3.1 Methodology
In this research, we demonstrate how subcarrier selection can be exploited to achieve reduction in average transmit optical power in MC-CDMA-based 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 MC-CDMA 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 post-equalization in MC-CDMA-based indoor optical wireless communications with IM/DD. Based on these upper bounds, we devise subcarrier selection criteria for both pre- and post-equalization 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 frequency-selective nature at high data rates for multi-carrier signals. To use the transmission bandwidth efficiently, subcarriers with better channel gains are used through an equalization-based criterion. For this, we investigate downlink optical wireless transmissions with the data rates of 10 Mbps, which is considered moderate for optical wireless-based industrial applications [1, 2]. The mentioned data transmission rate is used, keeping in view the two important factors in considering IR-based 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 post-equalization 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 MC-CDMA can be further advocated by the fact that different subcarriers contain information on the same data symbol; therefore, noise-dominated 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
-
Ncp=γ N, length of cyclic prefix where 0 ≤ γ ≤ 1
-
, set of active (selected) subcarrier indices
-
Na, number of active subcarriers indices, i.e.,
-
d
l
, data symbol from user l
-
cl,n, CDMA codeword component for user l on subcarrier
-
T, MC-CDMA symbol period
-
fc, electronic carrier (passband) frequency
-
p(t), transmit pulse shape
-
B, fixed bias of nonnegative passband signal for IM
-
Amax, maximum amplitude of QPSK symbol
-
sb(t), baseband MC-CDMA signal
-
sp(t), passband MC-CDMA signal
-
sopt(t), transmit optical signal
-
Es, QPSK symbol energy
-
H
n
, FFT of the discrete-time 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 σ2 (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 complex-valued symbols d
l
and spread in the frequency domain using a user-specific Walsh-Hadamard 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
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 N-subcarrier components. The N−Na components are discarded after FFT and the remaining Na components are despreaded to get each respective user-specific data symbol.
The above model is used to derive upper bounds for a fixed DC bias by employing pre-equalization and post-equalization schemes. For each case under investigation, CSI is assumed to be available at the transmitter or at the receiver as per post- or pre-equalization requirements, respectively. Then these fixed biases, denoted by Bpost and Bpre, 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 zero-mean unit-variance Gaussian random variable.
3.2.1 Post-equalization case
At the transmitter, regardless of the number of active subcarriers Na, the number of distinct symbols to be transmitted will always be N. The data symbols after the IFFT operation are represented as
(3)
The complex baseband transmitted signal is
(4)
where p(t) is the unit norm rectangular transmit pulse shape of width T/(1+γ)N
(5)
Hence, the real passband MC-CDMA signal to be used for IM is
(6)
while the transmitted optical signal will be
(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
For fc > > (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 can be taken as
It follows that the required bias during this interval is , which can be bounded as
(8)
We are interested in the maximum negative value of the signal. Hence, we set |d
l
|≤Amax, which, in turn, depends upon the constellation configuration and the allocated subcarrier powers. Moreover, we set . So inequality (8) becomes
To give an upper bound that is independent of k, i.e., fixed bias, the final expression is
(9)
Since only the data on the active subcarriers will be processed at the receiver
(10)
yielding an upper bound
(11)
The expression in (10) imposes an upper bound on the value of Cl,k, thus yielding a conservative value of the DC bias. Thus, from (9) and (11), we get
(12)
For the BER analysis, we shall focus on the equivalent baseband complex discrete-time system model. The received signals at the OFDM receiver can be expressed as vector r=(r0,…,rN−1) such that
(13)
where (H0,…,HN−1) is the FFT of the discrete-time 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 σ2 (when the noise PSD is given by N0/2, we have σ2=N0/N). For equalization as well as restoring the CDMA code orthogonality, we apply one-tap equalization to obtain such that
(14)
From the equalized received signals, the QPSK symbol d
l
can be obtained using the codeword c
l
, i.e., despreading. Let be the received signal after despreading. In particular,
(15)
For analyzing a user-specific performance, without loss of generality, we consider the received signal of user 0 ().
(16)
For QPSK, the BER can be found from the received symbol energy and the noise variance as
(17)
For QPSK, using in (12), the BER in terms of the DC bias is found to be
(18)
Subcarrier selection: post-equalization case. The BER expression in (18) yields the following theorem that provides a method to select active subcarriers. Let Nc(L) denote the length of CDMA codewords required to accommodate L users. Note that the value Nc(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 Nc(L)≤Na≤N is equivalent to solving the following optimization problem:
The value of ξ is decreasing with Na since |H
n
|’s are always positive. Consequently, the smallest possible value of Na is optimal. Since we need to use at least Nc(L) subcarriers, it follows that Na=Nc(L). This proves part 1.
Now, given that we must use Nc(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 Pre-equalization 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
(19)
After a similar treatment as in the case of post-equalization, we get the fixed DC bias Bpre as follows
(20)
(21)
We are interested in the maximum negative value of the signal. Hence, we set |d
l
|≤Amax, which, in turn, depends upon the constellation configuration and the allocated subcarrier power. Moreover, we set . So inequality (21) becomes
To give an upper bound that is independent of k, i.e., fixed bias, the final expression is
(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 post-equalization,
(23)
yielding an upper bound
(24)
The expression in (23) imposes an upper bound on the value of , thus yielding a conservative value of the DC bias. Thus, from (22) and (24), we get
(25)
By proceeding in the similar fashion as in the case of post-equalization and incorporating noise variance=σ2Na, we get the BER expression as
(26)
Using (25) and incorporating for QPSK, the BER expressing in terms of the DC bias is
(27)
Subcarrier selection: pre-equalization 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 Nc(L) ≤ Na≤ N is equivalent to solving the following optimization problem:
Since |H
n
|s are smaller than 1 (i.e., signal attenuation typically in the order of 10 −7), κ decreases with Na. Consequently, the smallest possible value of Na is optimal. Since we need to use at least Nc(L) subcarriers, it follows that Na=Nc(L). This proves part 1.
Now, given that we must use Nc(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
post-eq
and BER
pre-eq
From the previous discussions, we conclude that for both post-equalization and pre-equalization it is optimal to select Nc(L) subcarriers with the highest |H
n
|s. The next question is whether post-equalization or pre-equalization performs better in terms of the BER for a given DC bias Bpost=Bpre=B.
The next theorem shows that pre-equalization always performs no worse than post-equalization.
Theorem 3
Given that the DC biases in (12) and (25) are used for post-equalization and pre-equalization respectively, the corresponding BER for pre-equalization is no more than the BER for post-equalization.
Proof
We first rewrite the post-equalization BER expression of (18) as
(28)
We proceed in a similar fashion from the pre-equalization BER expression of (27) to obtain
(29)
In (28) and (29), the first two factors in the arguments of the Q function are identical (denoted by α). Hence,
From the direct consequence of the Cauchy-Schwarz inequality , the inequality holds. It follows that BERpre-eq ≤ BERpost-eq. □