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Investigating channel frequency selectivity in indoor visible‐light communication systems

2015; Institution of Engineering and Technology; Volume: 10; Issue: 3 Linguagem: Inglês

10.1049/iet-opt.2015.0015

1751-8776

Shihe Long, Mohammad‐Ali Khalighi, Mike Wolf, Salah Bourennane, Zabih Ghassemlooy,

Advanced Photonic Communication Systems

IET OptoelectronicsVolume 10, Issue 3 p. 80-88 Research ArticlesFree Access Investigating channel frequency selectivity in indoor visible-light communication systems Shihe Long, Shihe Long Institut Fresnel, UMR CNRS 7249, École Centrale Marseille, Marseille, FranceSearch for more papers by this authorMohammad Ali Khalighi, Corresponding Author Mohammad Ali Khalighi Ali.Khalighi@fresnel.fr Institut Fresnel, UMR CNRS 7249, École Centrale Marseille, Marseille, FranceSearch for more papers by this authorMike Wolf, Mike Wolf Communications Research Laboratory, Ilmenau University of Technology, Ilmenau, GermanySearch for more papers by this authorSalah Bourennane, Salah Bourennane Institut Fresnel, UMR CNRS 7249, École Centrale Marseille, Marseille, FranceSearch for more papers by this authorZabih Ghassemlooy, Zabih Ghassemlooy Optical Communications Research Group, NCR Lab, Faculty of Engineering and Environment, Northumbria University, Newcastle upon Tyne, UKSearch for more papers by this author Shihe Long, Shihe Long Institut Fresnel, UMR CNRS 7249, École Centrale Marseille, Marseille, FranceSearch for more papers by this authorMohammad Ali Khalighi, Corresponding Author Mohammad Ali Khalighi Ali.Khalighi@fresnel.fr Institut Fresnel, UMR CNRS 7249, École Centrale Marseille, Marseille, FranceSearch for more papers by this authorMike Wolf, Mike Wolf Communications Research Laboratory, Ilmenau University of Technology, Ilmenau, GermanySearch for more papers by this authorSalah Bourennane, Salah Bourennane Institut Fresnel, UMR CNRS 7249, École Centrale Marseille, Marseille, FranceSearch for more papers by this authorZabih Ghassemlooy, Zabih Ghassemlooy Optical Communications Research Group, NCR Lab, Faculty of Engineering and Environment, Northumbria University, Newcastle upon Tyne, UKSearch for more papers by this author First published: 01 June 2016 https://doi.org/10.1049/iet-opt.2015.0015Citations: 17AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Channel characterisation for indoor visible-light communication systems is revisited. The purpose of this study is to evaluate the channel frequency selectivity, or in other words, the significance of inter-symbol interference (ISI) at the receiver and the necessity of channel equalisation to recover the transmitted data. The authors focus on the effect of the indoor channel by assuming no bandwidth constraint on the light-emitting-diodes and by considering a simple intensity modulation technique, excluding discrete multi-tone modulation. The channel impulse response (CIR) is first simulated using the iterative site-based method. Then, conventional metrics for evaluating channel frequency selectivity, that is, the root-mean-square delay spread and channel frequency response are investigated and their practical interest is discussed. The authors also consider the signal-to-ISI ratio (SIR), which they define based on the sampled (i.e. discrete-time) signal at the receiver, and demonstrate its usefulness in determining the necessity of channel equalisation at the receiver. They consider several link scenarios in a medium-size and a large room, and show the significance of the LOS components of CIR in determining the channel frequency selectivity. They also discuss the choice of the receiver filter and explain how it affects the SIR. 1 Introduction Visible-light communications (VLC) have attracted particular attention in the research world due to their potential in providing very high-rate data transmission through the use of solid-state lighting devices, especially in indoor environments [1, 2]. VLC offer several advantages over the 'traditional' radio-frequency (RF) wireless technologies (e.g. WiFi connections) including the availability of a very large unlicensed spectrum (about hundreds of THz), spatial confinement leading to communication security, and immunity to RF interference. Moreover, by exploiting the already-installed light-emitting-diode (LED) lighting infrastructures for broadband data transmission, VLC can be considered as a good example of green communication for the next generation high-speed local area networks. The main constraint on the data rate in VLC systems arises from the limited bandwidth of commercially available LEDs. The 'classical' approach is to use separate red–green–blue (RGB) emitters and to combine them to produce white light. However, this approach is of limited use and is phasing out in the lighting industry due to difficulties in RGB balancing and the high cost, although it provides devices of relatively high bandwidth (a few tens of MHz) and gives the possibility of colour mixing [3]. A popular alternative to RGB lighting is to use the so-called white LEDs, where a blue LED is used together with yellowish phosphor coating to emit broad spectrum white light. The main drawback of commercial white LEDs is their slow modulation response due to the slow response time of the phosphorous that limits the modulation bandwidth of the device to a few MHz. One proposed solution is to use a blue filter at the receiver to remove this slow component from the modulated signal such that the modulation bandwidth can be increased. For instance, it was shown in [4] that the 6 dB modulation bandwidth can be increased to more than 170 MHz provided that the LED driver circuit is well designed. Very recently, researchers have considered the use of GaN-based micron-size LEDs (commonly called μLEDs) that have modulation bandwidths around 60 MHz [5, 6]. Over the past years, most of the research works have focused on increasing the data rate mainly by means of discrete multi-tone modulation (DMT) [7–12], and less attention has been devoted to the effect of the indoor channel and the limitations on the data rate that may arise from multipath propagation. As a matter of fact, DMT is a robust solution to the limited bandwidth of the LEDs and at the same time to the channel delay dispersion. Numerous experimental works have already demonstrated the potential of DMT to provide high data-rate VLC links [e.g. 13–15]. Although DMT has become very popular in VLC systems, its optimality is rather questionable, especially from the spectral and energy efficiency points of view. Compared with coherent field modulation, where two (optical) quadrature carriers are individually modulated, VLC based on DC-biased DMT suffers from a factor 2 bandwidth efficiency loss due to the intensity modulation constraint. If asymmetrically clipped DMT [16] is used to avoid the DC offset, the bandwidth efficiency is reduced by a further factor 2. Other disadvantages concern the requirement of highly linear LED-drivers, which have typically a low power efficiency. Furthermore, the bit error rate performance of DC-biased approaches is very sensitive to the clipping noise and the modulation index [17]. Some other modulation schemes have been under investigation recently such as pulse-amplitude modulation [18] and carrierless amplitude and phase modulation [19, 20] for use in VLC systems. While these schemes benefit from more implementation simplicity, they are more sensitive to channel delay dispersion as they do not benefit from the inherent dispersion mitigation property of DMT. Indeed, in some situations such as in relatively large rooms and under shadowed/blocked line-of-sight (LOS), the channel dispersion can be relatively large. In such cases, a channel equalisation step, implemented either in time or frequency domains, may be necessary at the receiver. The frequency-domain equalisation appears to be an interesting and relatively low-complexity solution as it effectively turns out to move the inverse fast-Fourier-transform (IFFT) operation (which is done at the transmitter in DMT-based signalling) to the receiver side [17, 18, 21]. Our purpose in this work is to quantify accurately the limitation arising from the indoor VLC multipath channel itself and to see in which situations it can effectively limit the transmission rate. In other words, we would like to investigate the limitation (arising solely from the channel) on the data rate before suffering from channel frequency selectivity, or in other words, inter-symbol interference (ISI). For this purpose, we will assume ideal optical components, for example, we neglect the bandwidth limitation of the LEDs, and focus on the optical channel and also exclude the case of DMT signalling. We simulate the aggregated channel impulse response (CIR) including both LOS and diffuse components by using the well-known iterative site-based method [22]. To investigate the channel frequency selectivity, we consider the channel delay dispersion, frequency response, and also the signal-to-ISI ratio (SIR). We show that the main factor that impacts the channel frequency selectivity is the asymmetry between the multiple LOS paths corresponding to the multiple LED emitters rather than multipath reflections. Moreover, we clarify the role of the receiver filter and its real impact on the ISI after signal sampling. The rest of the paper is organised as follows. We present the channel model and the simulation method that we use to obtain the CIR in Section 2. The definitions of different channel characterisation metrics including the SIR are provided in Section 3. Then, we state in Section 4 our main assumptions and describe the system configuration for the case studies that we consider in this paper. Next, some simulation-based numerical results are presented and discussed in Section 5 to study the VLC channel. Lastly, Section 6 concludes the paper. 2 Indoor optical channel modelling In VLC systems, intensity modulation and direct detection (IM/DD) is used as the LED source is non-coherent. At the transmitter, the information-bearing signal is DC-biased prior to IM of the LED [see Fig. 1 (the DC bias sets the initial illumination level)]. At the receiver, a photo-diode (PD) followed by a trans-impedance amplifier (TIA) are used to regenerate the electrical signal, which is then passed through the receiver filter and sampled prior to signal demodulation. The role of the receiver filter, which can be a matched filter (MF) or a simple low-pass filter, is to reduce the noise effect. Fig. 1Open in figure viewerPowerPoint General block diagram of an IM/DD-based VLC system Let us denote by x (t) and y (t) the emitted optical intensity at the transmitter and the generated photo-current at the output of the PD, respectively. We have (1)where is the PD responsivity in A/W, h (t) is the baseband CIR, ⊗ denotes convolution, and n (t) is the receiver noise, which is modelled as signal-independent additive white Gaussian noise (AWGN) with double-sided power spectral density (PSD) of N0 /2. h (t) that includes the contribution of both the LOS and non-LOS (diffuse) components is given by (2)Here δ (.) denotes the Dirac delta function, NLED denotes the number of LED emitters, and Vi is the visibility function corresponding to the i th emitter, which is set to 0 when the LOS path between the receiver and the i th source is blocked, and to 1 otherwise. Moreover, ϕi and φi denote the emitting and incident angles, respectively, as illustrated in Fig. 2, Ar (φi) is the effective receiving PD area, di is the link distance, c is the speed of light, and T (ϕi) is related to the i th source radiation pattern. Assuming Lambertian radiation pattern for LEDs, we have T (ϕ) = (m + 1)cosm(ϕ)/(2π), where m is the Lambertian order, which is related to the semi-angle at half power ϕ1/2 of the emitter: m = −ln2/ln(cos (ϕ1/2)). Note that in writing (2), we have implicitly assumed that the propagation delays between the electrical signals that modulate the different LEDs are negligible [7]. Fig. 2Open in figure viewerPowerPoint Propagation model for the indoor VLC channel The calculation of diffuse component in (2) is far to be an easy task, and hence, simulation techniques are usually employed to obtain an approximation of hk(t). As stated previously, in this work we have adopted the iterative site-based method for evaluating the non-LOS components. By this method, the inner surface of the room and the objects inside it are first decomposed into N tiny Lambertian reflecting elements with a given reflectivity. The k reflection response h(k) (t, S, R) for a given pair of source S (i.e. an LED or a reflecting surface) and receiver R (i.e. a surface or the PD) can be approximately evaluated as [22] (3)where and represent the n th element ɛn acting as a receiver and a source, respectively, and is the reflectivity of ɛn. Moreover, is the LOS CIR between the element n and R, which is simply a shifted Dirac delta function according to (2). This way, the convolution in (3) becomes rather a simple operation. Note that in order to reduce the simulation time, to determine the k -reflection response, we start by calculating (which is quite simple to do), and use them to compute according to (3). We repeat this procedure until we determine . 3 Channel characterisation metrics We consider three criteria to quantify the limitation on the transmission rate: the root-mean-square (RMS) delay spread μ, the channel frequency response, and the SIR. 3.1 Conventional metrics Two conventional channel characterisation metrics are the 3 dB cut-off frequency and μ. For instance, simulation results of channel delay spread were presented in [23], where the authors considered the wavelength dependence of the reflectivity of surfaces inside a room. Moreover, the channel frequency response was investigated in [7], where the integrating-sphere model [24] was used for the non-LOS component. Let us denote the channel frequency response by H (f). The 3 dB channel cut-off frequency, f−3dB corresponds to (4)However, because of the oscillating behaviour of the frequency response due to the presence of dominant LOS propagation components, this metric is of limited usage for determining the degree of channel frequency selectivity, as we will demonstrate in the next section. The RMS delay spread is given by [25] (5)where τ is the channel mean excess delay, defined as the square root of the second central moment of the CIR squared [25] (6) 3.2 Signal-to-ISI ratio In the case of a relatively large μ, the link may suffer from ISI, which can seriously impact the system performance. We propose here to quantify the amount of ISI by defining the metric of SIR as (7)where PR,sig and PR,ISI denote the received powers corresponding to the 'desired' signal and ISI, respectively. A high SIR corresponds to an almost frequency non-selective channel, whereas a relatively low SIR signifies the need to channel equalisation at the receiver. Let PR be the total received power corresponding to a transmitted symbol. In some previous works (e.g. in [26]), PR,sig is considered as the optical received power during the symbol period Ts and PR,ISI as the received power outside Ts. In other words, it is assumed that (8)Here we consider a more realistic definition for PR,sig and PR,ISI that is more appropriate for optical communication systems in practice. In fact, at the receiver, the electrical signal is filtered and then sampled prior to detection (see Fig. 1). Therefore, we should reasonably define the SIR after signal sampling. Let us consider the transmitted signal as follows (9)where ak denotes the k th transmitted symbol [equal to zero or one for the case of non-return-to-zero (NRZ) on–off-keying (OOK) modulation, for example] and g (t) is the pulse shaping filter. At the receiver, considering a sampling rate of 1/Ts and assuming negligible noise n (t), the signal corresponding to the time sample j is given by (10)where p (t) = g (t) ⊗ h (t) ⊗ r (t) with r (t) being the impulse response of the receiver filter. Note that in (10), aj is the desired signal at time sample j and the right-hand-side term is in fact the ISI. We accordingly define the SIR as follows (11)where E{.} stands for the expected value. Assuming power-normalised symbols and also normalised channel with respect to the main LOS path, (11) can be simplified as (12)Note that in our simulations we consider g (t) of a rectangular shape, which is quite rational for IM/DD signalling schemes. Concerning r (t), we may consider the optimal MF at the receiver, which allows to maximise the signal-to-noise ratio (SNR) at the sampling times. However, the design of the MF becomes a complex task when we should deal with non-white noise (note that we consider here an analogue receiver filter and a sampling rate of one sample per symbol duration). For instance, if a large area PD (of relatively large capacitance) is used, the f2 noise can be non-negligible [27]. This will necessitate a sharp roll-off of the filter transfer function in the stop-band. A suitable choice is then a Bessel filter (BF), which has a constant group delay in its pass-band. Alternatively, other filter types can be used such as a Butterworth filter, as suggested in [28], which has the advantage of providing a sharper transition between the pass-band and the stop-band for a given filter order, but causes a significant group delay distortion for high orders (typically larger than 7). 4 System configuration and assumptions We describe in this section the configuration of the indoor VLC systems that we consider as the case study and also specify the corresponding assumptions and parameters. We first consider the case of a medium-size room as shown in Fig. 3. An array of (2 × 2) LEDs is considered, positioned on the ceiling as shown in Fig. 4 a. We particularly study three receiver positions of R1, R2, and R3 with coordinates (2.5, 2.5 m), (1.25, 1.25 m) and (0.5, 0.5 m), respectively, at a height of 0.85 m above the floor. Receivers are pointing upward (i.e. vertically towards the ceiling) and we do not consider any receiver lens. The reflecting surfaces of walls, floor, and the ceiling are assumed of plastic materials and as Lambertian reflectors of order 1. Note that, although the surface reflectivity is usually wavelength dependent, it is quite difficult to perform the simulations in the whole visible spectrum by considering this dependency, from the point of view of computational complexity. To simplify the simulations, we have used the results of Fig. 1 in [23] and calculated the average reflectivity over the entire visible spectrum. This is a good approximation if the maximum number of reflections considered in the simulations is not too high [23]. The calculated average reflectivities together with the other parameters adopted in our simulations are specified in Table 1. Table 1. Simulation parameters Parameter Value ceiling reflectivity 0.38 floor reflectivity 0.61 wall reflectivity 0.74 transmitter Lambertian order 1 receiver field-of-view (half angle) 70° PD active area 1 cm2 Fig. 3Open in figure viewerPowerPoint Indoor VLC system configuration in a medium-size room Fig. 4Open in figure viewerPowerPoint Layout of LEDs on the ceiling for the two cases of a Medium-size room of dimension (5 × 5 × 3) m3 b Large room of dimension (10 × 10 × 4) m3 We also consider two un-typical yet realistic situations: when all LOS paths are blocked by an obstacle, and when the receiver is tilted with respect to the vertical axis. For the blocked LOS case, we consider an obstacle of length, width, and height of 1.75, 0.25, and 2 m, respectively, at the origin coordinates (0.75, 0.25, 0 m) as shown in Fig. 3. The surface reflectivity of this obstacle is set to the same as of the wall. For the titled receiver case, we tilt the R3 towards the centre of the ceiling by 30°. Furthermore, in order to investigate the impact of the room size, we consider a relatively large room of dimension (10 × 10 × 4) m3 with an array of (4 × 4) LED lamps on the ceiling. The layout of the LED lamps for this case is shown in Fig. 4 b. We investigate three receiver positions that we denote by R ′1, R ′2, and R ′3 at the coordinates (5, 5 m), (2, 2 m), and (0.5, 0.5 m), respectively. To simulate the CIR, we assume that the same signal is transmitted from all LEDs. For the case of a small-to-medium size room, the use of several LEDs has the advantage of offering space diversity in the sense of avoiding signal loss in the case of (LOS) beam blocking. For the case of relatively large rooms or halls, this assumption applies to the case where VLC is used for information broadcasting. However, it is not adapted to multiple access applications where the use of cellular configurations seems to be a more appropriate approach [29]. In such a case, the CIR of each cell should be investigated individually, which would rather correspond to the presented study for the case of small-to-medium size room. Moreover, if multiple-input multiple-output configurations are to be used [30], spatial multiplexing is likely to be performed on different LED chips inside a lamp, while all LED lamps transmit the same signals. As such, the results that we present in this paper will apply to this case as well. Lastly, concerning the iterative site-based method that we use for simulating the CIR, we consider a maximum reflection order of 3. Indeed, as it is shown in [22], the power contribution by considering more reflections is practically negligible. The spatial resolution of the simulation method, that is, the area of each reflecting surface, is set to 10 cm × 10 cm. Moreover, the temporal resolution, that is, the bin width of the simulated CIR is set to 0.1 ns. 5 Numerical results We present here some numerical results to investigate the optical propagation channel and the limitations arising from the channel on the maximum transmission rate. We investigate the three metrics of RMS delay spread, channel frequency response, and SIR, defined in Section 3. 5.1 Simulated CIR We have shown in Fig. 5 the simulated impulse responses of the LOS path and those corresponding to one, two, and three reflections for the case of medium-size room. Note that the levels of different LOS components are indicated by labels for the purpose of illustration. R1 and R3 positions are considered together with the two cases of blocked LOS and tilted receiver for the latter. For the tilted receiver case, we consider a tilting angle of 30° towards the centre of the ceiling (an untilted receiver points vertically towards the ceiling). This way, we have different LOS paths and also additional reflections from the floor and walls. Fig. 5Open in figure viewerPowerPoint Medium-size room case, CIR at receiver positions a R1 b R3 c R3, blocked LOS d R3, tilted receiver At R1, concerning the LOS component, we notice a single shifted Dirac delta function from Fig. 5 a, which is due to the symmetry of the LED arrangement with respect to the receiver position in this case (see Fig. 3). There are three shifted Diracs at R3 (except for the blocked-LOS case), where we have a symmetry with respect to LEDs 1 and 4. It is worth mentioning that since the LOS paths dominate the diffuse component in the CIR, they are the main factor that determines the channel frequency selectivity. Lastly, we have an almost similar behaviour for the cases of tilted and untilted receivers as in Figs. 5 b and d, apart from the received intensity level, as expected. 5.2 RMS delay spread To evaluate the channel frequency selectivity based on the delay spread criterion, we calculate for different scenarios the mean excess delay τ and the RMS delay spread μ using the simulated aggregated CIRs (i.e. taking both LOS and diffuse components into account). At the same time, since both the asymmetry between the LOS paths (if any) and the diffuse component (non-LOS) contribute to the channel frequency selectivity, in order to see the significance of the former factor, we have also calculated τ and μ based only on the LOS component. The results are summarised in Table 2 where the two cases of medium-size and large rooms are considered. We have furthermore shown in the last column of the table the ratio of the total power corresponding to the LOS component to that of the diffuse component; what is usually referred to as 'Rice-factor' in RF systems. Let us focus on the RMS delay spread and the case of medium-size room. At R1, there is no delay spread corresponding to the LOS component due to the symmetry of LEDs arrangement with respect to the receiver position. We notice a larger μ for R2 and R3 positions, compared with R1. The reason is that at these positions we have more asymmetry between the LOS paths and this results in a larger delay spread. We also note a slightly larger μ for the case of tilted receiver, which is quite logical. In fact, for a larger tilting angle, we receive more contribution from the LOS and also from the diffused light due to the reflections from the floor and the walls. Lastly, for the case of blocked LOS, we obtain a relatively large μ of more than 4 ns. Table 2. Mean excess delay and RMS delay spread for different link scenarios Receiver position RMS delay spread μ, ns Rice factor LOS Aggregate channel medium-size room R1 0 0.26 3.56 R2 1.53 1.58 2.90 R3 1.69 1.76 1.43 R3, blocked LOS 0 4.37 0 R3, tilted 2.08 2.15 2.59 large room R ′1 1.83 2.00 6.53 R ′2 2.43 3.07 2.96 R ′3 3.37 4.66 0.89 'LOS' refers to considering only the LOS component of CIR, whereas 'Aggregate channel' includes both LOS and diffuse components. On the other hand, for the case of a large room, we notice generally larger delay spreads. Interestingly, here the major part of μ arises from the contribution of LOS, and taking the diffuse component into account affects the delay spread only slightly. As a matter of fact, the more significant factor affecting μ is the asymmetry between the multiple LOS paths (16 in total for the large room case). As expected, again the largest delay spread corresponds to the room corner, that is, R ′3 position. As a matter of fact, although the study of the RMS delay spread seems to be useful in comparing the degree of channel frequency selectivity of the different link configurations, its absolute value cannot be used to determine the limitation on the transmission rate. For instance, for the case of the medium-size room at R3 position, a μ of 1.76 ns would suggest that we can transmit with up to a rate of ∼500 Mbps with OOK signalling. However, as we will show later in Section 5.4, the data rate is much more constrained. To this reason, we investigate other metrics in the following. 5.3 Channel frequency response Let us now investigate the usefulness of the channel frequency response for evaluating the degree of channel frequency selectivity. We have presented in Fig. 6 plots of normalised modulus of channel frequency response for the case of medium-size room at receiver positions R1, R2, and R3 (untilted and tilted receiver, and blocked LOS), as well as for the case of the large room at R ′3 position. Fig. 6Open in figure viewerPowerPoint Plots of normalised modulus of channel frequency response a Medium-size room with receiver at positions R1 and R2 b Medium-size room at R3 position for untilted and tilted receivers and blocked-LOS cases, and large room at position R ′3 First, we notice that for R1 position, the frequency response is almost flat except for data rates lower than about 30 MHz, which confirms the results of Fig. 5 : here, the dominant factor is the LOS component, which is a single shifted Dirac delta function. Thus, it is quite reasonable to have an almost flat frequency response. However, we cannot determine an explicit limitation on the data rate based on the frequency response. For the cases of R2 and R3 positions, we notice oscillations of a few decibels. Once again, the dominant factor is the LOS component of CIR, and since we have three shifted Dirac delta functions (see Fig. 5), we have an oscillating behaviour in the frequency response. From Fig. 6 b, we notice a somehow similar behaviour for the case of tilted receiver at R3. For the case of R ′3 position in the large room, we notice more significant fluctuations (in amplitude) in the frequency response. The reason is the contribution of the 16 LOS paths of different delays to the CIR; as a result, in certain frequencies we experience more severe 'fades', compared with the previous cases for the medium-size room. For all these studied cases, because of the oscillating behaviour of the frequency response (which is due to the contribution of the LOS component), fixing a 3 dB bandwidth for the channel is effectively useless. The 3 dB bandwidth becomes meaningful only for the purely diffuse channel in the case of blocked LOS at R3 position, which is about 43 MHz. In conclusion, due to the limited interest of the frequency response and the 3 dB channel bandwidth, we resort to the third criterion, that is, SIR, in the following subsection. 5.4 Signal-to-ISI ratio As mentioned previously, the interest of studying the SIR is that we can predict whether or not a channel equalisation is necessary for a given transmission rate. To evaluate the SIR, for the sake of simplicity, we consider the NRZ OOK modulation. We obtain the received signal by convolvin