The performance of Space-Time Block Codes combined with Discrete MultiTone modulation applied in a Large Core Step-Index POF link is examined theoretically. A comparative study is performed considering several schemes that employ multiple transmitters/receivers and a fiber span of 100 m. The performance enhancement of the higher diversity order configurations is revealed by application of a Margin Adaptive Bit Loading technique that employs Chow’s algorithm. Simulations results of the above schemes, in terms of Bit Error Rate as a function of the received Signal to Noise Ratio, are provided. An improvement of more than 6 dB for the required electrical SNR is observed for a 3 × 1 configuration, in order to achieve a 10−3 BER value, as compared to a conventional Single Input Single output scheme.
© 2011 OSA
Large Core Step-Index Plastic Optical Fibers (SI-POFs) have some tempting features, such as low cost, easy handling etc., making them an attractive medium for high speed interconnects. However, their large attenuation and modal dispersion limit their usage in short haul applications [1–5,7–9,13]. Despite the fact that Large Core SI-POFs suffer from small bandwidth, some attempts have been made to achieve Gbit/s rates using Large Core SI-POFs for distances up to 100 m. A common technique to squeeze information in little bandwidth is the use of Discrete Multitone (DMT) modulation technique combined with an algorithm that allocates the number of bits per subcarrier, according to the corresponding Signal-to-Noise Ratio (SNR), in order to maximize the transmission rate over the fiber. Such algorithms are known as Rate Adaptive (RA) bit loading algorithms [2,3]. Other ways to increase spectral efficiency are the use of baseband Multi-level Pulse Amplitude Modulation (M-PAM)  or single carrier Quadrature Amplitude Modulation (M-QAM) . From the previous modulation techniques, the more effective is the DMT justifying the complexity increase.
The technology, though, that can increase significantly the capacity of MMF links by inserting multiple streams in the fiber without applying wavelength multiplexing is the Multiple-Input Multiple-Output (MIMO) transmission technique. The multiple modes of the fiber can be exploited in order to increase capacity in a transmission medium whose bandwidth is limited due to modal dispersion. When examining MIMO systems two concepts have to be taken into consideration, diversity and spatial multiplexing :
- • Diversity is related to the emission of a number of replicas of the same signal, each suffering independent fading. The probability that the replicas fade severely and simultaneously is significantly reduced. Different diversity schemes can be realized in space, time or frequency domain.
- • Spatial multiplexing is the ability to transmit simultaneously different information over multiple sources, increasing this way the system capacity. A rule of thumb is that if N transmitters and M receivers exist in a MIMO system, then, for a rich scattering environment, the capacity will grow proportionally with min(N, M). Practically, all reported results in the literature, related to MIMO over MMF links fall within the “Spatial Multiplexing” technique [7–12], though some results in  concern diversity. The most common realization of spatial multiplexing is the Mode Group Diversity Multiplexing (MGDM), where different signals are transmitted on different mode groups, by selective excitation of the MMF .
In this paper, we focus on the diversity techniques and present calculations showing the potential of application of Space-Time Block Codes (STBCs) in combination with DMT for the performance enhancement of transmission systems over large core SI-POF. STBCs are transmission schemes that maximize diversity rather than capacity and are optimized for frequency flat fading channels. In terms of multipath, the large core SI-POF is a frequency selective channel. By using DMT in such a channel, each subcarrier can be considered a frequency flat fading channel and the application of STBCs for every subcarrier seems to be an ideal case. We evaluate, through detailed numerical simulations, different diversity configurations based on Multiple-Input Single-Output (MISO) and MIMO schemes assuming transmission over 100 m of large core SI-POFs with a core diameter of 980 μm. The paper is organized as follows. At first, the optical channel is described and the impulse response of the SI-POF is given. Then, STBC schemes are briefly described and their combination with DMT is examined. Afterwards, the MISO channel is presented and detailed simulation results are presented for different MISO and MIMO configurations. A Margin Adaptive (MA) version of the bit loading technique has been also considered.
2. Optical channel - fiber model
A SI-POF with 490 μm core radius, 10 μm cladding thickness and length equal to 100 m is considered. The above length is longer than the coupling length (Lc) and the length (zs, with zs > Lc) over which a Steady State Distribution (SSD) has been achieved, meaning that the output light distribution is independent from launch conditions . In the following, the core refractive index, the fiber attenuation and the Numerical Aperture (NA) are equal to 1.4893, 0.16 dB/m (at λ = 650 nm) and 0.5, respectively.
Many research groups have studied the frequency and/or the impulse response of SI-POFs [15–17]. Despite the different methodologies followed, the common starting point of these approaches is the time-dependent power flow equation presented by Gloge . All these approaches result in resembling but slightly different impulse response shapes, depending on the methodology applied and the specific fiber used. According to , the impulse response, obtained from the integral of the fiber output intensity over output angle, represents the pulse temporal spread. This means that the impulse response is a (real) function of power and depends on the fiber length. Here, the impulse response is the one presented in  and . More specifically, the impulse response is given by the following formula as a function of the fiber length (z) and time (t)
For z = 100 m and γ ∞ = 0.0368 m −1 – the attenuation coefficient –, the value of the coupling coefficient (D) was determined by matching the magnitude of the Fourier transform of the impulse response to the measured magnitude of the frequency response of a SI-POF link with the above properties. This resulted in a coupling coefficient value of D = 4.1 × 10−4 rad2/m. In Fig. 1(a) , the matched normalized frequency response is presented along with the respective measured one. It was decided to adopt a pessimistic approach; therefore, the selected 3 dB bandwidth value of the fiber was 73.75 MHz, almost 1 MHz less than the measured one. The mismatch is less than 1 dB for frequencies up to 150 MHz and less than 2 dB for frequencies in the range from 150 MHz to 300 MHz. The normalized impulse response of Eq. (1) is shown in Fig. 1(b).
The amplitude of the frequency response of the SI-POF drops steeply with frequency increase (Fig. 1(a)) and the bandwidth left for transmission is much less than that required for Gbit/s transmission, if a typical baseband NRZ-OOK modulation format is applied. DMT  seems to be a solution to these two problems. At first, a Cyclic Prefix (CP) with the proper duration is used for each real DMT symbol to overcome Inter-Symbol Interference (ISI). Theoretically, if the CP has a duration equal to or greater than the maximum Differential Mode Delay (DMD), then the ISI disappears. Secondly, DMT is a bandwidth-efficient modulation technique.
The following analysis is performed for the proper introduction of multiple receivers in the link and includes two stages: the first is related to the determination of the frequency response of the fiber assuming detectors with different surface areas and the second is the investigation on the stability of the determined frequency responses over time.
The frequency response of a POF link with a receiver having an active region smaller than the surface of the fiber end facet is analyzed in . Especially, from Fig. 5 of the same paper, if the radius of a circular detector is r and the radius of the SI-POF core is a (with r ≤ a), the mean DMD per unit length gets decreased by a factor r/a. However, the price paid is the power loss which is equal to 10·log10((r/a)2). The last observation is in agreement with . Taking into account the previous considerations for the mean DMD and power loss as a function of the detector radius, the amplitude of the frequency response of the system was calculated, as shown in Fig. 2 , for several values of the detector radius. The amplitude of each frequency response is normalized to the maximum value of the frequency response amplitude for r = a. It is obvious that the bandwidth gets larger by decreasing the detector radius, but the power loss becomes significant.
The degree of the temporal variation of the impulse response properties of the complete fiber span is of paramount importance for the system performance calculations, assuming that SSD has been achieved. Let the number of modes of the SI-POF be Mf, the number of modes that are received by the detector Md , the core surface of the fiber Af, the surface of the detector Ad, the overall integrated intensity at the fiber output Wf and the integrated intensity observed by the detector Wd. From , Mf = 2.802.048. The detector is considered circular with radius r. From expression (11) of  and assuming that Wf is constant, the mean value (μ) and the variance (σ2) of Wd are equal to
Such expressions correspond to a beta distribution. Figure 3(a) depicts the mean value and Fig. 3(b) depicts the standard deviation of Wd as functions of the percentage of modes collected by the detector (Md/Mf ·100) both normalized by the inverse of Wf. From Fig. 3(b), it can be observed that in the case of the detector area coincided with that of the fiber end facet (Ad = Af), the variation of the optical power incident to the detector over time is zero. When the detector area is smaller (the percentage of the total modes collected by the detector is less than 100%), the standard deviation of Wd remains very small and in all cases very close to zero. This last conclusion implies that the frequency response does not vary with time.
3. Space-time block codes
STBCs can be classified in transmit diversity techniques. The first STBC with two transmit antennas and M receive antennas was invented by S. Alamouti . The generalization to N transmit and M receive antennas was presented by V. Tarokh et. al .
Their main advantages are their simplicity, the maximum order of diversity they achieve and the linear processing at the receiver. Maximum diversity order enhances the transmission performance and is equal to N × M for an N × M STBC scheme. Linear processing at the receiver means that symbol-by-symbol decoding is done and combined detection for all the transmitted block of symbols is not required.
The streams of symbols from each source are divided into blocks. Each block has dimensions of time and space. For instance, the encoding of the Alamouti scheme for two sources is as follows: during a symbol period, two symbols are transmitted from the two sources simultaneously. Let the two symbols be s 0 and s 1. During the next symbol period, –(s 1)* is transmitted from the first source and (s 0)* from the second source. The channels must remain constant for the two symbol periods. A similar procedure is followed for the encoding of 3 × M and 4 × M STBC schemes. The decoding procedures for the Alamouti scheme are given in  and for the rest STBCs in the appendix of . STBCs are not “capacity–oriented”. Two symbols are transmitted in two symbol periods within the Alamouti scheme achieving unitary rate. The STBCs for three and four sources have a 3/4 rate, meaning that three symbols are transmitted as a block in 4 symbol periods. STBCs with rate greater than 1 don’t exist. STBCs for three and four sources for complex constellations with rate greater than 3/4 exist but they either don’t have linear decoding complexity or they don’t achieve the maximum diversity order or both.
STBCs are optimized for Rayleigh flat fading channels. Channels that are frequency selective, such as SI-POF, are not appropriate for STBCs. However, by applying DMT instead of plain NRZ-OOK, for each subcarrier the channel becomes a frequency non-selective (flat fading) and the STBC scheme can be applied in the subcarrier domain.
4. Space-time block codes - multiple input single output optical channel model
4.1 Channel description
Firstly, STBC schemes with N (N ∈ [2–4]) optical sources and a single receiver are being investigated. The detection area of the receiver is considered to be at least equal to the surface of the fiber end facet. It is assumed that SSD has been already achieved for a fiber length equal to 100 m, which is a realistic consideration. Assuming that the distance (a few tens of μms) between the different emitters located at the SI-POF front end facet is such (relative to the emission wavelength of 650 nm) that each beam excites different modes subjected to independent fading. In the following simulations, no geometric representation of the position of the sources is considered, and only the formula of the impulse response is used. A schematic of the simulation setup is shown in Fig. 4 .
For each source, direct intensity modulation is applied in the linear regime of the function P out = f (I in) of the optical source. Proper electrical bias is applied to ensure that the driving current is in all cases greater that the source’s threshold current, in order to avoid turn-on-delay effects. The slope of the curve is used in order to represent the conversion of current to optical power. Ideally, it is assumed that no saturation effects appear during modulation. The SI-POF can be considered a Low Pass Filter (LPF), as shown in Fig. 1(a). Different spectral components of a signal will be affected differently from the channel, because of the shape of the channel’s frequency response. DMT can alleviate the problems that appear in a frequency selective channel, as the SI-POF, by making the spectrum covered by each subcarrier a flat fading channel. Moreover, all the processing concerns optical power and not optical fields. Direct detection is considered at the reception side. A factor that represents responsivity is considered for the transformation of optical power to electrical current. The detector receives all the optical power that comes out from the SI-POF in this instance.
The encoding and decoding of STBCs is applied in the subcarrier domain before the Inverse Fast Fourier Transform (IFFT) at the two sources and after the FFT at the receiver, respectively [6,19]. In every transmitted DMT symbol, the CP with the proper length is appended. Channel estimation and synchronization are supposed to have been performed successfully. In our case the receiver has perfect knowledge of the channels.
The following analysis is for a 2 × 1 STBC scheme, but the same procedure is followed for the rest of the schemes, even in the case of the 2 × 2 STBC configuration . During two successive DMT symbol periods, for subcarrier k, the complex symbols S 0, k and S 1, k will be coded using the Alamouti STBC scheme in order to construct properly the DMT symbols that will be transmitted from each source. The channels must remain constant for the two DMT symbol periods. Let the duration of every DMT symbol be equal to Ts. From the procedure already described, at time t, two electrical currents, I 0(t) and I 1(t), are generated that correspond to the DMT symbols of the two sources, and at time t + Ts, two currents, Ĩ 0(t + Ts) and Ĩ 1(t + Ts), arise that correspond to the following DMT symbols. If the slope of the curve P out = f (I in) for each source is equal to β – in W/A units – then the instant transmitted optical power will be equal to P out = β·I in. Assuming uncorrelated channels and the linearity of the system, the instant power at the fiber end for the first and second DMT symbol durations will be21]:Eq. (4). N 0, k and N 1, k are complex numbers representing the thermal noise in the frequency domain. Combining R 0, k and R 1, k properly, the estimations of the complex symbols S 0, k and S 1, k can be produced . The two symbols will be decoded one by one.
The SISO and the 2 × 1 STBC schemes can achieve the same rate for a specific number of subcarriers that carry useful symbols that have been produced by the same QAM constellation. If the same rate is desired for the 3 × 1 and 4 × 1 STBC transmission schemes, either more subcarriers must be used than in the 2 × 1 case or denser QAM constellations or both.
4.2 Margin Adaptive (MA) bit loading
Algorithms that allocate the number of bits per subcarrier according to the corresponding SNR have been examined thoroughly. According to the results the algorithms achieve, they are classified in two categories : Rate and Margin Adaptive bit loading algorithms. RA algorithms are used for the maximization of the bit rate for a fixed BER and given power constraint, while MA loading algorithms achieve the minimization of the BER for a given bit rate. Here, Chow’s MA Loading algorithm  will be considered. The number of bits that can be carried by subcarrier k with SNR equal to SNRk is2]
The expressions (3) and (4) hold for uncorrelated channels. In wireless communications, correlation may appear due to the transmitter antennas environment, the receiver antennas environment and the characteristics of the MIMO channel itself. If the channels are correlated, then each channel is affected by all others. Following a similar approach in the SI-POF, the correlation related to the geometrical properties (distance) of multiple transmitters/receivers can be easily avoided since the cross section of the fiber is large enough to accommodate a number of transmitters/receivers sufficiently separated between each other. There is an open question related to the transmission channel itself. According to  it seems that there is no correlation in a SI-POF optical channel, whereas in  there is an indication that some degree of correlation could exist. To our opinion, the issue is open and only a detailed experimental investigation can give a safe conclusion. Therefore, for purpose of generality, correlation aspects were taken into account into our calculations following the formalism described below.
Here, hypothetical existence of correlation between multiple channels is taken into account by the insertion of a matrix that is called correlation matrix  (Section III.B). For a 2 × 1 STBC scheme, the introduction of channel correlation in Eqs. (3) and (4) leads to the following expressions for the frequency response matrix and the impulse responsesEq. (9)) are multiplicative coefficients. It should be pointed out that the type of correlation included in the simulations originates from the insufficient spatial displacement between the sources, a fact that can be avoided in an experimental fiber optic transmitter arrangement using large core POFs. However, the receiver(s) does/do not realize that the correlation may appear during propagation through the medium. Therefore, this is a first approximation that is used in order to examine the impact of the phenomenon.
5. Simulation results and discussion
In this section, the simulation results are presented. Several schemes, all using DMT in combination with STBC encoding, were considered. The evaluation of each scheme was performed in terms of Bit Error Rate (BER) versus the multi-channel electrical symbol SNR, as defined in , at the receiver. The subcarrier spacing (Δf) was set equal to 512 KHz and the CP was 5% of the 1/Δf duration. The duration of each DMT symbol was 2050.78 nsec, including a CP of 97.66 nsec. 2 × 1024 point IFFT and FFT were used. The first two subcarriers that were never used were followed by a number of information carrying subcarriers (NSC). Zero padding was applied whenever all 1024 subcarriers were not used. In all cases, the same bit rate was applied (~1.25 Gbit/s). More subcarriers in combination with denser constellations were used for the lower rate STBCs. The complex symbols in the constellations were not Gray-coded, while no clipping was applied to the DMT signal. More than 285000 complex symbols (106 bits) were transmitted in any case, in order to be able to measure BERs up to 10−4 accurately.
5.1 DMT with STBC encoding (without bit loading)
In Fig. 5, the BER versus the SNR is depicted after transmission over 100 m of SI-POF, without considering any channel correlation. The horizontal dashed line represents the 10−3 BER limit – over which error free (BER < 10−10) transmission can be achieved if Forward Error Correction is used  – and the vertical dashed line represents a reasonable SNR upper limit for a practical link. The average energy per subcarrier is the same for all cases. The diversity order of 2 of the Alamouti scheme achieves the lower BER (better performance) for any SNR value in this instance, while it requires a 3 dB smaller SNR value as compared to the SISO scheme in order to achieve a 10−3 BER, for both given configurations. The corresponding 3 × 1 and 4 × 1 STBC schemes achieve worse results than the 2 × 1 scheme for all transmission cases, despite the greater diversity order provided by these two STBCs. This behavior is attributed to the addition of extra subcarriers and the higher constellations required to compensate their lower rate (3/4), in order to achieve the same bit rate. By adding more subcarriers, the symbols that correspond to the final subcarriers (larger frequencies) are severely affected by the channel’s frequency response. In addition, the denser constellations leave little “space” for the symbols to be corrupted by noise. Subsequently, the worse performance of 3 × 1 and 4 × 1 STBC schemes is justified. For large SNR values, the noise is not a crucial parameter for the performance degradation of the system. Thus, using less subcarriers with denser constellations gives better results.
In presence of correlation the 10−3 BER limit is hard to meet for reasonable SNR values, for the 3 × 1 and 4 × 1 STBC schemes, as shown in Fig. 6 . This is also the case for the Alamouti scheme for ρ greater than 0.1. It seems that if correlation exists between the channels, the 3 × 1 and 4 × 1 STBC schemes are meaningless, as they increase the complexity and they also perform worse than the SISO scheme. The degradation of the Alamouti scheme is tolerable for small values of ρ. However, this model is pessimistic, especially as the number of sources increase, meaning that the existence or inexistence of correlation is crucial and in case of existence, a realistic representation model would be very helpful.
The impact of the radius decrease of a circular detector to the BER is depicted in Fig. 7 , for a specific SNR value of a detector that covers the entire fiber core output facet (r = a). The SNR chosen (~28 dB) is the value for which almost 10−3 BER is achieved for the Alamouti case in Fig. 6. For this specific SNR value, the noise was produced for each scheme separately and the BERs for r = a were first obtained for each transmission. The noise that has been produced for each scheme remains constant for the rest of the BER computations for r < a, leading to a reduction of SNR. The 2 × 2 Alamouti scheme is also depicted in the same figure. The two detectors were identical and circular. Their radius was such that both detectors can be contained over the fiber surface. Furthermore, some distance between the detectors was considered for the same reason that some distance is needed between the places where the optical beams from each source enter the SI-POF. That’s why the BER for the 2 × 2 Alamouti scheme has been computed only for 4 percentage values of the detector radius. Because of the SSD being achieved, the impulse responses were the same to each other and their amplitude was considered constant as well.
From Fig. 7 and taking into account  and , for every transmission scheme with a single detector, it is obvious that the best results are obtained when using a photodetector with an active area equal to that of the fiber facet. There is no part on the curve of each transmission scheme where the use of a detector with a radius smaller than a gave better BER results than the case of r = a.
Taking into account the second detector, for any radius value, the 2 × 2 STBC scheme achieves worse results than the 2 × 1 scheme with r = a. This means that the power loss outperforms not only the bandwidth enhancement, but also the diversity order increase . However, for the same detector radius and a specific SNR (per detector), the 2 × 2 scheme achieves lower BER than the 2 × 1 Alamouti scheme, which is a reasonable result. The insertion of probable correlation is also involved in the results presented in Fig. 7. For every case, the correlation leads to a performance degradation. Moreover, no great improvement is provided by employing a second detector (for the Alamouti scheme), which additionally will increase the complexity of the entire system. Thus, the single detector is the best candidate for transmission in SI-POFs using STBC encoding schemes.
5.2 DMT with STBC encoding and margin adaptive bit loading
For each transmission scheme examined in the previous section, the QAM constellation was the same and the average energy per subcarrier was kept constant for all subcarriers. By doing so, the form of the frequency response of the SI-POF is not taken into account. The goal in this section is to use the number of subcarriers needed in order to achieve the 10−3 BER for smaller SNR values by allocating the bits and the energy per subcarrier according to the SNR of each subcarrier and keeping the 1.25 Gbit/s rate constant.
In the previous section, a combination of 513 subcarriers with 32-QAM was needed to achieve the target rate for the SISO and 2 × 1 STBC schemes. For the same bit rate, 571 subcarriers and 64-QAM were needed for the 3 × 1 and 4 × 1 STBC schemes. The algorithm utilized 686 (⌈1.2 × 573⌉ – 2) subcarriers, 16-QAM modulation format for each subcarrier, and was executed two times for the SISO scheme and for every SNR value, each time taking into account the combination of the number of subcarriers and the QAM constellation of each couple of transmission schemes. Nevertheless, the reallocation of the bits per subcarrier is done for the initial number of subcarriers. From , the previous action doesn’t seem to create any violation in the algorithm. Furthermore, up to 256-QAM is used in the simulation. Parameter Γ in (7) was set equal to 4.2 dB.
As shown in Fig. 8 , with bit loading, the desired BER is met for smaller SNR values for all the schemes and the gain due to diversity order is recovered for the lower rate STBC schemes as the SNR increases. The improvement is obvious, especially for the 3 × 1 and 4 × 1 STBC schemes that require over 6 dB lower SNR values in order to achieve the same BER value (10−3) with the respective SISO configuration.
The characteristic fluctuations at certain parts of the curves are related to the algorithm parameters: i.e. the number of subcarriers, the target bit rate and the value of Γ. These fluctuations are the result of the suboptimal energy allocation to each subcarrier. The resultant energy distribution to the subcarriers had the anticipated “saw tooth” shape and the bit allocation had the “stair-like” shape for all SNR values, though the balance of the algorithm that would lead to constant slope of the curves is not achieved due to the harsh rounding procedure followed during the algorithm. The diversity gain is a factor that causes the appearance of the fluctuation of BER at smaller BER values. This is clear for the 3 × 1 and 4 × 1 STBC schemes which achieve greater diversity order than the Alamouti scheme. If Γ gets another value and the rest of the parameters are kept constant, then some fluctuations will appear again, but at different values of BER and for different SNR values.
Once again, the impact of suppositional correlation on the system is disastrous. The change of each curve due to correlation insertion is intent and the performance deterioration is severe, even for small values of the correlation coefficient, leading to a worse performance for all the STBCs schemes when compared to the SISO one.
A more practical approach for the application of the MA bit loading in a real system would be to run the algorithm for one SNR value only, as this task takes place at the transmitter side of the link in a real system, where information regarding the SNR at the receiver is not always available. In Fig. 9 the BER curves, after running the Chow’s MA algorithm for an SNR value of 20 dB and keeping the resulting bit allocation for every SNR value at the receiver, are shown. No fluctuations appear in this instance, as expected, while the diversity gain of each STBC scheme against SISO is clearly depicted. It is worth mentioning that if a lower SNR value for the algorithm execution was used, the diversity gain was reduced and the absolute value of the slope of the BER curves was decreased for the 3 × 1 and 4 × 1 STBC schemes, as regards the other two schemes.
In this paper, the application of STBC encoding in a Large Core SI-POF transmission system has been numerically evaluated. If no bit loading techniques are used, only the 2x1 STBC scheme proposed by Alamouti improves the performance of the transmission system, compared to the conventional SISO scheme. In contrast, the application of Chow’s MA bit loading algorithm showed that a 10−3 BER is achieved for smaller SNR values for all the schemes in comparison to the case without bit loading. Moreover, the diversity gain is recovered for the 3 × 1 and the 4 × 1 STBC schemes. Channel correlation is of crucial importance, since even for small values of the correlation coefficient the calculated system performance is strongly degraded. However, the existence of such correlation is not clearly documented in the literature. In addition, there is strong evidence that such correlation can be avoided by a careful design of the system. Finally, the detector radius reduction doesn’t offer any improvement because the bandwidth increase is overwhelmed by the power loss. This restricts the useful configurations to schemes with a single detector.
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