Effects of frequency and timing offsets in DCT-based multiple access system for VLC

Optics Communications 435 (2019) 297–310

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Optics Communications journal homepage: www.elsevier.com/locate/optcom

Effects of frequency and timing offsets in DCT-based multiple access system for VLC Suseela Vappangi ∗, V.V. Mani Electronics and Communication Engineering National Institute of Technology Warangal, India

ARTICLE Keywords: VLC CFO STO ICI MUI CRLB

INFO

ABSTRACT Visible light communication (VLC) has gained a significant momentum in the recent times due to its distinctive characteristics of imparting concurrent ‘illumination’ and ‘communication’ by exploiting sustainable and energyefficient opto-electronic devices like light emitting diodes (LEDs). Consequently, by utilizing the already installed LED lighting fixtures a small scale cellular communication network can be created within an indoor environment imparting economic and ubiquitous data transmission services to multitude of roaming mobile stations within the proximity of LEDs. Therefore, it is vital to address the detrimental aspects like carrier frequency offset (CFO) and symbol time offset (STO) which emanates predominantly in an uplink scenario. This paper evaluates the performance of DC offset optical fast orthogonal frequency division multiple access (DCO-FOFDMA) system in the presence of both CFO and STO and accomplishes a thorough mathematical analysis emphasizing the different interferences like inter carrier (ICI) and multi user interferences (MUI) emanating due to different timing disparities and frequency discrepancies. The simulation results evidences that, with the increase in sensitivity of CFO and timing misalignments, the system performance drastically deteriorates thereby hindering the detection capability of multiple users operating at the same time. In order to estimate the STO, synchronization algorithms like Maximum Likelihood (ML) and Classen are imposed to the developed system model. Additionally, Cramer Rao Lower Bound (CRLB) is derived for the estimation of CFO and STO.

1. Introduction The perpetual augmentation in the insistence of data hungry applications imposes an unprecedented growth in the mobile data traffic [1]. This eventually makes the radio frequency (RF) spectrum to become overcrowded thereby curtailing the reliability of services. To alleviate the RF spectrum scarcity problem, alternative communication technologies are envisioned to impart high data rate communication in the order of several Gbps to multitude of subscribers throughout the globe. Moreover, significant amount of interest has been devoted towards the visible light portion of the electromagnetic spectrum leading to the emergence of visible light communication (VLC) which offers huge, unregulated and licensed-free bandwidth in the range of several THz frequency. Towards this end, VLC exploiting cost-effective, energyefficient and sustainable opto-electronic devices like light emitting diodes (LEDs) will assuredly be a promising revolutionary to RF based wireless communication [2]. VLC utilizes the advantages of LEDs to facilitate contemporaneous ‘illumination’ and high-speed wireless access in an indoor room environment. Additionally, since visible light is used as a medium for data transmission, a high degree of secure communication can be guaranteed. This is certain because, since light signals cannot penetrate

through walls or any other non-transparent objects, the data can be protected against eavesdropping. In general, a fundamental VLC system can be realized by employing LED at the transmitting end and simple photosensitive device like photodiode (PD) is utilized at the receiving end for the detection of the transmitted data signal as well as for the conversion of the light signal into electrical form [3]. VLC can also be referred to as a Green Communication technology furnishing both economical and ubiquitous data transmission aid by utilizing the available lighting infrastructure as access points rendering simultaneous ‘illumination’ and ‘communication’. Moreover, this technology can be easily interfaced with the existing lighting infrastructure without the requirement of additional energy. Consequently, its energy efficient nature will reduce the emission of green house gases. In addition, the omnipresence of LEDs replenished a way to establish a small scale cellular communication network in an indoor room environment by relying on the already installed lighting fixtures where each LED is acting as an Access Point (AP) or as a Base Station (BS) imparting services to several roaming user equipments (UEs)/mobile stations within its vicinity [4,5]. This sort of communication reduces the burden of installation costs of expensive RF base stations. It is obvious that, the non-coherent emission characteristics of LEDs make intensity modulation (IM) and direct detection (DD) as the most

∗ Corresponding author. E-mail addresses: [email protected] (S. Vappangi), [email protected] (V.V. Mani).

https://doi.org/10.1016/j.optcom.2018.11.023 Received 26 September 2018; Received in revised form 7 November 2018; Accepted 8 November 2018 Available online 16 November 2018 0030-4018/© 2018 Elsevier B.V. All rights reserved.

S. Vappangi and V.V. Mani

Optics Communications 435 (2019) 297–310

Inevitably, this kind of solution offers the flexibility for integration of VLC with almost every road-side units which includes street lightings, vehicular infrastructural units, etc. In general, it is widely known that in order for OFDM to be used as a multiple access scheme it needs to be interfaced with existing multiple access techniques like frequency division multiple access (FDMA), time division multiple access (TDMA), etc. Therefore, the integration of OFDM with Frequency Division Multiple Access (FDMA) has given rise to a predominant multiple access scheme called Orthogonal Frequency Division Multiple Access (OFDMA). It is imperative that, OFDMA is one of the promising multiple access technique which is extensively exploited in a RF based cellular network. However, the same can be incorporated in VLC under the constraint that the signal transmission is assured of its real and unipolar nature. Moreover, the advantages of OFDM can be utilized only if the orthogonality among the subcarriers is preserved. In case if the orthogonality is not warranted by any means then a high data rate communication aid can no longer be aided. Nevertheless, despite being acquired with renowned dominance when compared with RF, there are several issues which hinders the deployment of high speed VLC-based multiple access systems and therefore it necessitates to effectively resolve such issues. Investigations on the most detrimental aspects of carrier frequency offsets (CFO) and symbol timing offset (STO) in uplink scenario stems out to be as one of the most crucial challenge in VLC. Moreover, the frequency offset arises due to Doppler shift. It is obvious that, the misalignments between transmitter and receiver leads to the emergence of frequency offsets, which disrupts the orthogonality among the subcarriers thereby resulting in inter carrier interference (ICI). Additionally, timing synchronization i.e., frame detection stems out to be the most vital aspect which needs to be addressed in VLC. This is because, a timing induced error may cause a fraction of a fast Fourier transform (FFT) window to occur in an extended region of adjacent symbol thereby leading to the occurrence of ICI [19]. However, this is even more assertive in an uplink scenario when the number of users increase, hindering the detection capability of multiple users in such scenarios, and accordingly making the system prone to multi-user/multi access interference (MUI/MAI). To the best of the authors knowledge, this paper for the first time accomplishes a thorough mathematical analysis of the received signal pertaining to the desired subscriber which is affected with CFO as well as highlights different possibilities of timing errors emanating in an uplink scenario in a DCO-Fast Orthogonal Frequency Division Multiple Access (DCO-FOFDMA) system. Furthermore, it is vital to estimate CFO and STO in order to combat the effects of ICI and MUI/MAI. Hence, it is necessary to impose synchronization algorithms like Classen/minimum difference method and Maximum correlation method/Maximum Likelihood estimate method. In particular, these methods play a significant role to estimate the STO. Furthermore, this work derives the Cramer Rao Lower Bound (CRLB) for the estimation of CFO using the joint parameter estimation approach and compares with the Classen approach for the estimation of CFO. In addition, CRLB is derived for the estimation of STO. In general, these algorithms cannot be enforced in a straightforward manner due to real and positive constraint of the time-domain signal. Therefore, this work incorporates the methodology of these algorithms for the estimation of STO and CFO for DCT-based multiple access system i.e., DCO-FOFDMA system. Starting with the introduction, the remainder of this paper is organized as follows: Section 2, details the proposed system model and elaborates through mathematical analysis the different scenarios for the occurrence of interference in DCOFOFDMA system. Section 3 derives the CRLB for the estimation of CFO and Section 4 gives the synchronization algorithms for the estimation of STO. Section 5 presents the results and discussions of this work and Section 6 draws the conclusion of this work.

appropriate modulation scheme for VLC [6]. In particular, the data is encoded by varying the intensity of light which is referred to as IM and then at the receiving end, the data is detected by means of a simple photodiode and this process is termed as DD. Thereupon, this necessitates that the transmitted signal to be both real and positive [7]. In spite of offering remarkable benefits, the limited modulation bandwidth of white phosphorescent LEDs (WLEDs) makes VLC to adopt the multicarrier and spectral-efficient modulation technique like orthogonal frequency division multiplexing (OFDM), one of the significant modulation scheme of RF [8,9]. This is due to its indispensable notability of being robust to frequency-selective fading channel environment, utilizing simple equalizers, imparting high data rate communication, etc. However, OFDM can be incorporated in VLC-based multicarrier system by imposing a constraint on the transmitted signal where it has to be both real and positive-valued. For the purpose of accomplishing a real and unipolar signal transmission, diverse variants of optical OFDM which are complying with the requirements of IM/DD systems can be evidenced from the literature [10]. It is apparent that inverse fast Fourier transform (IFFT) and fast Fourier transform (FFT) blocks play a vital role for the modulation and demodulation in OFDM-based multicarrier system. Moreover, when the input to the IFFT is mapped by employing a complex modulation scheme like quadrature amplitude modulation (QAM), the signal which is attained at the output of the IFFT will be a complex and bipolar signal. Therefore, for attaining LED-compatible real-valued signal, it necessitates to enforce the input to the IFFT module to satisfy Hermitian symmetry criteria. The fundamental methodology involved behind the usage of Hermitian symmetry criteria is that only half of the subcarriers are utilized for the transmission of the data and the rest half are flipped complex conjugate versions of the previous ones. Consequently, a realvalued signal is obtained at the expense of reduced throughput [11]. Accordingly, in order to circumvent the Hermitian symmetry criteria, it entails to exploit real transformation techniques like discrete cosine transform (DCT), discrete Hartley transform (DHT) and fast Walsh Hadamard Transform (FWHT) [12–14]. However, this work focuses on DCT where the real signal processing of the cosine transform replaces the Fourier transform. It is appealing to note that upon utilizing the real transformation technique like DCT for IM/DD-based VLC system, simple, real and one-dimensional mapping strategies like Binary Phase Shift Keying (BPSK), M-ary Pulse Amplitude Modulation (M-PAM) can be employed for mapping the input stream of data. In addition, its eminent energy concentration property unveils a considerable amount of robustness against frequency offsets. Moreover, in the signal processing perspective, it has less computational complexity as it requires fewer number of additions and multiplications when compared with DFT/FFT [15]. Much relevant work pertaining to the exploitation of DCT for IM/DD systems can be found in [16,17] where the authors named it as Fast OFDM (FOFDM). By imposing either Hermitian symmetry criteria or by employing real transformations a real valued signal can be attained, but the obtained signal cannot be assured of its positivity. So, this obliges to depend on different variants of OFDM like DC-biased Optical OFDM (DCO-OFDM), Asymmetrically Clipped Optical OFDM (ACO-OFDM), Flip OFDM, Pulse Amplitude Modulated-Discrete Multitone Modulation (PAM-DMT), Asymmetrically clipped DC biased OFDM (ADO-OFDM), Layered ACO-OFDM (LACO-OFDM), Spectral and energy efficient OFDM (SEE-OFDM), hybrid diversity combined OFDM (HDC-OFDM), etc. In this work, we have included the spectral efficient DCO-OFDM methodology which consists of adding a DC bias to the negative signal to fetch a positive signal [18]. The noteworthy feature of this growing technology (VLC) is to bestow with a significant opportunity by expediting an extensive multiple access setup within an indoor as well as outdoor environment by utilizing the already deployed lighting fixtures to render simultaneous ‘illumination’ and wide range of data communication by reducing the burden of installation costs of expensive RF infrastructural units. 298

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Optics Communications 435 (2019) 297–310

2. System model

DC bias added signal is given by √ 𝑁−1 ) ( 𝜋 (2𝑛 + 1) 𝑘 2 ∑ = 𝑥(𝑟) + 𝛽𝐷𝐶 𝐶𝑘 𝑋𝑘(𝑟) 𝑐𝑜𝑠 𝑛 𝑁 𝑘=0 2𝑁 𝐷𝐶

The simple schematic of uplink scenario in DCO-FOFDMA system in which the mobile and the base stations are designed in accordant with IM/DD systems for VLC is delineated in Fig. 1, where 𝑃 subscribers/users are carrying on their transactions with the base stations through their relevant independent multipath optical channels. Looking into the details of transmitter and receiver blocks, at the transmitting terminal huge sets of incoming data streams are mapped by exploiting simple real, one dimensional mapping formats like binary phase shift keying (BPSK) and M-ary Pulse Amplitude Modulation (M-PAM). Later on, this serialized mapped data is transmitted in parallel by employing Serial to Parallel (S/P) converter. In order to emphasize the detrimental aspects of CFO and STO on the desired subscriber 𝑟, we surmise that the total number of subcarriers present in one OFDM symbol are 𝑁. Therefore, the mapped data corresponding to the 𝑟th subscriber on the 𝑘th subcarrier be represented as 𝑋𝑘(𝑟) , 𝑘 ∈ 𝑍𝑟 , where 𝑍𝑟 is the set of subcarriers appropriated to the 𝑟th subscriber and ∪𝑃𝑟=1 𝑍𝑟 = {0, 1, 2, … , 𝑁 − 1}. Unlike, FFTbased optical OFDM (OOFDM), FOFDMA does not require Hermitian symmetry criteria. Hence, 𝑋𝑘(𝑟) is loaded into the 𝑁-Point Inverse DCT (IDCT) block to yield the corresponding time-domain signal 𝑥(𝑟) 𝑛 as follows [20] √ 𝑁−1 ( ) 𝜋 (2𝑛 + 1) 𝑘 2 ∑ 𝐶𝑘 𝑋𝑘(𝑟) 𝑐𝑜𝑠 , 0≤𝑘≤𝑁 −1 (1) = 𝑥(𝑟) 𝑛 𝑁 𝑘=0 2𝑁

𝑘∈𝑍𝑟

And now, this signal corresponding to the 𝑟th subscriber i.e., 𝑥(𝑟) 𝑛 is 𝐷𝐶

subsequently passed through the channel with an impulse response ℎ(𝑟) 𝑛 . Hence, the signal invading the receiver can be put up as (𝑟) (𝑟) (𝑟) 𝑦(𝑟) 𝑛 = 𝑥𝑛 ∗ ℎ𝑛 + 𝑤𝑛

(6)

𝐷𝐶

It is to be noted that, the channel response corresponding to the 𝑟th subscriber is non-zero only for the values 𝑙 = 0, 1, 2, 3, … , 𝐿 − 1. Hence, the frequency response of the channel corresponding to the 𝑟th subscriber is given as √ ) ( 𝐿−1 ∑ (𝑟) 𝜋 (2𝑙 + 1) 𝑘 2 (𝑟) (7) 𝐶𝑘 ℎ𝑙 𝑐𝑜𝑠 𝐻𝑘 = 𝑁 2𝑁 𝑙=0 In addition, it is deduced that, the channels belonging to all the subscribers are statistically independent. In order to elucidate the deleterious aspects of CFO. We assume that, the received signal is affected with CFO of 𝛽. The presence of CFO leads to loss of orthogonality among the subcarriers and results in inter carrier interference (ICI). Besides, multi user interference (MUI) or multi access interference (MAI) emanates due to frequency discrepancies occurring among the uplink subscribers and even there are chances for dissimilarities to emerge between the uplink users and the BS. At the receiver terminal, reverse operations are incorporated like removal of cyclic prefix and DC bias. In order to highlight, the various interferences, we proceed further by stressing on the frequency domain signal. Accordingly, the frequency domain representation of 𝑦(𝑟) 𝑛 is given as √ ( ) 𝑁−1 ∑ 𝜋 (2𝑛 + 1) 𝑘 2 𝑐𝑜𝑠 𝑦(𝑟) 𝐶𝑘 (8) 𝑌𝑘(𝑟) = 𝑛 𝑁 2𝑁 𝑛=0 ] [ (𝑟) and using the fundamental It is well known that, 𝑦(𝑟) 𝑛 = 𝐼𝐷𝐶𝑇 𝑌𝑘 signal processing operation that convolution in time domain is multiplication in frequency domain i.e., 𝑌𝑘(𝑟) = 𝑋𝑘(𝑟) 𝐻𝑘(𝑟) . ] [ ] [ (𝑟) = 𝐼𝐷𝐶𝑇 𝑋𝑘(𝑟) 𝐻𝑘(𝑟) . Hence, it is apt to express 𝑦(𝑟) 𝑛 = 𝐼𝐷𝐶𝑇 𝑌𝑘

𝑘∈𝑍𝑟

and from (1), 𝐶𝑘 denotes ⎧ 1 ⎪√ , 𝐶𝑘 = ⎨ 2 ⎪1, ⎩

(5)

𝑘=0 (2) 𝑘 = 1, 2, 3, … , 𝑁 − 1

The Eq. (1), clearly depicts the ease and simplicity of incorporation of DCT. In addition, there is an enhancement in the spectral efficiency as all of the subcarriers are utilized for data transmission. Generally, in an indoor room environment, inter symbol interference (ISI) emanates due to the multipath dispersion as well as sudden blockage of signal due to presence of several obstacles like furniture, etc. Consequently, in order to prohibit inter symbol interference (ISI) a suitable amount of cyclic prefix or guard interval of length 𝑁𝑐𝑝 is added to 𝑥(𝑟) 𝑛 . The amount of cyclic prefix to be added is chosen as 41 th of the subcarriers’ size and must be larger than the channel delay spread 𝐿. Therefore, the cyclic prefix added signal can be represented as √ 𝑁−1 ( ) 𝜋 (2𝑛 + 1) 𝑘 2 ∑ (𝑟) 𝑥(𝑟) = 𝐶 𝑋 𝑐𝑜𝑠 , 0≤𝑘≤𝑁 −1 𝑛 𝑁 𝑘=0 𝑘 𝑘 2𝑁 (3)

Consequently, upon incorporating in (8), the following frequency domain signal is attained ( ⎡√ ( ) )⎤ 𝑁−1 ∑ ⎢ 2 𝑁−1 ∑ 𝜋 (2𝑛 + 1) 𝑘 + 𝛽𝑟 ⎥ 2 (𝑟) 𝐶𝑘 𝑐𝑜𝑠 𝐶 𝑋 ⎢ 𝑁 ⎥ 𝑘 𝑘 𝑁 2𝑁 ⎥ 𝑛=0 ⎢ 𝑘=0 ⎣ ⎦ 𝑘∈𝑍𝑟 ( ) 𝜋 (2𝑛 + 1) 𝑘 (𝑟) (𝑟) × 𝑐𝑜𝑠 𝐻𝑘 + 𝑊 𝑘 2𝑁

√ 𝑌𝑘(𝑟) =

𝑘∈𝑍𝑟

−𝑁𝑐𝑝 < 𝑛 ≤ 𝑁 − 1

(9)

(9), can be further solved to attain [√ ( )] 𝐿−1 ∑ (𝑟) 𝜋 (2𝑙 + 1) 𝑘 2 (𝑟) 𝑌𝑘(𝑟) = 𝐶𝑘2 𝑋𝑘(𝑟) 𝐶𝑘 ℎ𝑙 𝑐𝑜𝑠 𝛤𝑘𝑘 + 𝑁 2𝑁 𝑙=0 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

Eq. (3), clearly emphasizes a real valued signal, further this can be transformed into a positive signal by adding a suitable amount of DC bias. Generally, the amount of DC bias to be added is equal to the absolute value of the maximum negative amplitude of the signal. In addition, the amount of DC bias to be added also has a direct relation with the constellation size of the modulation schemes. Too much amount of DC bias leads to power inefficiency, but however this is desired for illumination requirement in VLC. Minimal addition of DC bias leads to clipping of the peak values, thereby leading to the occurrence of clipping noise. Hence, an optimal choice of DC bias is desired to transform the negative real valued signal to positive signal. In the literature, it is specified that, the amount of DC bias to be added is given as √ (4) 𝛽𝐷𝐶 = 𝑘 𝐸{𝑥2 [𝑛]}

𝐷𝑒𝑠𝑖𝑟𝑒𝑑 𝑆𝑖𝑔𝑛𝑎𝑙 𝑜𝑓 𝐶𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑖𝑛𝑔 𝑈 𝑠𝑒𝑟 𝑟

𝐶𝑘

𝑁−1 ∑

[√

𝐶𝑝 𝑋𝑝(𝑟)

𝑝=0 𝒑∈𝒁 𝒓 𝒑≠𝒌

∑ (𝑟) 2 𝐶𝑝 ℎ𝑙 𝑐𝑜𝑠 𝑁 𝑙=0 𝐿−1

(

𝜋 (2𝑙 + 1) 𝑝 2𝑁

)]

(𝑟) 𝛤𝑝𝑘 +

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ 𝐼𝐶𝐼 𝑒𝑚𝑎𝑛𝑎𝑡𝑖𝑛𝑔 𝑑𝑢𝑒 𝑡𝑜 𝑠𝑢𝑏𝑐𝑎𝑟𝑟𝑖𝑒𝑟 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡𝑠 𝑎𝑙𝑙𝑜𝑐𝑎𝑡𝑒𝑑 𝑡𝑜 𝑟

𝐶𝑘

𝑁−1 ∑ 𝑝=0 𝒑∈𝒁 𝒔 𝒑≠𝒌

[√

𝐶𝑝 𝑋𝑝(𝑠)

∑ (𝑠) 2 𝐶𝑝 ℎ𝑙 𝑐𝑜𝑠 𝑁 𝑙=0 𝐿−1

(

𝜋 (2𝑙 + 1) 𝑝 2𝑁

)]

(𝑠) 𝛤𝑝𝑘

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ 𝑀𝑈 𝐼∕𝑀𝐴𝐼

From (4), the factor 𝑘 specifies the clipping factor. A fixed amount of ( ) bias which is to be added is given by 10𝑙𝑜𝑔10 𝑘2 + 1 dB. Therefore, the

+𝑊𝑘(𝑟) 299

(10)

S. Vappangi and V.V. Mani

Optics Communications 435 (2019) 297–310

Fig. 1. Illustration of Uplink transmitter and receiver in DCO-FOFDMA in accordant with IM/DD systems for VLC. (𝑟) (𝑟) (𝑠) can be expressed as From (10), 𝛤𝑘𝑘 , 𝛤𝑝𝑘 and 𝛤𝑝𝑘 (𝑟) 𝛤𝑘𝑘 =

(𝑟) 𝛤𝑝𝑘

2.1. Mathematical illustration of combined effects of CFO and STO

[ ( ( )) ( )] 𝑁−1 𝜋 (2𝑛 + 1) 2𝑘 + 𝛽𝑟 𝜋 (2𝑛 + 1) 𝛽𝑟 1 ∑ 𝑐𝑜𝑠 + 𝑐𝑜𝑠 𝑁 𝑛=0 2𝑁 2𝑁

𝑁−1 [ 1 ∑ = 𝑐𝑜𝑠 𝑁 𝑛=0

(

( )) 𝜋 (2𝑛 + 1) 𝑝 + 𝛽𝑟 + 𝑘 2𝑁

( + 𝑐𝑜𝑠

(11) This section evaluates the different possibilities of timing discrepancies along with the existence of frequency offset. It is to be noted that, in the absence of other users, self interference i.e, ICI arises among the subcarrier components which are allocated to the desired subscriber 𝑟. Whereas, in the presence of other users MUI/MAI originates due to misalignments among the users, which manifests the previous frame or the next frame to overlap with the current frame. These different scenarios of timing errors are taken from [21] where analysis has been carried in the RF domain. In this work we have modeled fulfilling the prerequisite of IM/DD systems for VLC i.e, ensuring a real and positive signal transmission. The different scenarios of timing errors is summarized in Fig. 2. Additionally, we assume that 𝛼𝑢 and 𝛽𝑢 , 𝑢 = {1, 2, 3, … , 𝑃 } denotes the 𝑢th subscriber’s CFO and STO. It is likely that the timing errors to be both negative as well as positive.

( )) ] 𝜋 (2𝑛 + 1) 𝑝 + 𝛽𝑟 − 𝑘 2𝑁

(12) (𝑠) = 𝛤𝑝𝑘

𝑁−1 [

1 ∑ 𝑐𝑜𝑠 𝑁 𝑛=0

(

)) 𝜋 (2𝑛 + 1) 𝑝 + 𝛽𝑠 + 𝑘

(

(

2𝑁

+ 𝑐𝑜𝑠

( )) ] 𝜋 (2𝑛 + 1) 𝑝 + 𝛽𝑠 − 𝑘 2𝑁

(13) Accordingly, the interference components in (11)–(13) can be solved to attain ( ) ( ) 𝑠𝑖𝑛𝜋 2𝑘 + 𝛽𝑟 𝑠𝑖𝑛𝜋 𝛽𝑟 (𝑟) 𝛤𝑘𝑘 = (14) ( )+ ( ) 2𝐾+𝛽𝑟 𝛽 2𝑁𝑠𝑖𝑛𝜋 2𝑁𝑠𝑖𝑛𝜋 2𝑁𝑟 2𝑁

(𝑟) 𝛤𝑝𝑘

( ) ( ) 𝑠𝑖𝑛𝜋 𝑝 + 𝛽𝑟 + 𝑘 𝑠𝑖𝑛𝜋 𝑝 + 𝛽𝑟 − 𝑘 = ( )+ ( ) 𝑝+𝛽𝑟 +𝑘 𝑝+𝛽𝑟 −𝑘 2𝑁𝑠𝑖𝑛𝜋 2𝑁𝑠𝑖𝑛𝜋 2𝑁 2𝑁

(15)

( ) ( ) 𝑠𝑖𝑛𝜋 𝑝 + 𝛽𝑠 + 𝑘 𝑠𝑖𝑛𝜋 𝑝 + 𝛽𝑠 − 𝑘 ( )+ ( ) 𝑝+𝛽𝑠 +𝑘 𝑝+𝛽𝑠 −𝑘 2𝑁𝑠𝑖𝑛𝜋 2𝑁𝑠𝑖𝑛𝜋 2𝑁 2𝑁

(16)

(𝑠) 𝛤𝑝𝑘 =

• Scenario 1: This scenario is illustrated in Fig. 2a, which indicates negative timing error, i.e., 𝛼𝑢 < 0 and the interval where the offset lies is 0 ≤ −𝛼𝑢 ≤ 𝑁𝑐𝑝 − 𝐿 + 1 [21]. Therefore, by using the aforesaid conditions, the frequency domain representation of the users 𝑢 = 1, 2, 3, … , 𝑃 on the 𝑘th subcarrier can be formulated as √ 𝑌𝑘(𝑢) =

The first term in (10) specifies the desired signal, while the second term represents the ICI originating from the subcarrier components which are allocated to the desired user 𝑟, for the ease of depiction we showed as the interference occurring between 𝑘 and 𝑝 subcarriers, and the third term signifies the MUI/MAI arising due to user 𝑠. The terms 𝛽𝑟 and 𝛽𝑠 corresponds to the offset associated with user 𝑟 and 𝑠.

𝑁−1 ∑[ 2 𝐶𝑘 𝑁 𝑛=0

√

𝑁−1 2 ∑ 𝐶 𝑋 (𝑢) 𝑁 𝑘=0 𝑘 𝑘 𝑘∈𝑍𝑢

( ( ( ) )( )) ( ) 𝜋 2 𝑛 + 𝛼𝑢 + 1 𝑘 + 𝛽𝑢 ] 𝜋 (2𝑛 + 1) 𝑘 × 𝑐𝑜𝑠 𝑐𝑜𝑠 𝐻𝑘(𝑢) 2𝑁 2𝑁 + 𝑊𝑘(𝑢) 300

(17)

S. Vappangi and V.V. Mani

Optics Communications 435 (2019) 297–310

Fig. 2. Illustration of different timing discrepancies in DCO-FOFDMA in accordant with IM/DD systems for VLC.

(17), signifies the DCT output of the 𝑢th user on the 𝑘th subcarrier. Inevitably, in order to illustrate the effects of mismatching timing interval, we confine our discussion with desired user as 𝑟. Hence, the frequency domain output of the dedicated user 𝑟 on the 𝑘th subcarrier, in the presence of all the signals invading from all the paths belonging to different subscribers can be derived as

domain output of the appropriate subscriber 𝑟 as follows: [ 𝑌𝑘(𝑟) = 𝐶𝑘2 𝑋𝑘(𝑟) 𝑐𝑜𝑠

𝑐𝑜𝑠

𝐶𝑘

𝑝=0 𝒑∈𝒁 𝒔 𝒑≠𝒌

+

𝐶𝑘

𝑝=0 𝒑∈𝒁 𝒔 𝒑≠𝒌

𝑁

(

− 𝑠𝑖𝑛

𝑁−1 [ 1 ∑ 𝑐𝑜𝑠 𝑁 𝑛=0

( )) ] 𝜋𝛼𝑟 𝑝 + 𝛽𝑟 (𝑟) 𝐻𝑝(𝑟) 𝛶𝑝𝑘 + 𝑁

[ 𝐶𝑝 𝑋𝑝(𝑠) 𝑐𝑜𝑠

(

( )) 𝜋𝛼𝑠 𝑝 + 𝛽𝑠 𝑁

(

− 𝑠𝑖𝑛

( )) ] 𝜋𝛼𝑠 𝑝 + 𝛽𝑠 (𝑠) 𝐻𝑝(𝑠) 𝛶𝑝𝑘 𝑁

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ 𝑀𝑈 𝐼∕𝑀𝐴𝐼 𝑎𝑟𝑖𝑠𝑖𝑛𝑔 𝑓 𝑟𝑜𝑚 𝑠𝑢𝑏𝑠𝑐𝑟𝑖𝑏𝑒𝑟 𝑠 𝑡𝑜 𝑟

+𝑊𝑘(𝑟)

(19) (𝑟) 𝛶𝑝𝑘

(𝑠) 𝛶𝑝𝑘

can be solved From (19), the interference terms and to attain ( ( ) ) ( ) 𝑁−1[ 𝜋 2𝑘 + 𝛽𝑟 (2𝑛 + 1) 𝜋𝛽𝑟 (2𝑛 + 1) 1 ∑ (𝑟) 𝛶𝑘𝑘 = 𝑐𝑜𝑠 + 𝑐𝑜𝑠 + 𝑁 𝑛=0 2𝑁 2𝑁 ) ( ( ) ( ) 𝜋 2𝑘 + 𝛽𝑟 (2𝑛 + 1) 𝜋𝛽𝑟 (2𝑛 + 1) ] + 𝑠𝑖𝑛 (20) 𝑠𝑖𝑛 2𝑁 2𝑁

( )) ] 𝜋𝛼𝑟 𝑝 + 𝛽𝑟 𝐻𝑝(𝑟) + 𝑁 ( ( ) 𝜋 𝑝 + 𝛽𝑠 + 𝑘 (2𝑛 + 1) +

2𝑁 (𝑟) 𝛶𝑝𝑘 =

+ 𝑁 ( ( ) 𝜋 𝑝 + 𝛽𝑠 − 𝑘 (2𝑛 + 1)

+𝑊𝑘(𝑟)

𝐶𝑝 𝑋𝑝(𝑟) 𝑐𝑜𝑠

𝐷𝑒𝑠𝑖𝑟𝑒𝑑 𝑆𝑖𝑔𝑛𝑎𝑙 𝑜𝑓 𝑠𝑢𝑏𝑠𝑐𝑟𝑖𝑏𝑒𝑟 𝑟

( )) 𝜋𝛼𝑟 𝑝 + 𝛽𝑟

𝐼𝐶𝐼 𝑒𝑚𝑎𝑛𝑎𝑡𝑖𝑛𝑔 𝑎𝑚𝑜𝑛𝑔 𝑡ℎ𝑒 𝑠𝑢𝑏𝑐𝑎𝑟𝑟𝑖𝑒𝑟𝑠 𝑎𝑙𝑙𝑜𝑐𝑎𝑡𝑒𝑑 𝑡𝑜 𝑠𝑢𝑏𝑠𝑐𝑟𝑖𝑏𝑒𝑟 𝑟

∑

𝑁−1

( ( ) ) 𝑁−1 [ 𝜋 𝑝 + 𝛽𝑟 + 𝑘 (2𝑛 + 1) 1 ∑ 𝑐𝑜𝑠 𝑁 𝑛=0 2𝑁 ( ( ) ) 𝜋 𝑝 + 𝛽𝑟 − 𝑘 (2𝑛 + 1) +𝑐𝑜𝑠 + 2𝑁 ( ( ) ) 𝜋 𝑝 + 𝛽𝑟 + 𝑘 (2𝑛 + 1) 𝑠𝑖𝑛 2𝑁 ( ( ) ) 𝜋 𝑝 + 𝛽𝑟 − 𝑘 (2𝑛 + 1) ] +𝑠𝑖𝑛 2𝑁

( )) 𝜋𝛼𝑠 𝑝 + 𝛽𝑠

𝑐𝑜𝑠

( )) ] 𝜋𝛼𝑟 𝑘 + 𝛽𝑟

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

+

2𝑁 𝐶𝑝 𝑋𝑝(𝑠)

( − 𝑠𝑖𝑛

(𝑟) 𝛶𝑘𝑘 ,

𝑁 ( ( ) 𝜋 𝑝 + 𝛽𝑟 − 𝑘 (2𝑛 + 1)

𝑁−1 ∑

(

[

𝑝=0 𝒑∈𝒁 𝒓 𝒑≠𝒌

𝒑∈𝒁 𝒓 𝒑≠𝒌

+

∑

𝑁−1

( ( ) ( )) 𝑁−1 ∑[ 𝜋 2𝑘 + 𝛽𝑟 (2𝑛 + 1) 𝜋𝛼𝑟 𝑘 + 𝛽𝑟 2 (𝑟) 1 = 𝐶𝑘 𝑋𝑘 𝑐𝑜𝑠 + + 𝑁 𝑛=0 2𝑁 𝑁 ( ) ( ) 𝜋𝛽𝑟 (2𝑛 + 1) 𝜋𝛼𝑟 𝑘 + 𝛽𝑟 ] (𝑟) 𝑐𝑜𝑠 + 𝐻𝑘 + 2𝑁 𝑁 ( ( ) 𝑁−1 𝑁−1 [ ∑ 𝜋 𝑝 + 𝛽𝑟 + 𝑘 (2𝑛 + 1) 1 ∑ 𝐶𝑝 𝑋𝑝(𝑟) 𝐶𝑘 𝑐𝑜𝑠 𝑁 𝑛=0 2𝑁 𝑝=0 ( )) 𝜋𝛼𝑟 𝑝 + 𝛽𝑟

( )) 𝜋𝛼𝑟 𝑘 + 𝛽𝑟

(𝑟) 𝐻𝑘(𝑟) 𝛶𝑘𝑘 + 𝑁 𝑁 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

𝐶𝑘

𝑌𝑘(𝑟)

(

2𝑁

+

( )) ] 𝜋𝛼𝑠 𝑝 + 𝛽𝑠 𝐻𝑝(𝑠) 𝑁 (18)

Therefore, the scenario highlighted by (18) signifies that the received signal seriously suffers form ICI and MUI/MAI thereby resulting in the degradation of the Signal to Interference Noise Ratio (SINR). Furthermore, (18), can be solved to attain the frequency

(𝑠) 𝛶𝑝𝑘 =

301

(21)

S. Vappangi and V.V. Mani

Optics Communications 435 (2019) 297–310

( ( ) ) 𝜋 𝑝 + 𝛽𝑠 + 𝑘 (2𝑛 + 1)

𝑁−1 [ 1 ∑ 𝑐𝑜𝑠 𝑁 𝑛=0 2𝑁 ) ( ( ) 𝜋 𝑝 + 𝛽𝑠 − 𝑘 (2𝑛 + 1) + +𝑐𝑜𝑠 2𝑁 ( ( ) ) 𝜋 𝑝 + 𝛽𝑠 + 𝑘 (2𝑛 + 1) 𝑠𝑖𝑛 2𝑁 ( ( ) ) 𝜋 𝑝 + 𝛽𝑠 − 𝑘 (2𝑛 + 1) ] +𝑠𝑖𝑛 2𝑁

( × 𝑐𝑜𝑠 ( × √ (22) ×

Further, (20)–(22) can be solved to yield the interference coefficients as follows ( ) 𝑠𝑖𝑛𝜋 2𝑘 + 𝛽𝑟 𝑠𝑖𝑛𝜋𝛽𝑟 (𝑟) 𝛶𝑘𝑘 = ( ( )+ )+ 2𝑘+𝛽 𝛽 2𝑁𝑠𝑖𝑛𝜋 2𝑁 𝑟 2𝑁𝑠𝑖𝑛𝜋 2𝑁𝑟 ( ) ( ) 2𝑘+𝛽𝑟 𝛽 𝑠𝑖𝑛2 𝜋 𝑠𝑖𝑛2 𝜋 2𝑟 2 (23) ( ( ) )+ 2𝑘+𝛽 𝛽 2𝑁𝑠𝑖𝑛𝜋 2𝑁 𝑟 𝑁𝑠𝑖𝑛𝜋 2𝑁𝑟

(𝑟) = 𝛶𝑝𝑘

(𝑠) 𝛶𝑝𝑘

( ) ( ) 𝑠𝑖𝑛𝜋 𝑝 + 𝛽𝑟 + 𝑘 𝑠𝑖𝑛𝜋 𝑝 + 𝛽𝑟 − 𝑘 ( ( )+ )+ 𝑝+𝛽𝑟 +𝑘 𝑝+𝛽𝑟 −𝑘 2𝑁𝑠𝑖𝑛𝜋 2𝑁𝑠𝑖𝑛𝜋 2𝑁 2𝑁 ( ) ) ( 𝑝+𝛽𝑟 +𝑘 2 𝜋 𝑝+𝛽𝑟 −𝑘 𝑠𝑖𝑛 𝑠𝑖𝑛2 𝜋 2 2 ( )+ ( ) 𝑝+𝛽𝑟 +𝑘 𝑝+𝛽𝑟 −𝑘 2𝑁𝑠𝑖𝑛𝜋 𝑁𝑠𝑖𝑛𝜋 2𝑁 2𝑁

( ) ( ) 𝑠𝑖𝑛𝜋 𝑝 + 𝛽𝑠 + 𝑘 𝑠𝑖𝑛𝜋 𝑝 + 𝛽𝑠 − 𝑘 = ( ( )+ )+ 𝑝+𝛽𝑠 +𝑘 𝑝+𝛽𝑠 −𝑘 2𝑁𝑠𝑖𝑛𝜋 2𝑁𝑠𝑖𝑛𝜋 2𝑁 2𝑁 ( ( ) ) 𝑝+𝛽𝑠 −𝑘 𝑝+𝛽𝑠 +𝑘 𝑠𝑖𝑛2 𝜋 𝑠𝑖𝑛2 𝜋 2 2 ( )+ ( ) 𝑝+𝛽𝑠 +𝑘 𝑝+𝛽𝑠 −𝑘 2𝑁𝑠𝑖𝑛𝜋 𝑁𝑠𝑖𝑛𝜋 2𝑁 2𝑁

𝜋 (2𝑛 + 1) 𝑘 2𝑁

) ] ⎡√ 𝐿−1 ∑ ⎢ 2 𝐶𝑘 ℎ(𝑢) 𝑐𝑜𝑠 𝑙 ⎢ 𝑁 𝑙=𝑁𝑐𝑝 +𝛼𝑢 −1 ⎣

)⎤ ⎥+ ⎥ ⎦ 𝑁−1 [√ ∑

𝜋 (2𝑙 + 1) 𝑘 2𝑁

2 𝐶 𝑁 𝑘 𝑛=−𝛼

𝑢 −𝑁𝑐𝑝 +𝑙

𝑁−1 2 ∑ (𝑢) 𝑋 𝐶 𝑐𝑜𝑠 𝑁 𝑘=0 𝑘 𝑘 𝑘∈𝑍𝑢

( ( )( ( ) )) 𝜋 𝑘 + 𝛽𝑢 2 𝑛 + 𝛼𝑢 + 1 2𝑁 (

× 𝑐𝑜𝑠

𝜋 (2𝑛 + 1) 𝑘 2𝑁

) ] [√

∑ (𝑢) 2 𝐶 ℎ 𝑐𝑜𝑠 𝑁 𝑘 𝑙=0 𝑙 𝐿−1

(

𝜋 (2𝑙 + 1) 𝑘 2𝑁

)]

+𝑊𝑘(𝑢)

(26)

(26), specifies the general scenario, and now the DCT output of the desired user 𝑟 can be deduced as 𝑌𝑘(𝑟) =

( ( ) 𝑁−1 [ ∑ 𝜋 2𝑘 + 𝛽𝑟 (2𝑛 + 1) 1 𝑐𝑜𝑠 𝑁 𝑛=−𝛼 −𝑁 +𝑙 2𝑁 𝑟 𝑐𝑝 ( )) 𝜋𝛼𝑟 𝑘 + 𝛽𝑟 + + 𝑁 ( ( )) ] 𝜋𝛽𝑟 (2𝑛 + 1) 𝜋𝛼𝑟 𝑘 + 𝛽𝑟 𝑐𝑜𝑠 + 𝐻𝑘(𝑟) + 2𝑁 𝑁 ( ( ) 𝑁−1 𝑁−1 [ ∑ ∑ 𝜋 𝑝 + 𝛽𝑟 + 𝑘 (2𝑛 + 1) (𝑟) 1 𝐶𝑘 𝐶𝑝 𝑋𝑝 𝑐𝑜𝑠 𝑁 𝑛=−𝛼 −𝑁 +𝑙 2𝑁 𝑝=0 𝐶𝑘2 𝑋𝑘(𝑟)

(24)

𝑟

𝑐𝑝

𝒑∈𝒁 𝒓 𝒑≠𝒌

(25) +

The first term under the brace in (19), elucidates the signal of the corresponding subscriber 𝑟, while the second term specifies the amount of ICI generated among the subcarrier components 𝑘 and 𝑝 which are allocated to subscriber 𝑟. Whereas, the third term under the brace signifies MUI/MAI emanated from subscriber 𝑠 to the dedicated subscriber 𝑟. Thus the corresponding interferences associated with them are categorized and solved in (20)–(25) respectively.

( )) 𝜋𝛼𝑟 𝑝 + 𝛽𝑟

𝑐𝑜𝑠

+ 𝑁 ( ( ) 𝜋 𝑝 + 𝛽𝑟 − 𝑘 (2𝑛 + 1) 2𝑁

∑

𝑁−1

𝐶𝑘

𝑝=0 𝒑∈𝒁 𝒔 𝒑≠𝒌

+

• Scenario 2: This is also the scenario which is depicting negative STO. Here, the STO 𝛼𝑢 lies in the interval 𝑁𝑐𝑝 − 𝐿 + 1 < 𝛼 ≤ 𝑁𝑐𝑝 . This phenomena illustrates that only some of the paths experience interference from previous paths due to overlap with the previous frame. Thereupon, this clearly emphasizes the deterioration in the strength of the received signal, because the previous frame overlaps with the present frame thereby leading to loss of orthogonality which further leads to ICI and MUI/MAI exists. This scenario is clearly depicted in Fig. 2b, where some of the samples of the current frame are lost in the processing window due to overlap with the previous frame. The frequency domain signal at the output of 𝑁-Point DCT of the dedicated subscriber 𝑟 on the 𝑘th subcarrier can be derived as follows:

𝐶𝑝 𝑋𝑝(𝑠)

1 𝑁

( )) ] 𝜋𝛼𝑟 𝑝 + 𝛽𝑟

[ 𝑐𝑜𝑠

∑

𝑁−1

𝐻𝑝(𝑟) + 𝑁 ( ( ) 𝜋 𝑝 + 𝛽𝑠 + 𝑘 (2𝑛 + 1) 2𝑁

𝑛=−𝛼𝑠 −𝑁𝑐𝑝 +𝑙

( )) 𝜋𝛼𝑠 𝑝 + 𝛽𝑠

𝑐𝑜𝑠

+ 𝑁 ( ( ) 𝜋 𝑝 + 𝛽𝑠 − 𝑘 (2𝑛 + 1) 2𝑁

∑

𝑁−1

𝐶𝑘

𝐶𝑝 𝑋𝑝(𝑟)(𝑃 𝑟𝑒𝑣𝑖𝑜𝑢𝑠)

𝑝=0 𝒑∈𝒁 𝒓 𝒑≠𝒌

+

+

1 𝑁

+

( )) ] 𝜋𝛼𝑠 𝑝 + 𝛽𝑠 𝑁

−𝛼𝑟 −𝑁𝑐𝑝 +𝑙−1 [

∑ 𝑛=0

𝑐𝑜𝑠

𝐻𝑝(𝑠) +

( ( ) 𝜋 𝑝 + 𝛽𝑟 + 𝑘 (2𝑛 + 1) 2𝑁

( )( )) 𝜋 𝑁𝑐𝑝 + 𝛼𝑟 𝑝 + 𝛽𝑟

+ 𝑁 ( ( ) 𝜋 𝑝 + 𝛽𝑟 − 𝑘 (2𝑛 + 1)

( )( )) ] ′ 𝜋 𝑁𝑐𝑝 + 𝛼𝑟 𝑝 + 𝛽𝑟 + 𝐻𝑝(𝑟) + 𝑐𝑜𝑠 2𝑁 𝑁 ( ( ) −𝛼𝑠 −𝑁𝑐𝑝 +𝑙−1 [ 𝑁−1 ∑ ∑ 𝜋 𝑝 + 𝛽𝑠 + 𝑘 (2𝑛 + 1) (𝑠)(𝑃 𝑟𝑒𝑣𝑖𝑜𝑢𝑠) 1 𝑐𝑜𝑠 𝐶𝑘 𝐶𝑝 𝑋𝑝 𝑁 2𝑁 𝑛=0 𝑝=0

𝑌𝑘(𝑢) = √ −𝛼𝑢 −𝑁𝑐𝑝 +𝑙−1 [√ 𝑁−1 ∑ 2 2 ∑ (𝑢) 𝐶𝑘 𝑋 𝐶 𝑐𝑜𝑠 𝑁 𝑁 𝑘=0 𝑘 𝑘 𝑛=0

𝒑∈𝒁 𝒔 𝒑≠𝒌

+

( )( )) 𝜋 𝑁𝑐𝑝 + 𝛼𝑠 𝑝 + 𝛽𝑠

𝑘∈𝑍𝑢

( ( )( ( ) )) 𝜋 𝑘 + 𝛽𝑢 2 𝑛 + 𝛼𝑢 + 𝑁𝑐𝑝 + 1

𝑐𝑜𝑠

2𝑁 302

𝑁

+

( ( ) 𝜋 𝑝 + 𝛽𝑠 − 𝑘 (2𝑛 + 1) 2𝑁

+

( )( )) ] ′ 𝜋 𝑁𝑐𝑝 + 𝛼𝑠 𝑝 + 𝛽𝑠 𝐻𝑝(𝑠) 𝑁

S. Vappangi and V.V. Mani

Optics Communications 435 (2019) 297–310

+𝑊𝑘(𝑟)

(27)

𝑐𝑜𝑠

𝑋𝑝(𝑠)(𝑃 𝑟𝑒𝑣𝑖𝑜𝑢𝑠)

the term in (27), enumerates the 𝑠th subscriber’s data symbol on the 𝑝th subcarrier of the previous frame. Further, from (27) it is evident that, the received signal of the desired user 𝑟 is severely degraded due to the prevalence of ICI among the subcarrier components which are allocated to it and this is can be categorized as self-interference between subcarriers 𝑘 and 𝑝. [ 𝑌𝑘(𝑟) = 𝐶𝑘2 𝑋𝑘(𝑟) 𝑐𝑜𝑠

( ( ) ) 𝜋 𝑘 + 𝛽𝑟 𝛼𝑟

+𝑠𝑖𝑛

[ 𝐶𝑝 𝑋𝑝(𝑟) 𝑐𝑜𝑠

∑

( ( ) ) 𝜋 𝑝 + 𝛽𝑟 𝛼𝑟 𝑁

𝑝=0 𝒑∈𝒁 𝒓 𝒑≠𝒌

𝛹 𝑝𝑘(𝑠) =

− 𝑠𝑖𝑛

( ( ) ) 𝜋 𝑝 + 𝛽𝑟 𝛼𝑟 ] 𝑁

(𝑟) 𝐻𝑝(𝑟) 𝛹𝑝𝑘 +

𝐼𝐶𝐼

∑

[

𝐶𝑘

𝐶𝑝 𝑋𝑝(𝑠) 𝑐𝑜𝑠

( ( ) ) 𝜋 𝑝 + 𝛽𝑠 𝛼𝑠 𝑁

𝑝=0 𝒑∈𝒁 𝒔 𝒑≠𝒌

− 𝑠𝑖𝑛

( ( ) ) 𝜋 𝑝 + 𝛽𝑠 𝛼𝑠 ] 𝑁

1 𝑁

(𝑠) + 𝐻𝑝(𝑠) 𝛹𝑝𝑘

𝑠𝑖𝑛

[ 𝑐𝑜𝑠

𝑁−1 ∑

( ( ) ) 𝜋 𝑝 + 𝛽𝑠 + 𝑘 (2𝑛 + 1)

𝑛=−𝛼𝑠 +𝑁𝑐𝑝 +𝑙

+𝑠𝑖𝑛

( ( ) 𝜋 𝑝 + 𝛽𝑠 − 𝑘 (2𝑛 + 1)

2𝑁 ( ( ) ) 𝜋 𝑝 + 𝛽𝑠 − 𝑘 (2𝑛 + 1) ]

[

𝐶𝑘

𝐶𝑝 𝑋𝑝(𝑟)(𝑃 𝑟𝑒𝑣𝑖𝑜𝑢𝑠) 𝑐𝑜𝑠

( ( )( )) 𝜋 𝑝 + 𝛽𝑟 𝑁𝑐𝑝 + 𝛼𝑟 𝑁

𝑝=0 𝒑∈𝒁 𝒓 𝒑≠𝒌

𝑠𝑖𝑛

( ( )( )) ] 𝜋 𝑝 + 𝛽𝑟 𝑁𝑐𝑝 + 𝛼𝑟

(𝑟)(𝑃 𝑟𝑒𝑣𝑖𝑜𝑢𝑠) = 𝜒𝑝𝑘

−

∑

𝑁−1

[ 𝐶𝑝 𝑋𝑝(𝑠)(𝑃 𝑟𝑒𝑣𝑖𝑜𝑢𝑠) 𝑐𝑜𝑠

𝑁

( ( )( )) ] 𝜋 𝑝 + 𝛽𝑠 𝑁𝑐𝑝 + 𝛼𝑠

′

(𝑠) 𝐻𝑝 (𝑠) 𝜒𝑝𝑘 𝑁 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

−

𝑠𝑖𝑛

+𝑊𝑘(𝑟)

1 𝑁

𝑀𝑈 𝐼 𝑎𝑟𝑖𝑠𝑖𝑛𝑔 𝑓 𝑟𝑜𝑚 𝑠𝑢𝑏𝑠𝑐𝑟𝑖𝑏𝑒𝑟 𝑠 𝑡𝑜 𝑟 𝑓 𝑟𝑜𝑚 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 𝑠𝑦𝑚𝑏𝑜𝑙

From (28), the frequency response of the channel when the signal is prone to interference with the previous symbol is given as [𝐿−1 ( )] ∑ (𝑟) 𝜋 (2𝑙 + 1) 𝑘 (𝑟) ℎ𝑙 𝑐𝑜𝑠 𝐻𝑘 = (29) 2𝑁 𝑙=0 𝐻𝑝(𝑟) =

[𝐿−1 ∑

ℎ𝑙(𝑟) 𝑐𝑜𝑠

(

𝜋 (2𝑙 + 1) 𝑝 2𝑁

𝑙=0

𝐻𝑝(𝑠)

=

[𝐿−1 ∑

ℎ(𝑠) 𝑐𝑜𝑠 𝑙

(

𝑙=0

𝜋 (2𝑙 + 1) 𝑝 2𝑁

𝑐𝑜𝑠

𝑠𝑖𝑛

)]

)] (31)

(32)

( )⎤ ⎡ 𝐿−1 ∑ 𝜋 (2𝑙 + 1) 𝑝 ⎥ ′ 𝐻𝑝(𝑠) = ⎢ ℎ(𝑠) 𝑐𝑜𝑠 ⎢𝑙=𝑁 +𝛼 +1 𝑙 ⎥ 2𝑁 ⎣ 𝑐𝑝 𝑠 ⎦

(33)

1 𝑁

[ 𝑐𝑜𝑠

𝑁−1 ∑

( ( ) ) 𝜋 2𝑘 + 𝛽𝑟 (2𝑛 + 1) 2𝑁

𝑛=−𝛼𝑟 +𝑁𝑐𝑝 +𝑙

+𝑠𝑖𝑛

( ( ) ) 𝜋 2𝑘 + 𝛽𝑟 (2𝑛 + 1) 2𝑁

( + 𝑠𝑖𝑛

∑

𝛹 𝑝𝑘(𝑟)

1 = 𝑁

𝑛=−𝛼𝑟 +𝑁𝑐𝑝 +𝑙

[

𝑐𝑜𝑠

+

+

(37)

2𝑁

𝑐𝑜𝑠

( ( ) ) 𝜋 𝑝 + 𝛽𝑠 + 𝑘 (2𝑛 + 1)

𝑛=0

2𝑁

+ 𝑠𝑖𝑛 2𝑁 ( ( ) ) 𝜋 𝑝 + 𝛽𝑠 − 𝑘 (2𝑛 + 1) ]

+

( ( ) ) 𝜋 𝑝 + 𝛽𝑠 + 𝑘 (2𝑛 + 1) 2𝑁

( + 𝑐𝑜𝑠

( ( )( ( ) )) 𝜋 𝑘 + 𝛽𝑢 2 𝑛 + 𝛼𝑢 + 1

𝜋𝛽𝑟 (2𝑛 + 1) 2𝑁

( ( ) ) 𝜋 𝑝 + 𝛽𝑟 + 𝑘 (2𝑛 + 1)

+

(38)

2𝑁

𝑘∈𝑍𝑢

) 𝜋𝛽𝑟 (2𝑛 + 1) ] 2𝑁

2𝑁

2𝑁 ( ( ) ) 𝜋 𝑝 + 𝛽𝑟 − 𝑘 (2𝑛 + 1) ]

) ( ( ) 𝜋 𝑝 + 𝛽𝑠 − 𝑘 (2𝑛 + 1)

2𝑁

)

) [√ ( )] 𝐿−1 ∑ (𝑢) 𝜋 (2𝑛 + 1) 𝑘 ] 𝜋 (2𝑙 + 1) 𝑘 2 𝐶𝑘 ℎ𝑙 𝑐𝑜𝑠 + 2𝑁 𝑁 2𝑁 𝑙=0 √ √ 𝑁−1 𝑁−1 ∑ 2 2 ∑ (𝑢) 𝐶𝑘 𝑋 𝐶 𝑁 𝑁 𝑘=0 𝑘 𝑘 𝑛=𝑁−𝛼 +𝑙 (

× 𝑐𝑜𝑠

(34)

𝑢

𝑁−1 ∑

2𝑁

𝑌𝑘(𝑢) = √ 𝑁−𝛼𝑢 +𝑙−1 [√ 𝑁−1 ∑ 2 2 ∑ (𝑢) 𝐶𝑘 𝑋 𝐶 𝑐𝑜𝑠 𝑁 𝑁 𝑘=0 𝑘 𝑘 𝑛=0

The interference components in (28) can be derived as 𝛹 𝑘𝑘(𝑟) =

−𝛼𝑠 +𝑁𝑐𝑝 +𝑙−1 [

( ( ) ) 𝜋 𝑝 + 𝛽𝑟 + 𝑘 (2𝑛 + 1)

• Scenario 3: Similar to scenario 2, this case also depicts the condition where the current symbol of the existing frame is overlapped with the next frame’s symbol indicating that the present frame experiences interference from all the paths belonging to the next frame and this is detailed in Fig. 2c. Unlike, scenario 1 and 2, the timing error 𝛼 is positive in this case and lies in the interval 0 < 𝛼 < 𝐿. Therefore, the DCT output in general (pertaining to any subscriber) can be represented as

(30)

( )⎤ ⎡ 𝐿−1 ∑ 𝜋 (2𝑙 + 1) 𝑝 ⎥ ′ 𝐻𝑝(𝑟) = ⎢ ℎ𝑙(𝑟) 𝑐𝑜𝑠 ⎢𝑙=𝑁 +𝛼 +1 ⎥ 2𝑁 ⎣ 𝑐𝑝 𝑟 ⎦

(36)

2𝑁 ) ( ( ) 𝜋 𝑝 + 𝛽𝑟 + 𝑘 (2𝑛 + 1)

(𝑠)(𝑃 𝑟𝑒𝑣𝑖𝑜𝑢𝑠) 𝜒𝑝𝑘 =

(28)

𝑐𝑜𝑠

( ( ) ) 𝜋 𝑝 + 𝛽𝑟 − 𝑘 (2𝑛 + 1)

+𝑠𝑖𝑛

( ( )( )) 𝜋 𝑝 + 𝛽𝑠 𝑁𝑐𝑝 + 𝛼𝑠

+

𝑛=0

′

𝑝=0 𝒑∈𝒁 𝒔 𝒑≠𝒌

𝑠𝑖𝑛

∑

(𝑟) 𝐻𝑝 (𝑟) 𝜒𝑝𝑘 +

𝐼𝐶𝐼 𝑒𝑚𝑎𝑛𝑎𝑡𝑖𝑛𝑔 𝑓 𝑟𝑜𝑚 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 𝑓 𝑟𝑎𝑚𝑒 𝑠𝑦𝑚𝑏𝑜𝑙

𝐶𝑘

−𝛼𝑟 +𝑁𝑐𝑝 +𝑙−1 [

1 𝑁 𝑐𝑜𝑠

𝑁 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

+

2𝑁

𝑀𝑈 𝐼∕𝑀𝐴𝐼 𝑎𝑟𝑖𝑠𝑖𝑛𝑔 𝑓 𝑟𝑜𝑚 𝑠𝑢𝑏𝑠𝑐𝑟𝑖𝑏𝑒𝑟 𝑠 𝑡𝑜 𝑟

∑

2𝑁 )

2𝑁 ( ( ) ) 𝜋 𝑝 + 𝛽𝑠 + 𝑘 (2𝑛 + 1)

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ 𝑁−1

(35)

2𝑁

𝑐𝑜𝑠

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ 𝑁−1

+ 2𝑁 ( ( ) ) 𝜋 𝑝 + 𝛽𝑟 − 𝑘 (2𝑛 + 1) ]

𝑠𝑖𝑛

𝐷𝑒𝑠𝑖𝑟𝑒𝑑 𝑆𝑖𝑔𝑛𝑎𝑙 𝑐𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑖𝑛𝑔 𝑡𝑜 𝑟𝑡ℎ 𝑢𝑠𝑒𝑟 𝑁−1

2𝑁 ( ( ) ) 𝜋 𝑝 + 𝛽𝑟 + 𝑘 (2𝑛 + 1)

( ( ) ) 𝜋 𝑘 + 𝛽𝑟 𝛼𝑟 ]

(𝑟) − 𝑠𝑖𝑛 + 𝐻𝑘(𝑟) 𝛹𝑘𝑘 𝑁 𝑁 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

𝐶𝑘

( ( ) ) 𝜋 𝑝 + 𝛽𝑟 − 𝑘 (2𝑛 + 1)

+

× 𝑐𝑜𝑠

303

𝑘∈𝑍𝑢

( ( )( ( ) )) 𝜋 𝑘 + 𝛽𝑢 2 𝑛 + 𝑁𝑐𝑝 − 𝛼𝑢 + 1 2𝑁

( 𝑐𝑜𝑠

𝜋 (2𝑛 + 1) 𝑘 2𝑁

)

S. Vappangi and V.V. Mani

[√

Optics Communications 435 (2019) 297–310

∑ (𝑢) 2 𝐶𝑘 ℎ𝑙 𝑐𝑜𝑠 𝑁 𝑙=0 𝐿−1

(

𝜋 (2𝑙 + 1) 𝑘 2𝑁

)]

subscriber 𝑟 can be derived to yield:

(39)

[ 𝑌𝑘(𝑟) = 𝐶𝑘2 𝑋𝑘(𝑟) 𝑐𝑜𝑠

1 𝐶𝑘2 𝑋𝑘(𝑟) 𝑁

𝑁−𝛼𝑟 +𝑙−1 [

∑

𝑐𝑜𝑠

2𝑁 )) ( ] 𝜋𝛼𝑟 𝑘 + 𝛽𝑟

𝑛=0

(

+

)) ( 𝜋𝛼𝑟 𝑘 + 𝛽𝑟 𝑁

𝐶𝑘

+

𝑐𝑜𝑠

𝐶𝑘

×

[𝐿−1 ∑ 𝑙=0

𝐶𝑘

𝑁−1 ∑

𝐶𝑘

𝐶𝑝 𝑋𝑝(𝑟)(𝑁𝑒𝑥𝑡)

1 𝑁

𝑁

𝑐𝑜𝑠

×

[𝐿−1 ∑ 𝑙=0

𝐶𝑘

𝑁−1 ∑

𝐶𝑝 𝑋𝑝(𝑠)(𝑁𝑒𝑥𝑡)

𝑐𝑜𝑠

2𝑁 ( ( ) 𝜋 𝑝 + 𝛽𝑠 − 𝑘 (2𝑛 + 1)

[𝐿−1 ∑ 𝑙=0

2𝑁 ℎ𝑙(𝑠) 𝑐𝑜𝑠

(

𝜋 (2𝑙 + 1) 𝑝 2𝑁

)]

𝑁

−

2𝑁

𝑁 ( )( )) ] 𝜋 𝑁𝑐𝑝 − 𝛼𝑠 𝑝 + 𝛽𝑠

𝑁

( ( )( )) ] 𝜋 𝑁𝑐𝑝 − 𝛼𝑠 𝑝 + 𝛽𝑠

−

+𝑊𝑘(𝑟)

∑

𝑐𝑜𝑠 2𝑁 𝑛=0 ) ( ( ) 𝜋 𝑝 + 𝛽𝑟 − 𝑘 (2𝑛 + 1) 𝑐𝑜𝑠 + 2𝑁 ( ( ) ) 𝜋 𝑝 + 𝛽𝑟 + 𝑘 (2𝑛 + 1) 𝑠𝑖𝑛 2𝑁 ( ( ) ) 𝜋 𝑝 + 𝛽𝑟 − 𝑘 (2𝑛 + 1) ] +𝑠𝑖𝑛 2𝑁 𝑁−𝛼𝑠 +𝑙−1 [

∑

1 𝑁

(41)

𝑠𝑖𝑛

2𝑁 ( ( ) ) 𝜋 𝑝 + 𝛽𝑠 + 𝑘 (2𝑛 + 1)

+𝑠𝑖𝑛

(40), can be further rearranged by segregating the desired signal as well as the interference components pertaining to the subscriber 𝑟 and finally, the expression for the DCT output for the desired

(𝑟) 𝛾𝑝𝑘

304

1 = 𝑁

2𝑁 )

+

(43)

( ( ) ) 𝜋 𝑝 + 𝛽𝑠 + 𝑘 (2𝑛 + 1)

( ( ) 𝜋 𝑝 + 𝛽𝑠 − 𝑘 (2𝑛 + 1)

𝑁 (40)

𝑐𝑜𝑠

𝑛=0

𝑐𝑜𝑠

+

( ( ) ) 𝜋 𝑝 + 𝛽𝑟 + 𝑘 (2𝑛 + 1)

𝑁−𝛼𝑟 +𝑙−1 [

1 = 𝑁

𝑁

( )( )) 𝜋 𝑁𝑐𝑝 − 𝛼𝑠 𝑝 + 𝛽𝑠

( ( )( )) 𝜋 𝑁𝑐𝑝 − 𝛼𝑠 𝑝 + 𝛽𝑠

𝑀𝑈 𝐼∕𝑀𝐴𝐼 𝑎𝑟𝑖𝑠𝑖𝑛𝑔 𝑓 𝑟𝑜𝑚 𝑠𝑢𝑏𝑠𝑐𝑟𝑖𝑏𝑒𝑟 𝑠 𝑡𝑜 𝑟 𝑑𝑢𝑒 𝑡𝑜 𝑁𝑒𝑥𝑡 𝐹 𝑟𝑎𝑚𝑒

( )( )) ] 𝜋 𝑁𝑐𝑝 − 𝛼𝑟 𝑝 + 𝛽𝑟

+ 𝑊𝑘(𝑟)

+ 𝐻𝑝(𝑠) ℘(𝑠) 𝑝𝑘

The interference components in (41), can be further solved to attain ( ( ) ) 𝑁−𝛼𝑟 +𝑙−1 [ ∑ 𝜋 2𝑘 + 𝛽𝑟 (2𝑛 + 1) 1 = ℘(𝑟) 𝑐𝑜𝑠 + 𝑘𝑘 𝑁 2𝑁 𝑛=0 ( ( ) ) ( ) 𝜋 2𝑘 + 𝛽𝑟 (2𝑛 + 1) 𝜋𝛽𝑟 (2𝑛 + 1) 𝑐𝑜𝑠 + 𝑠𝑖𝑛 2𝑁 2𝑁 ( ) 𝜋𝛽𝑟 (2𝑛 + 1) ] +𝑠𝑖𝑛 (42) 2𝑁

℘(𝑠) = 𝑝𝑘

+

𝑁

( ( )( )) 𝜋 𝑁𝑐𝑝 − 𝛼𝑟 𝑝 + 𝛽𝑟

(𝑠) 𝐻𝑝(𝑠) 𝛾𝑝𝑘 𝑁 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

𝑠𝑖𝑛

℘(𝑟) 𝑝𝑘

+

[ 𝐶𝑝 𝑋𝑝(𝑠)(𝑁𝑒𝑥𝑡) 𝑐𝑜𝑠

𝑝=0 𝒑∈𝒁 𝒔 𝒑≠𝒌

𝑁−1 1 ∑ 𝑁 𝑛=0

( ( ) 𝜋 𝑝 + 𝛽𝑠 + 𝑘 (2𝑛 + 1)

( )) ] 𝜋𝛼𝑠 𝑝 + 𝛽𝑠

( ( )( )) ] 𝜋 𝑁𝑐𝑝 − 𝛼𝑟 𝑝 + 𝛽𝑟

∑

𝐶𝑘

( ( ) 𝜋 𝑝 + 𝛽𝑟 + 𝑘 (2𝑛 + 1)

2𝑁 ( )] 𝜋 (2𝑙 + 1) 𝑝 ℎ𝑙(𝑟) 𝑐𝑜𝑠 + 2𝑁

𝑝=0 𝒑∈𝒁 𝒔 𝒑≠𝒌

[ × 𝑐𝑜𝑠

+

( − 𝑠𝑖𝑛

𝐼𝐶𝐼 𝑑𝑢𝑒 𝑡𝑜 𝑁𝑒𝑥𝑡 𝐹 𝑟𝑎𝑚𝑒

+

( ( ) 𝜋 𝑝 + 𝛽𝑟 − 𝑘 (2𝑛 + 1)

[ 𝐶𝑝 𝑋𝑝(𝑟)(𝑁𝑒𝑥𝑡) 𝑐𝑜𝑠

𝑁−1

𝑁

𝑛=𝑁−𝛼𝑟 +𝑙

( )( )) 𝜋 𝑁𝑐𝑝 − 𝛼𝑟 𝑝 + 𝛽𝑟

𝐻𝑝(𝑟) ℘(𝑟) + 𝑝𝑘

(𝑟) 𝐻𝑝(𝑟) 𝛾𝑝𝑘 + 𝑁 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

( )) ] 𝜋𝛼𝑠 𝑝 + 𝛽𝑠

[ 𝑐𝑜𝑠

𝑁−1 ∑

𝑁

𝑝=0 𝒑∈𝒁 𝒓 𝒑≠𝒌

( )) ] 𝜋𝛼𝑟 𝑝 + 𝛽𝑟

2𝑁 ( )] 𝜋 (2𝑙 + 1) 𝑝 ℎ(𝑠) 𝑐𝑜𝑠 + 𝑙 2𝑁

𝑝=0 𝒑∈𝒁 𝒓 𝒑≠𝒌

+

+

𝑁

𝑀𝑈 𝐼∕𝑀𝐴𝐼 𝑎𝑟𝑖𝑠𝑖𝑛𝑔 𝑓 𝑟𝑜𝑚 𝑠𝑢𝑏𝑠𝑐𝑟𝑖𝑏𝑒𝑟 𝑠 𝑡𝑜 𝑟

∑

𝑁−1

( )) 𝜋𝛼𝑠 𝑝 + 𝛽𝑠

𝑐𝑜𝑠

( )) ] 𝜋𝛼𝑟 𝑝 + 𝛽𝑟

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

𝒑∈𝒁 𝒔 𝒑≠𝒌

+ 𝑁 ( ( ) 𝜋 𝑝 + 𝛽𝑠 − 𝑘 (2𝑛 + 1)

𝐶𝑝 𝑋𝑝(𝑠) 𝑐𝑜𝑠

𝐼𝐶𝐼

( )) 𝜋𝛼𝑠 𝑝 + 𝛽𝑠

𝑝=0 𝒑∈𝒁 𝒔 𝒑≠𝒌

2𝑁 𝑁 [𝐿−1 ( )] ∑ (𝑟) 𝜋 (2𝑙 + 1) 𝑝 + × ℎ𝑙 𝑐𝑜𝑠 2𝑁 𝑙=0 ( ( ) 𝑁−𝛼𝑠 +𝑙−1 [ 𝑁−1 ∑ ∑ 𝜋 𝑝 + 𝛽𝑠 + 𝑘 (2𝑛 + 1) (𝑠) 1 𝐶𝑝 𝑋 𝑝 𝐶𝑘 𝑐𝑜𝑠 𝑁 2𝑁 𝑝=0 𝑛=0

+

(

− 𝑠𝑖𝑛

𝑁

(

[

∑

𝑠𝑖𝑛

+

𝐶𝑝 𝑋𝑝(𝑟) 𝑐𝑜𝑠

𝑁−1

+

𝑁 ( ( ) 𝜋 𝑝 + 𝛽𝑟 − 𝑘 (2𝑛 + 1)

( )) ] 𝜋𝛼𝑟 𝑘 + 𝛽𝑟

⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

𝒑∈𝒁 𝒓 𝒑≠𝒌

( )) 𝜋𝛼𝑟 𝑝 + 𝛽𝑟

( )) 𝜋𝛼𝑟 𝑝 + 𝛽𝑟

𝑝=0 𝒑∈𝒁 𝒓 𝒑≠𝒌

𝜋𝛽𝑟 (2𝑛 + 1) 𝑐𝑜𝑠 + 2𝑁 𝑁 [𝐿−1 )] ( ∑ (𝑟) 𝜋 (2𝑙 + 1) 𝑘 × + ℎ𝑙 𝑐𝑜𝑠 2𝑁 𝑙=0 ( ( ) 𝑁−𝛼𝑟 +𝑙−1 [ 𝑁−1 ∑ ∑ 𝜋 𝑝 + 𝛽𝑟 + 𝑘 (2𝑛 + 1) (𝑟) 1 𝑐𝑜𝑠 𝐶𝑘 𝐶𝑝 𝑋𝑝 𝑁 2𝑁 𝑛=0 𝑝=0

+

( − 𝑠𝑖𝑛

𝐷𝑒𝑠𝑖𝑟𝑒𝑑 𝑢𝑠𝑒𝑟’𝑠 𝑆𝑖𝑔𝑛𝑎𝑙

(

[

∑

𝑁−1

( ( ) 𝜋 2𝑘 + 𝛽𝑟 (2𝑛 + 1)

( )) 𝜋𝛼𝑟 𝑘 + 𝛽𝑟

𝐻 (𝑟) + ℘(𝑟) 𝑘𝑘 𝑘 𝑁 𝑁 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟

Thus, the DCT output of the dedicated user 𝑟 on the 𝑘th subcarrier can be evaluated as 𝑌𝑘(𝑟) =

(

+

+

2𝑁 ( ( ) ) 𝜋 𝑝 + 𝛽𝑠 − 𝑘 (2𝑛 + 1) ]

(44)

2𝑁 𝑁−1 ∑

𝑛=𝑁−𝛼𝑟 +𝑙

[ 𝑐𝑜𝑠

( ( ) ) 𝜋 𝑝 + 𝛽𝑟 + 𝑘 (2𝑛 + 1) 2𝑁

+

S. Vappangi and V.V. Mani

𝑐𝑜𝑠

𝑠𝑖𝑛

2𝑁 ( ( ) ) 𝜋 𝑝 + 𝛽𝑟 + 𝑘 (2𝑛 + 1)

+𝑠𝑖𝑛

(𝑠) 𝛾𝑝𝑘

1 = 𝑁

Optics Communications 435 (2019) 297–310

) ( ( ) 𝜋 𝑝 + 𝛽𝑟 − 𝑘 (2𝑛 + 1)

+

2𝑁 ( ( ) ) 𝜋 𝑝 + 𝛽𝑟 − 𝑘 (2𝑛 + 1) ]

(45)

2𝑁 𝑁−1 ∑

[ 𝑐𝑜𝑠

( ( ) ) 𝜋 𝑝 + 𝛽𝑠 + 𝑘 (2𝑛 + 1)

2𝑁 ( ( ) ) 𝜋 𝑝 + 𝛽𝑠 − 𝑘 (2𝑛 + 1) 𝑐𝑜𝑠 + 2𝑁 ( ( ) ) 𝜋 𝑝 + 𝛽𝑠 + 𝑘 (2𝑛 + 1) 𝑠𝑖𝑛 2𝑁 ( ( ) ) 𝜋 𝑝 + 𝛽𝑠 − 𝑘 (2𝑛 + 1) ] +𝑠𝑖𝑛 2𝑁

+

𝑛=𝑁−𝛼𝑟 +𝑙

Fig. 3. Comb type pilot arrangement.

(46) of ICI and MUI/MAI. Hence, it is necessary to estimate the CFO. This section derives Cramer Rao Lower Bound (CRLB) for the estimation of CFO over additive white Gaussian noise (AWGN) channel. Therefore, the received signal pertaining to the 𝑟th subscriber can be put up as ] [ (𝑟) (48) 𝑦(𝑟) 𝑛 = 𝐼𝐷𝐶𝑇 𝑌𝑘

Therefore, the aforementioned mathematical analysis on the frequencydomain received signal emphasizes the need to compensate these offsets by incorporating suitable synchronization algorithms which are compatible with IM/DD systems for VLC. The simulation results are detailed in the next section.

where 𝑌𝑘(𝑟) = 𝑋𝑘(𝑟) + 𝑊𝑘(𝑟) and upon substitution in (48), the following expression is obtained

3. Algorithms for the estimation of Carrier Frequency offset (CFO) 3.1. Classen

(𝑟) (𝑟) 𝑦(𝑟) 𝑛 = 𝑥𝑛 + 𝑤𝑛

This algorithm utilizes the pilot tones for the estimation of CFO because, in the recent times, VLC can be even utilized in the outdoor environment to provide Internet hot spots using street lighting and mobile access. In such kind of dynamic environment, the communication link may experience shadowing or temporary blocking thereby degrading the quality of service [22]. In general, pilot tones are inserted along with the data based on a specific pilot arrangement called comb type pilot arrangement as shown in Fig. 3. Therefore, the transmitted signal comprises of both data as well as pilot tones. In general, this algorithm as proposed by Classen comprises of two modes for the estimation of CFO: acquisition and tracking, where the former deals with large range of CFO while the latter involves only small frequency fluctuations. Let the time-domain signal corresponding to the desired subscriber be denoted as 𝑦(𝑟) 𝑛 and the mathematical expression is given in (6) and let 𝑓 denote the pilot insertion interval, therefore the pilot added time domain signal be represented as 𝑦(𝑟) . According to the principality of this algorithm, 𝑛+𝑓 these two signals are saved in the memory and then they are transformed into frequency domain by computing the DCT operation. In general, the pilot tones are extracted and then the CFO is estimated. This estimated CFO is used for compensation in the time domain. Accordingly, the estimated CFO of the desired subscriber 𝑟 can be expressed as

Furthermore, the received signal which is affected with CFO of 𝛽𝑟 can be expressed as ( ( )) √ 𝑁−1 𝜋 (2𝑛 + 1) 𝑘 + 𝛽𝑟 2 ∑ (𝑟) = 𝑦(𝑟) (50) 𝑐𝑜𝑠 + 𝑤(𝑟) 𝐶 𝑋 𝑛 𝑛 𝑁 𝑘=0 𝑘 𝑘 2𝑁 𝑘∈𝑍𝑟

The parameter 𝛽𝑟 to be estimated is hidden in the argument of cosine term. This necessitates to assume 𝜋(2𝑛+1)(𝑘+𝛽𝑟) as 𝜇. 2𝑁 √ 𝑦(𝑟) 𝑛 =

𝑁−1 2 ∑ 𝐶 𝑋 (𝑟) 𝑐𝑜𝑠𝜇 + 𝑤(𝑟) 𝑛 𝑁 𝑘=0 𝑘 𝑘

(51)

𝑘∈𝑍𝑟

Therefore, the Fisher Information matrix pertaining to the estimation of any parameter 𝛩 is given as [𝐼 (𝛩)]𝑖𝑗 =

𝑁−1 𝜕 1 ∑ 𝜕 𝑆 [𝑛, 𝛩] 𝑆 [𝑛, 𝛩] 𝜕𝛩𝑗 𝜎 2 𝑛=0 𝜕𝛩𝑖

(52)

Here, √ 𝑆 [𝑛, 𝛩] =

1 𝑚𝑎𝑥 𝛽̂ = 2𝜋𝑇𝑠𝑢𝑏 𝛽 ⎧|𝑁𝑝 −1 |⎫ |⎪ ⎪|| ∑ (𝑟) ∗(𝑟) ∗(𝑟) (𝑟) |⎬ 𝑌 , 𝛽] 𝑌 , 𝛽] 𝑋 , 𝛽] 𝑋 , 𝛽] [𝑞 [𝑑] [𝑞 [𝑑] [𝑞 [𝑑] [𝑞 [𝑑] ⎨| | 𝑘+𝑓 𝑘 𝑘+𝑓 𝑘 |⎪ ⎪|| 𝑑=0 |⎭ ⎩

(49)

𝑁−1 2 ∑ 𝐶 𝑋 (𝑟) 𝑐𝑜𝑠𝜇 𝑁 𝑘=0 𝑘 𝑘

(53)

𝑘∈𝑍𝑟

(47) [𝐼 (𝛽)]𝑖𝑗 =

From (47), 𝑁𝑝 denotes the total number of pilot tones, 𝑞 [𝑑] specifies the location of the 𝑑th pilot tone and 𝑋𝑘(𝑟) [𝑞 [𝑑]] signifies the pilot tone in the frequency domain which corresponds to the 𝑘th symbol period of the desired user 𝑟

𝑁−1 𝜕 1 ∑ 𝜕 𝑆 [𝑛, 𝛽] 𝑆 [𝑛, 𝛽] 𝜕𝛽𝑗 𝜎 2 𝑛=0 𝜕𝛽𝑖

(54)

𝜕 It is to be noted that 𝜇 is a function of 𝛽. Hence, 𝜕𝛽 (𝑐𝑜𝑠𝜇) = − (𝑠𝑖𝑛𝜇) ( ) 𝜋(2𝑛+1) . Therefore, the fisher information matrix can be solved to 2𝑁 attain [ ] [ ( ) ] 𝐼 (𝛩)11 = 𝐼 𝛽𝑟 11

3.2. Cramer Rao Lower Bound (CRLB)

2

⎡ ( )⎤ 𝑁−1 √ 𝑁−1 𝜋 (2𝑛 + 1) ⎥ 1 ∑⎢ 2 ∑ = 𝐶𝑘 𝑋𝑘(𝑟) 𝑠𝑖𝑛𝜇 ⎢ ⎥ 2𝑁 𝜎 2 𝑛=0 ⎢ 𝑁 𝑘=0 ⎥ ⎣ ⎦ 𝑘∈𝑍𝑟

3.2.1. CRLB for the estimation of CFO The aforementioned mathematical analysis as dealt in Section 2, clearly depicts the fact that the presence of CFO leads to the emergence 305

(55)

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Optics Communications 435 (2019) 297–310

upon substitution of (65) in (63), the following expression can be attained

(55) can be further solved to attain 2

⎡ 𝑁−1 ⎤ 𝑁−1 [ ] [ ( ) ] ⎥ 1 ∑ 2 𝜋 2 (2𝑛 + 1)2 ⎢ ∑ (𝑟) 𝐶 𝑋 𝑠𝑖𝑛𝜇 𝐼 (𝛩)11 = 𝐼 𝛽𝑟 11 = ⎢ ⎥ 𝑘 𝑘 𝜎 2 𝑛=0 𝑁 4𝑁 2 ⎢ 𝑘=0 ⎥ ⎣𝑘∈𝑍𝑟 ⎦

2

(56) [𝐼 (𝛼)]11 =

Using paraxial approximations where 𝑠𝑖𝑛𝜇 = 𝜇, (56) can be reduced to obtain [ ] [ ( ) ] 𝐼 (𝛩)11 = 𝐼 𝛽𝑟 11 =

⎡ 𝑁−1 ⎤ 𝑁−1 ∑ )⎥ 𝜋4 ∑ (𝑟) ( 4⎢ 𝐶𝑘 𝑋𝑘 𝑘 + 𝛽𝑟 ⎥ (2𝑛 + 1) ⎢ 8𝜎 2 𝑁 5 𝑛=0 ⎢ 𝑘=0 ⎥ ⎣𝑘∈𝑍𝑟 ⎦

[𝐼 (𝛼)]11

( ( ( ⎡ ) ) )⎤ 𝑁−1 𝑁−1 𝜋 2 𝑛 + 𝛼𝑟 + 1 𝑘 ⎥ 2𝜋 2 ∑ ⎢ ∑ (𝑟) = 𝑘𝐶 𝑋 ⎢ ⎥ 𝑘 𝑘 2𝑁 𝜎 2 𝑁 3 𝑛=0 ⎢ 𝑘=0 ⎥ ⎣𝑘∈𝑍𝑟 ⎦ 2

[𝐼 (𝛼)]11

⎡ ⎤ 𝑁−1 ∑ ) )2 ⎢ 𝑁−1 ⎥ 𝜋4 ∑ ( ( = 2 𝑛 + 𝛼𝑟 + 1 ⎢ 𝑘2 𝐶𝑘 𝑋𝑘(𝑟) ⎥ 2𝜎 2 𝑁 5 𝑛=0 ⎢ 𝑘=0 ⎥ ⎦ ⎣𝑘∈𝑍𝑟

[𝐼 (𝛼)]11 =

(58)

(𝑁+1)(48𝑁 3 +192𝑁 2 +248𝑁+112) 15

+1

𝑘=0 𝑘∈𝑍𝑟

]2 ( ) 𝐶𝑘 𝑋𝑘(𝑟) 𝑘 + 𝛽𝑟

⎡ 𝑁−1 ⎤ ⎢∑ 2 ⎥ ×⎢ 𝑘 𝐶𝑘 𝑋𝑘(𝑟) ⎥ ⎢ 𝑘=0 ⎥ ⎣𝑘∈𝑍𝑟 ⎦

𝑘∈𝑍𝑟

It is to be noted that the parameter 𝛼𝑟 which is to be estimated is hidden in the argument of cosine term. Therefore, the argument term 𝜋 (2(𝑛+𝛼𝑟 )+1)𝑘 i.e., can be assumed as 𝜇. Therefore, (61) can be reduced 2𝑁 as √ 𝑁−1 2 ∑ 𝑦(𝑟) = 𝐶 𝑋 (𝑟) 𝑐𝑜𝑠𝜇 + 𝑤(𝑟) (62) 𝑛 𝑛 𝑁 𝑘=0 𝑘 𝑘

⎡ 𝑁−1 ⎤ ⎢∑ 2 ⎥ ×⎢ 𝑘 𝐶𝑘 𝑋𝑘(𝑟) ⎥ ⎢ 𝑘=0 ⎥ ⎣𝑘∈𝑍𝑟 ⎦

It is to be noted that 𝜇 is a function of 𝛼𝑟 and making use of (52), the Fisher Information Matrix can be expressed as

𝑆 [𝑛, 𝛼] =

𝜕 𝑆 [𝑛, 𝛼] = − 𝜕𝛼

(71)

2

(72)

(63) 𝑉 [𝛼] ̂ ≥ [𝐼 (𝛼)]−1 11

(73)

Finally,

𝑁−1 2 ∑ 𝐶 𝑋 (𝑟) 𝑐𝑜𝑠𝜇 𝑁 𝑘=0 𝑘 𝑘

√

2

Therefore, the variance of the estimate 𝛼 which is denoted as 𝑉 (𝛼) ̂ can be expressed as

(64)

𝑉 [𝛼] ̂ ≥

𝑘∈𝑍𝑟

and

(70)

Finally, the Fisher Information Matrix can be derived as [ ( ) ( )2 ] 𝜋4 4 (𝑁 + 1) 𝑁 + 3𝛼𝑟 + 2 + 3 2𝛼𝑟 + 1 [𝐼 (𝛼)]11 = 2 4 6𝜎 𝑁

𝑘∈𝑍𝑟

√

2

(70) can be further solved as [ ( ) 4𝑁 (𝑁 + 1) (2𝑁 + 1) 4𝑁 (𝑁 + 1) 2𝛼𝑟 + 1 𝜋4 + [𝐼 (𝛼)]11 = 6 2 2𝜎 2 𝑁 5 ( )2 ] +𝑁 2𝛼𝑟 + 1

3.2.2. CRLB for the estimation of STO In the similar manner, when the received signal pertaining to the desired subscriber 𝑟 is effected with a timing offset i.e., STO of 𝛼𝑟 , then the expression can be formulated as ( ( ( ) ) ) √ 𝑁−1 𝜋 2 𝑛 + 𝛼𝑟 + 1 𝑘 2 ∑ (𝑟) = 𝑦(𝑟) 𝑐𝑜𝑠 𝐶 𝑋 (61) + 𝑤(𝑟) 𝑛 𝑛 𝑁 𝑘=0 𝑘 𝑘 2𝑁

where

(69)

𝑁−1 ( ( ) ( )2 ) 𝜋4 ∑ 4𝑛2 + 4𝑛 2𝛼𝑟 + 1 + 2𝛼𝑟 + 1 2 5 2𝜎 𝑁 𝑛=0

⎤ ⎡ 𝑁−1 ⎥ ⎢∑ 2 ×⎢ 𝑘 𝐶𝑘 𝑋𝑘(𝑟) ⎥ ⎢ 𝑘=0 ⎥ ⎣𝑘∈𝑍𝑟 ⎦

(60)

𝑁−1 ] 𝜕 [ ] 1 ∑ 𝜕 [ 𝑆 𝑛, 𝛼𝑖 𝑆 𝑛, 𝛼𝑗 [𝐼 (𝛼)]𝑖𝑗 = 2 𝜕𝛼𝑗 𝜎 𝑛=0 𝜕𝛼𝑖

2

Upon solving the summation in (69), [𝐼 (𝛼)]11 =

[ } ∑ 𝑁−1

𝑁−1 ( ( ) ( )2 ) 𝜋4 ∑ 4𝑛2 + 4𝑛 2𝛼𝑟 + 1 + 2𝛼𝑟 + 1 2 5 2𝜎 𝑁 𝑛=0

⎡ 𝑁−1 ⎤ ⎢∑ 2 ⎥ ×⎢ 𝑘 𝐶𝑘 𝑋𝑘(𝑟) ⎥ ⎢ 𝑘=0 ⎥ ⎣𝑘∈𝑍𝑟 ⎦

Finally,

𝜋4

(68)

(68) can be further solved to attain

2

8𝜎 2 𝑁 4

(67)

(67) can be further solved as

( ) Therefore, the variance of the estimate 𝛽 which is denoted as 𝑉 𝛽̂ can be expressed as [ ] (59) 𝑉 𝛽̂ ≥ [𝐼 (𝛽)]−1 11

{

(66)

2

Furthermore, upon solving the summation in (57), the following expression for the Fisher Information matrix can be derived as [ ] [ ( ) ] 𝐼 (𝛩)11 = 𝐼 𝛽𝑟 11 { } ( ) (𝑁 + 1) 48𝑁 3 + 192𝑁 2 + 248𝑁 + 112 𝜋4 +1 = 15 8𝜎 2 𝑁 4

[ ] 𝑉 𝛽̂ ≥

𝑛=0

⎤ ⎡ √ 𝑁−1 ( )⎥ ⎢ 2 ∑ 𝜋𝑘 (𝑟) 𝐶𝑘 𝑋𝑘 𝑠𝑖𝑛𝜇 ⎢− 𝑁 𝑁 ⎥⎥ ⎢ 𝑘=0 ⎦ ⎣ 𝑘∈𝑍𝑟

Furthermore, by employing paraxial approximations where 𝑠𝑖𝑛𝜇 ≃ 𝜇, the Fisher Information matrix corresponding to (66) can be reduced as

2

(57)

⎡ 𝑁−1 ⎤ )⎥ ⎢∑ (𝑟) ( ×⎢ 𝐶𝑘 𝑋𝑘 𝑘 + 𝛽𝑟 ⎥ ⎢ 𝑘=0 ⎥ ⎣𝑘∈𝑍𝑟 ⎦

1 𝜎2

𝑁−1 ∑

6𝜎 2 𝑁 4 [ ]2 { ( ) ( )2 } ∑𝑁−1 2 𝐶 𝑋 (𝑟) 𝜋 4 4 (𝑁 + 1) 𝑁 + 3𝛼𝑟 + 2 + 3 2𝛼𝑟 + 1 𝑘 𝑘 𝑘 𝑘=0 𝑘∈𝑍𝑟

2 𝑁

𝑁−1 ∑ 𝑘=0 𝑘∈𝑍𝑟

𝐶𝑘 𝑋𝑘(𝑟) 𝑠𝑖𝑛𝜇

(

𝜋𝑘 𝑁

)

(74)

(65)

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Fig. 4. Cyclic Prefix (CP) based STO estimation.

4. Synchronization algorithms for the estimation of STO The most detrimental aspects of STO is seen prior where, the signal is prone to severe ICI and ISI due to loss of orthogonality. Hence, it necessitates to estimate the starting point of the OOFDM symbol i.e., accurately estimating the STO by incorporating a synchronization algorithm at the receiving end. This section revisits the STO estimation techniques reconcilable with IM/DD systems for VLC. It is apparent that, cyclic prefix (CP)/guard interval is obtained by appending the last part of the symbol to the front of the symbol. Hence, it is evident that first and the last portion are associated with each other, and therefore can be used for STO estimation. It is evident from Fig. 4 that, the two windows designated as 𝐴 and 𝐴′ delineates the CP and the last portion of the data samples respectively of length 𝛿𝑁𝑐𝑝 . Therefore, this intensifies that, periodic information can be employed for effective frame synchronization. Hence, in order to detect the identical nature of samples existing within these two windows 𝐴 and 𝐴′ , the windows can be slided. Therefore, this particular fact can be highlighted to search a point where the deviation between the samples of data in these two windows is minimum. But, this constraint has got nothing to deal with CFO. In spite of the existence of CFO, the most efficacious way of estimation of STO is to minimize the squared difference between the identical sets of samples in windows 𝐴 and 𝐴′ . Subsequently, the STO estimate for the subscriber 𝑟 on the 𝑘th subcarrier for the 𝑞th symbol can be obtained as [23] ⎫ ⎧𝑁𝑐𝑝 −1+𝛼 ) ⎪ ∑ (| (𝑟) | 2⎪ | | + 𝑁 + 𝑗) 𝛼̂ = 𝑎𝑟𝑔 𝑚𝑖𝑛 ⎨ (𝑛 | |𝑦𝑞 (𝑛 + 𝑗)| − |𝑦∗(𝑟) | ⎬ | | 𝑞 | 𝛼 ⎪ ⎪ 𝑗=𝛼 ⎭ ⎩

Fig. 5. Performance Analysis of BER vs SNR in DCO-FOFDMA system employing BPSK modulation with the existence of CFO for single user scenario.

Fig. 6. Illustration of BER vs SNR in DCO-FOFDMA system employing 4-PAM modulation in the presence of CFO for single user scenario.

(75)

Secondly, like calculation of correlation between two identical sets of samples present in the two windows 𝐴 and 𝐴′ can be pronounced. Therefore, the STO estimate under this scenario can be formulated as [23,24] ⎧𝑁𝑐𝑝 −1+𝛼 ⎫ ⎪ ∑ | (𝑟) |⎪ 𝛼̂ = 𝑎𝑟𝑔 𝑚𝑎𝑥 ⎨ + 𝑁 + 𝑗) (𝑛 |𝑦𝑞 (𝑛 + 𝑗) 𝑦∗(𝑟) | 𝑞 | |⎬ 𝛼 ⎪ 𝑗=𝛼 ⎪ ⎩ ⎭

(76)

5. Results and discussions This section presents the simulated results for DCO-FOFDMA system employing 256 subcarriers. The length of cyclic prefix is taken as 1 th the subcarriers size. Following the literature, the amount of DC 4 bias which is added to assure the positivity of signal is 7 dB. Since, DCT is employed, the system model emphasizes the dependency on real modulation schemes like M-PAM and BPSK, the same has been illustrated in the results. The performance analysis of DCO-FOFDMA using BPSK modulation is interpreted in Fig. 5. To emphasize the sensitivity of CFO on the system performance, different range of CFO values from 0.05, 0.1, 0.15, 0.2, 0.25, 3 are taken into consideration. The simulated result as shown in Fig. 5, evidences that the increase in the range of CFO, worsens the system performance. At an SNR of 4 dB, the attained error floor is 4.57×10−4 for BPSK modulation in DCO-FOFDMA system without the presence of CFO.

Fig. 7. BER vs CFO in DCO-FOFDMA system employing BPSK modulation.

While, for CFO values of 0.05, 0.01, 0.15 and 0.2, the obtained probability of error floor is 5.82 × 10−4 , 0.0014 and 0.009348 respectively. For higher order of modulation like M-PAM, the performance of the system is more sensitive to CFO and the same can be interpreted from Fig. 6. A similar inference can be drawn pertaining to the performance of the system as that of the aforementioned case. Here, it is evident that, for higher values of CFO like 0.15, it is impossible to achieve a reduced error floor. Consequently, this hinders the detection capability in case of multi 307

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in Fig. 8, indicating the presence of different values of CFO degrades the performance of system thereby hindering the detection capability of multiple users. This can be evidenced in Fig. 8, because with the increase in number of subscribers, the BER performance deteriorates for higher values of CFO. It can be interpreted that when there is no CFO i.e., for CFO=0, the achievable BER is 7.813 × 10−5 , while, upon increasing the range of CFO from 0.05, 0.1, 0.15, 0.2, 0.25, 0.3 the obtained probability of error is 3.906 × 10−4 , 5.469 × 10−4 , 7.813 × 10−4 , 3.828 × 10−3 , 6.477 × 10−2 and 2.291 × 10−1 respectively. The detrimental aspects of CFO and STO on the received signal is clearly illustrated through mathematical analysis in the aforementioned sections. Moreover, different possible scenarios of timing mismatch intervals are also illustrated to clearly highlight the impact of negative and positive time error. Therefore, in order to estimate these timing errors, it is vital to employ synchronization algorithms. Consequently, two synchronization algorithms such as Classen which is also known as minimum difference method and Maximum Likelihood estimate or maximum correlation method are exploited and these are imposed to the developed DCO-FOFDMA system model. The simulation results as shown in Fig. 9a and 9b clearly depicts the scenario of negative and positive timing error. It indicates the scenario where the actual sample value arrives too late and little early at the receiving end. However, these scenarios are considered without the existence of CFO. It is interesting to note that, the actual and the estimated samples upon imposing the aforesaid algorithms are

Fig. 8. BER vs CFO in DCO-FOFDMA system illustrating multi-user scenario.

user scenario, thereby making MUI inevitable. Fig. 7, delineates the variation of BER with respective to CFO for different values of SNR. From the figure, it can be inferred that, the performance of the system deteriorates as the range of CFO values increase from 0, 0.05, 0.1, 0.15, 0.2. The same inference can be drawn with multi user scenario which is elucidated

Fig. 9. Different Timing Error estimations using Cyclic Prefix (CP) Based methods.

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Fig. 10. Cramer Rao Lower Bound (CRLB) for CFO estimation with varying number of subcarriers in DCO-FOFDMA system pertaining to desired subscriber.

Fig. 12. Cramer Rao Lower Bound (CRLB) for STO estimation with varying number of subcarriers in DCO-FOFDMA system pertaining to desired subscriber.

Unlike RF, handling these issues is not straight forward, due to real and positive nature of the signal transmission. Consequently, this work evaluates the effects of CFO and different scenarios of timing induced disparities (STO) on the received signal in DCO-FOFDMA system. Furthermore, a thorough mathematical analysis is accomplished highlighting the different forms of interference like ICI which emanates especially due to overlap of different subcarrier components allocated to the corresponding subscriber and MUI/MAI also occur in the presence of multiple users. The simulation results infers that the system performance is seriously deteriorated due to the presence of these offsets, thereby hindering the detection capability of multiple users in uplink scenario. Furthermore, it becomes difficult to achieve a reduced error floor when the sensitivity of these offsets increases. In this work we have incorporated the real trigonometric transform like DCT in order to relieve the burden of Hermitian Symmetry criteria which is mandatory in FFT based optical OFDM (OOFDM). In doing so, there is an increase in spectral efficiency when compared with conventional OOFDM. Hence, the proposed multiple access system provides a flexibility to impart high data rate communication to the end-users without relying on RF counterparts. The analytical analysis provided in this work emphasizes the necessity to estimate these offsets and then to compensate them by employing suitable synchronization algorithms which are compatible with IM/DD systems for VLC. Consequently, this work revisits the synchronization algorithms like Classen/Minimum difference method and Maximum Likelihood method/Maximum correlation method for the estimation of different scenarios of timing mismatches. Furthermore, the simulation results emphasizes that, the presence of CFO hinders the estimation capability of Maximum correlation method to accurately estimate the STO. This work derives the CRLB for the estimation of CFO and STO.

Fig. 11. Mean Square Error (MSE) for Classen vs CRLB.

in a good agreement. Furthermore, the exact peak and the minimum value coincides with each other. In order to further elucidate the detrimental aspects of CFO on the received signal, Fig. 9c and Fig. 9d illustrates the scenario of estimation of the negative timing error with and without the existence of CFO. As evident, the presence of CFO hinders the estimation capability of the sample values upon using maximum correlation method while, the estimated and actual sample values coincides with each other upon enforcing minimum difference method. Fig. 10 depicts the bound vs SNR for the estimation of CFO of the corresponding subscriber in DCO-FOFDMA system. As depicted in Fig. 10, the error reduces upon increasing the SNR. For 𝑁 = 128 subscribers, at SNR of 50 dB the error achieved was 10−14 . Fig. 11 depicts the performance comparison of Classen algorithm against CRLB where, the simulated result analysis emphasizes that Classen algorithm attains a better reduction in mean square error (MSE) for higher values of SNR. Even though, there is a significant improvement in the accuracy of the MSE curve of Classen algorithm, it is far from the CRLB and exhibits high error floor when compared with CRLB. The CRLB for the estimation of STO pertaining to desired subscriber is delineated in Fig. 12. It can be inferred that, higher amount of SNR is desired to attain a reduced variance of error. In particular, at SNR of 15 dB, the achieved error is 6.233 × 10−11 while at SNR of 50 db, the obtained error is 1.971 × 10−14 .

Acknowledgments The authors would like to thank Science and Engineering Research Board (SERB), New Delhi, India for the assistance of this research work under grant no: EMR/2016/007687. References [1] C.V.N. Index, Global mobile data traffic forecast update, 2015–2020 white paper, link: http://goo.gl/ylTuVx.

6. Conclusion

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