {\mathcal{N}}_{\mathrm{a}}}1/\left{H}_{n}\right}\right).\phantom{\rule{2em}{0ex}}\end{array}
(27)
Subcarrier selection: preequalization case. The BER expression in (27) yields the following theorem that provides a method to select active subcarriers.
Theorem 2
Given that we use the DC bias in (25), the set of active subcarriers that yields the minimum BER is selected as follows:

1.
The number of active subcarriers is N _{c}(L).

2.
The selected active subcarriers are the N _{c}(L) subcarriers with the highest gain magnitudes i.e., highest H _{
n
}s.
Proof
Minimizing the BER expression in (27) over N_{c}(L) ≤ N_{a}≤ N is equivalent to solving the following optimization problem:
\begin{array}{l}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\text{maximize}\phantom{\rule{2em}{0ex}}\kappa =\frac{\sqrt{{N}_{\mathrm{a}}}}{\sum _{n\in {\mathcal{N}}_{\mathrm{a}}}1/\left{H}_{n}\right}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\text{subject to}\phantom{\rule{1em}{0ex}}{N}_{\mathrm{c}}\left(L\right)\le {N}_{\mathrm{a}}\le N\phantom{\rule{2em}{0ex}}\end{array}
Since H_{
n
}s are smaller than 1 (i.e., signal attenuation typically in the order of 10 ^{−7}), κ decreases with N_{a}. Consequently, the smallest possible value of N_{a} is optimal. Since we need to use at least N_{c}(L) subcarriers, it follows that N_{a}=N_{c}(L). This proves part 1.
Now, given that we must use N_{c}(L) subcarriers, the best choice is to select the subcarriers with the highest magnitude gains to maximize κ. This proves part 2. □
3.3 Comparison between BER_{
posteq
}and BER_{
preeq
}
From the previous discussions, we conclude that for both postequalization and preequalization it is optimal to select N_{c}(L) subcarriers with the highest H_{
n
}s. The next question is whether postequalization or preequalization performs better in terms of the BER for a given DC bias B_{post}=B_{pre}=B.
The next theorem shows that preequalization always performs no worse than postequalization.
Theorem 3
Given that the DC biases in (12) and (25) are used for postequalization and preequalization respectively, the corresponding BER for preequalization is no more than the BER for postequalization.
Proof
We first rewrite the postequalization BER expression of (18) as
\begin{array}{l}{\text{BER}}_{\text{posteq}}=\mathcal{Q}\left(\frac{B}{\mathrm{L\sigma}}\sqrt{\frac{T}{(1+\gamma ){N}_{\mathrm{a}}}}\sqrt{\frac{{N}_{\mathrm{a}}}{\sum _{n\in {\mathcal{N}}_{\mathrm{a}}}1/{H}_{n}{}^{2}}}\right).\phantom{\rule{2em}{0ex}}\end{array}
(28)
We proceed in a similar fashion from the preequalization BER expression of (27) to obtain
\begin{array}{l}{\text{BER}}_{\text{preeq}}=\mathcal{Q}\left(\frac{B}{\mathrm{L\sigma}}\sqrt{\frac{T}{(1+\gamma ){N}_{\mathrm{a}}}}\frac{{N}_{\mathrm{a}}}{\sum _{n\in {\mathcal{N}}_{\mathrm{a}}}1/\left{H}_{n}\right}\right).\phantom{\rule{2em}{0ex}}\end{array}
(29)
In (28) and (29), the first two factors in the arguments of the Q function are identical (denoted by α). Hence,
\begin{array}{l}\left(28\right)\phantom{\rule{2.77626pt}{0ex}}\Rightarrow \phantom{\rule{2.77626pt}{0ex}}{\text{BER}}_{\text{posteq}}=\mathcal{Q}\left(\alpha \sqrt{\frac{{N}_{\mathrm{a}}}{\sum _{n\in {\mathcal{N}}_{\mathrm{a}}}1/{H}_{n}{}^{2}}}\right)\phantom{\rule{2em}{0ex}}\\ \text{and}\left(29\right)\phantom{\rule{2.77626pt}{0ex}}\Rightarrow \phantom{\rule{2.77626pt}{0ex}}{\text{BER}}_{\text{preeq}}=\mathcal{Q}\left(\alpha \frac{{N}_{\mathrm{a}}}{\sum _{n\in {\mathcal{N}}_{\mathrm{a}}}1/\left{H}_{n}\right}\right).\phantom{\rule{2em}{0ex}}\end{array}
From the direct consequence of the CauchySchwarz inequality \frac{\sum _{i=0}^{n}{x}_{i}}{n}\le \sqrt{\frac{\sum _{i=0}^{n}{x}_{i}^{2}}{n}}, the inequality \frac{{N}_{\mathrm{a}}}{\sum _{n\in {\mathcal{N}}_{\mathrm{a}}}1/\left{H}_{n}\right}\ge \sqrt{\frac{{N}_{\mathrm{a}}}{\sum _{n\in {\mathcal{N}}_{\mathrm{a}}}1/{H}_{n}{}^{2}}} holds. It follows that BER_{preeq} ≤ BER_{posteq}. □