Performance of modified and low complexity pulse shaping filters for IEEE 802.11 OFDM transmission

ABSTRACT The most commonly used multicarrier modulation method in 802.11 wireless communication systems is Orthogonal Frequency Division Multiplexing (OFDM). However, OFDM is easily affected by Inter Symbol Interference (ISI). In this paper, different pulse shaping filters are employed in the OFDM system and three new filters are presented to further reduce ISI effects. The first one is a Modified Parametric Exponential Pulse (MPEXP) filter which employs a different transfer function that is implemented at the transmitter side. The second one is a Better Than Modified Flipped Exponential Pulse (BTMFEXP) filter which is derived by modifying the transfer function of Modified Flipped Exponential Pulse (MFEXP). The third one is a hybrid of BTMFEXP and MPEXP (HBTMFPEXP). Lastly, low complexity forms of the MFEXP and BTMFEXP, derived from Taylor Series, are implemented. The OFDM system was tested with the existing and proposed pulses over AWGN and Fading channels. The proposed filters show better Bit Error Rate (BER) performance. Under a high ISI level, BTMFEXP and HBTMFPEXP showed a gain of 0.23 and 0.57 dB respectively over MFEXP with AWGN channel. With a fading channel at high ISI level, the proposed BTMFEXP and HBTMFPEXP demonstrated a gain of 2.33 and 5.33 dB respectively over MFEXP.


Introduction
Inter Symbol Interference (ISI) is a problem that is encountered in most communication systems including 802.11a/g.ISI occurs when the symbols in OFDM overlap with each other in bandlimited channels.As a result, the probability of error due to distortion effects increases and the performance of the OFDM system degrades.Pulse shaping filters are therefore used to limit the effects of ISI and improve the error performance of the OFDM system.Pulse shaping filters have several conditions to satisfy to curb the ISI effects (Nyquist, 1928).The main condition is that the equivalent impulse response of the transmitting and receiving filters should have zero crossings at multiples of the symbol period, T (Nyquist, 1928).Several works have proposed enhanced pulse shaping filters to solve the problem of ISI and to improve the performance of OFDM.An overview of the current pulse shaping filters is given next.
The authors in (Fowdur & Jhengree, 2017) presented two new filters, Modified Flipped Exponential (MFEXP) pulse and Hybrid of MFEXP and PEXP (HFPEXP) which demonstrated better Bit Error Rate (BER) performance.The lower side lobes of the filters made the system less affected to ISI and Inter Carrier Interference (ICI).The authors in (Alexandru & Balan, 2016) demonstrated that with Piecewise Flipped Exponential (PFE) pulses, better BER performances are obtained.In (Mohammad, Abdelhamid, & Hafez, 2015), New Nyquist Linear Combination Pulse (NNLCP) which effectively controlled high peak to average power ratio (PAPR) of IFDMA while keeping the same bandwidth compared to Raised Cosine (RC) was presented.The authors of (Azurdia-Meza, 2013) implemented Parametric Exponential Pulse (PEXP).PEXP showed a lower PAPR compared to Raised Cosine (RC) since it has a decay rate of t −2 whereas RC has a decay rate of t −3 .Therefore, PEXP has smaller side lobes compared to RC.In (Gandhi, Gupta, & Dalal, 2013), the study demonstrated an improvement in the BER performance while using pulse shaping schemes.It was also observed that filters with lower side lobe levels and larger main lobe width in the frequency domain have made the system less affected by ISI and ICI effects.In (Kaur & Banga, 2013), results showed that the Square Root Raised Cosine (SRRC) filter improved the BER performance of the OFDM system compared to Finite Impulse Response (FIR) Nyquist and RC filters.The implementation of pulse shaping design for OFDM systems in 5G and 4G networks has been worked on by the authors in (Nagapushpa & Chitra, 2017), (Eeckhaute, Bourdoux, De Doncker, & Horlin, 2017), (Zhao et al., 2017).
This paper examines the performance of an OFDM system, in the presence of AWGN and Fading, in accordance with the IEEE 802.11WLAN standard with current pulse shaping filters as well as three novel filters which are suggested in this project.The first unmatched filter, Modified Parametric Exponential Pulse (MPEXP) is designed by changing the impulse response of the PEXP filter.The Better Than Modified Flipped Exponential Pulse (BTMFEXP), which is the second filter, is derived by making slight modifications to the impulse response equation of the MFEXP pulse.The third filter, which is a Hybrid of BTMFEXP and MPEXP (HBTMFPEXP) is formed by the linear combination of the BTMFEXP and MPEXP pulses.BTMFEXP and HBTMFEXP can be implemented at both transmitter and receiver.Low complexity forms of MFEXP and BTMFEXP were also designed.The response of the different filters was compared and examined in time and frequency domains.Simulations based on Matlab TM were carried out.The MPEXP pulse produced a better BER performance compared to the unmatched filters (RC, FEXP, NNLCP, PEXP and Flipped Hyperbolic Secant (FSECH)).Among the matched filters (SRRC, MFEXP, HFPEXP), BTMFEXP and HBTMFPEXP filters resulted in a better BER performance when examined under a noisy channel together with ISI and Fading.
This paper is structured as follows.Section 2 discusses the earlier studies that have been carried out on the pulse shaping filters.Section 3 describes the proposed new pulse shaping schemes.Section 4 explains the OFDM system model employed.Section 5 presents the results.Finally, Section 6 concludes the paper.

Raised cosine (RC) pulse
The RC filter follows two criteria to avoid interference of pulses.Firstly, the pulse should have zero crossing at each sampling instant except at its own.Secondly, the side lobes' amplitude should decay rapidly.The bandwidth occupied, and the decay rate depends on the roll-off factor.The RC pulse decays asymptotically as t −3 (Assalini & Tonello, 2004).Therefore, ISI effects are reduced since the side lobes decay fast enough.

Square root raised cosine (SRRC) pulse
One way to form an SRRC pulse is to take square root of the raised cosine filter in frequency domain.It is implemented at both the transmitter and receiver as a matched filter.The SRRC pulse decays slower compared to RC pulse (Sklar, 2001).

Flipped exponential (FEXP) pulse
FEXP filter has a better eye diagram thus giving a smaller equivalent noise power.As a result, lower BER performance and smaller symbol timing errors are obtained as compared to the RC pulse.The impulse response is as follows: where T s is the sampling time of value 16.67 ns It is noticed in Figure 1 that, the FEXP filter has weaker side lobes compared to the RC pulse.The use of the FEXP filter makes the system more robust against ISI.The pulse has a decay rate of t −2 (Assalini & Tonello, 2004).It was proved that pulses with t −2 decay rate have smaller side lobes than those with t −3 decay rate.The first main side lobe was observed at the 55th sample for an alpha value of 1. Considering alpha = 0.5, the first main side lobe is observed at the 53rd sample and for and alpha value of 0.1, the main side lobe is recorded at the 50th sample.As observed, the response is characterized by the alpha known as the roll-off factor.The roll off factor impacts the bandwidth of the transmitted spectrum and the lobes amplitudes.As the value of alpha increases from 0.1 to 1, the main side lobe weakens thus rendering the system more effective in reducing ISI.

Flipped hyperbolic secant (FSECH) pulse
FSECH is derived by expanding a series of exponentials and has a decay rate of t −3 as RC.The FSECH filter behaves as the RC filter.One small difference is that the FSECH pulse has smaller main side lobe level compared to the RC pulse.FSECH pulse decays at a rate of t −3 same as the RC pulse (Assalini & Tonello, 2004).

Parametric exponential (PEXP) pulse
The PEXP pulse belongs to the parametric group of Nyquist pulses and was chosen to reduce Peak-to-Average Power Ratio (PAPR) in an SC-FDMA signal.The PEXP pulse has a decay rate of t −2 (Azurdia-Meza, 2013).Its impulse is as follows: where where T s is the sampling time of value 16.67 ns The PEXP pulse has a decay rate of t −2 .It has the same time and frequency domain plots as the FEXP pulse.For the alpha values of 0.1, 0.5 and 1, the main side lobe occurs at the samples 50, 53 and 55 respectively.For the increasing roll-off factor, the main side lobe weakens thus making the system robust against ISI.The increasing alpha brings in a rise in cost but since the system gets better, the tradeoff needs to be maintained.The reason is that the receiver is usually not able to sample exactly at the midpoint of each pulse.This is because the lobes only exhibit zero amplitude at the middle of adjacent pulse intervals as in Figure 2, therefore a receiver sample that is not coincident with the midpoint of a pulse interval necessarily samples some lobes from adjacent pulses.

New Nyquist linear combination (NNLCP) pulse
The NNLCP pulse is a combination of RC and PEXP.It was observed that the NNLCP can effectively reduce PAPR OF IFDMA while keeping the same bandwidth compared to RC.Therefore, the NNLCP filter will be more effective in reducing the effects of ISI in a system compared to the RC filter.NNLCP pulse has smaller main side lobe levels compared to the RC pulse.

Hybrid of FEXP and PEXP (HFPEXP) pulse
The HFPEXP pulse is a linear combination of the PEXP and FEXP pulse.The HFPEXP filter was designed by modifying the FEXP pulse's transfer function and combining it with the PEXP pulse.(Fowdur & Jhengree, 2017) where b 2 = ln 4/(2.5aB) and the values ln 4 and 2.5 were experimentally determined.
where τ = nT s /T is the normalized time, m(nT s ) is taken from Equation (3) h fpexp (nT s ) is provided in Equation (2) and T s is the sampling time of value 16.67 ns.
From Figure 3, it is observed that as α increases, the main side lobes weaken more in comparison to the original FEXP and PEXP filter Therefore, the HFPEXP filter will be more effective in reducing the effects of ISI in a system.

Modified flipped exponential (MFEXP) pulse
The MFEXP filter uses a different transfer function and it can be employed at both transmitter and receiver (Fowdur & Jhengree, 2017).Its impulse response is as follows: where b 1 = ln3 aB , T s is the sampling time of value 16.67 ns.The MFEXP pulse shown in Figure 4 has lower side lobe levels as α increases.However, there is a tradeoff between side lobe amplitude and main lobe width for larger values of α.If the main lobe width becomes too small, more excess bandwidth will be needed and the pulse will not so effective since the cost of implementation will rise.

Proposed pulse shaping filters
In this section, the new filters proposed in this work have been described.The pulses are shown in discrete time, i.e. the plot gives the amplitude against sample number (n) instead of amplitude against continuous time (t).The relationship between n and t is t = nT s where T s is the sampling instant and is given as T s = 1/f s where f s is the sampling frequency.Suppose each sample,n, is coded with N b bits then the bit-rate, R b , will be, R b = N b .fs = N b /Ts.The bandwidth in Hertz, W, is then given as W = R b /2(α + 1) (Sklar, 2001).With this relationship it is possible to analyse the effect of the bandwidth while alpha is varied in each of the graphs in Sections 3.1-3.3.

Modified parametric exponential (MPEXP) pulse
This novel filter is a modified form of the PEXP filter.It is designed to further reduce ISI by decreasing the side lobes power.The MPEXP pulse follows a decay rate of t −2 same as the PEXP pulse.From Equation (2), The value of ln2 is changed to (ln2)/3 which was experimentally determined.The impulse response of the MPEXP filter is given below: τ =nT s /T is the normalized time.The value 5.63 was used to normalize the response.A range of values from 0 to 1 was tested and Table 1 below shows five of the values used in the range along with their sidelobe amplitudes and bandwidths.The value of 1 3 ln2 as observed to produce the best tradeoff that is a good bandwidth along with weak side lobes power, thus, improving the system's robustness against ISI.The values that were chosen for the pulse were determined experimentally via several simulations in order to obtain the appropriate pulses.Values outside the range considered lead to distorted pulse shape.
In Table 1 the bandwidth is given in terms of the number of samples.Assuming that the total number of samples is N T and N b is the number of bits per sample, then W represents the total bandwidth for N T *N b bits.From Figure 5(a), it is observed that the MPEXP pulse has even lower side lobe levels as α increases.However, for larger values of α, the main lobe width becomes too narrow, more excess bandwidth is required.Hence, the filter will not be effective enough.From Figure 5(b), the MPEXP shows weaker sidelobe levels compared to MFEXP for α = 0.5.At the 27th sample, the MPEXP shows weaker side lobe power by 13.25% compared to MFEXP.

Better than modified flipped exponential (BTMFEXP) pulse
This proposed filter is a modified form of the MFEXP filter which is employed at both transmitter and receiver.It is designed to further reduce ISI by decreasing the side lobes power.From Equation (7).The value of b1 is changed to the experimentally determined value b 2 to obtain the impulse response of BTMFEXP.The impulse of the BTMFEXP filter is specified as follows: where B = 1/2 T, and b 2 = 2 3 ln3 aB .
A range of values from 0 to 5 was tested to obtain the coefficient 2 3 ln3 which yields the best response with main side lobe amplitude of 0.0226 and bandwidth of 16 samples.In Table 2 below, some of the values used are presented along with their sidelobe amplitudes and bandwidths.
As observed from Figure 6(a), the BTMFEXP pulse has even lower side lobe levels as α increases.However, even in this scheme, the tradeoff between side lobe amplitude and main lobe width for larger values of α is present.If the main lobe width becomes too narrow, more excess bandwidth will be required.From Figure 6(b), for α = 0.5, the main sidelobes' amplitudes of BTMFEXP are lower compared to MFEXP by an amplitude of 0.0169.Consequently, BTMFEXP handles ISI better than MFEXP.

Proposed hybrid of BTMFEXP and MPEXP (HBTMFPEXP) pulse
This novel filter is formed by the linear combination of the BTMFEXP and MPEXP pulse.The filter is employed at both the transmitter and receiver.The equation of BTMFEXP is normalized accordingly using a constant of 2.27, which was experimentally determined, as shown where τ =nT s /T is the normalized time and h mpexp (nT s ) is provided in Equation ( 8) and h btmfexp (nT s ) is provided in Equation ( 9).The value of C is taken as 1.1 to evaluate the impulse response of this filter since it gives weaker side lobes.In Table 3, the responses for the different values of C used are shown.
As observed in Figure 7(a), the side lobe levels amplitude of HBTMFPEXP is very small for larger values of α.For α = 0.1, the first sidelobes have amplitudes of 0.045 and for α = 1, the main sidelobes have almost zero amplitude.This aspect makes the pulse more robust against ISI and helps in reducing the PAPR.From Figure 7(b), at the first side lobes MPEXP and BTMFEXP showed higher amplitudes of 0.01625 and 0.006698 respectively over HBTMFPEXP which has flat main side lobe.

Low complexity filters (MFEXP and BTMFEXP)
In this sub-section, an approach to reduce the complexity of the high complexity pulse shaping filters (MFEXP, BTMFEXP) is proposed.The approach is based on Taylor series  3 that is 43 RMs and 4 RAs.The same method is used to calculate the number of computations for the term cos(2βπαt).β 2 yields 7 RMs.The term 4π 2 t 2 + β 2 have 13 RMs

I. Low complexity MFEXP
From Equation ( 7): The sine term and cosine term were expanded using Taylor series expansion.
where x = F 3 = 2pb 1 nT s It is observed that expansion up to the first term itself is sufficient to produce the same impulse response as the high complexity MFEXP.The impulse response of the low complexity MFEXP pulse is as follows: where b 1 = ln 3/aB and T s is the sampling time of value 16.67 ns.

II. Low Complexity BTMFEXP
From Equation (9), The complexity of the BTMFEXP filter is reduced using the same principle as MFEXP.The impulse response of the low complexity BTMFEXP pulse is where b 2 = 2 3 ln3 aB and T s is the sampling time of value 16.67 ns.The impulse plots are given in Figures 8(a,b).
From Figure 8(a,b), it observed that the low complexity forms of MFEXP and BTMFEXP produce the same impulse responses as their respective high complexity forms.However, the low complexity forms lead to less computation time compared to the high complexity forms.

Complexity analysis of MFEXP and BTMFEXP
Equations ( 7) and ( 9) are used to determine the number of computations for the High Complexity, MFEXP and BTMFEXP.Since BTMFEXP is derived from MFEXP by slightly modifying its equation, the two are computed in the same way as shown below and the only difference is that β changes.The number of computations for the different parts of the Equations ( 7), ( 9), ( 14) and ( 15) are shown in Table 5.
The total number of computations was calculated per sample first then it was calculated for the 41 samples.The choice of this parameter (n = 41) only affects the simulation time and the resolution of the graphs.It can be chosen in a range 30-100.However, it does not affect the performance of the schemes. High For n = 41,    For computing low complexity, MFEXP and BTMFEXP, Equations ( 14) and ( 15) are used along with Tables 4 and 5.Even for the less complex filters the number of computations is the same.
Total number of RMs and RAs for low complexity MFEXP and low complexity BTMFEXP per sample, TR per sample is given as Total number of RMs and RAs for low complexity MFEXP and low complexity BTMFEXP for the 41 samples, TR is given as

Complexity evaluation
The chart in Figure 9 is plotted to illustrate clearly the reduction in complexity of the high complexity MFEXP and the high complexity BTMFEXP.
From the analysis of the complexity, the low complexity MFEXP and low complexity BTMFEXP are much less complex compared to high complexity MFEXP and high complexity BTMFEXP.The latter performs 634844 RMs and 369 RAs more compared to the low complexity forms of MFEXP and BTMFEXP.Thus, the low complexity forms of MFEXP and BTMFEXP take less computation time and above all, the complexity reduction leads to reduced implementation cost.

System model
The complete system model is shown in Figure 10.On the transmitter side, random bits are generated and are modulated by BSPK Modulation (BSPK Mod).Data subcarriers are then assigned to modulated bits as per IEEE 802.11 WLAN standard (IEEE, 1999).The signal undergoes IFFT operation.After the addition of a cyclic prefix (CP), the data is converted into pulses by using Baseband signalling.Pulse shaping is then implemented.Each pulse-shaped signal is passed through a Butterworth filter which acts as a band limited channel.It has a defined length of 7 and a variable cut-off frequency is used to produce different levels of ISI.Then AWGN noise is added and flat Rayleigh fading is added separately to the channel.For the fading, the channel has a greater bandwidth with constant gain and linear phase response than the transmitted stream.
where N(t) represents the random values, N. At the receiver in Figure 10, the received signal again undergoes pulse shaping.The signal is then sent through the Receiver Correlator.The FFT operation is executed followed by the extraction of the data subcarriers.Finally, the signal received on extracting the data subcarriers, passes through the demodulation process and the output bits are received.

Results
The system in Figure 10 was implemented and simulated on Matlab TM .For AWGN, the range of values chosen for E b /N o was from 0 to 15 dB and for Fading, the range of values chosen was from 16 to 42 dB.The OFDM parameters as per IEEE 802.11WLAN were used (IEEE, 1999).The oversampling factor was 10 and all the pulse shaping filters' length was kept constant at 41 taps.An optimum value of α = 0.6 was used and the code rate was taken as 1 throughout the simulations.A modulation order of 2 was used and the sampling time, T s is 16.67 ns.A bandwidth of 20 MHz was used for a lower ISI level, one of 12 MHz was maintained for a higher ISI level and for an almost zero ISI level a bandwidth of 48 MHz was used.
The existing pulses that have been compared are: 1  11(a), the RC pulse has the strongest side lobe levels.FSECH has weaker sidelobes compared to RC but compared to NNLCP, the latter has lower sidelobe amplitudes.FEXP and PEXP have the same impulse responses with even weaker sidelobes compared to NNLCP.Among all, the proposed MPEXP pulse handles ISI more efficiently since it has the weakest side lobes.From Figure 11    high complexity BTMFEXP surpass SRRC and MFEXP at a gain of 2 and 0.94 dB respectively in E b /N o at a BER of 10 −4 .The proposed HBTMFPEXP has the lowest BER performance and outperforms BTMFEXP, MFEXP and SRRC at a gain of 0.55, 1.65 and 2.5 dB respectively in E b /N o at a BER of 10 −4 .

Conclusion
The aim of this paper was to investigate the performance of an OFDM system as per IEEE 802.11 using BPSK Modulation under AWGN channel and Fading channel with existing and new pulse shaping filters.The proposed MPEXP pulse showed a better BER performance among the unmatched filters.At a high ISI level, the proposed HBTMFPEXP pulse surpasses MFEXP and HFPEXP by a gain of 0.57 dB in E b /N o at a BER of 10 −4 in AWGN channel.Under the Fading channel, the proposed HBTMFPEXP pulse outperforms both MFEXP and HFPEXP at a gain of 5.33 dB in E b /N o at a BER close to 10 −4 .The newly proposed filters demonstrate almost flat side lobe levels thus they handle ISI better compared to existing filters.The low complexity filters moreover lead to less computation time and reduction in the implementation cost.

Disclosure statement
No potential conflict of interest was reported by the authors.
Since b 2 gives the same number of computations as b 1 , the total number of RMs and RAs for High Complexity BTMFEXP is the same as for High Complexity MFEXP.Therefore, Total number of RMs and RAs = 715204RMs + 615RAs
Figure 11.(a) Impulse responses of unmatched filters.(b) Impulse responses of matched filters.

Figure 12 .
Figure 12.(a) Unmatched filters at ISI level of bandwidth 12 MHz.(b) Matched filters at ISI level of bandwidth 12 MHz.

Figure 14 .
Figure 14.(a) Unmatched filters at ISI level of bandwidth 12 MHz.(b) Matched filters at ISI level of bandwidth 12 MHz.

Figure 15 .
Figure 15.(a) Unmatched filters at ISI level of bandwidth 20 MHz.(b) Matched filters at ISI level of bandwidth 20 MHz.
Dr. Tulsi Pawan Fowdur received his BEng (Hons) degree in Electronic and Communication Engineering with first class honours from the University of Mauritius in 2004.He was also the recipient of a Gold medal for having produced the best degree project at the Faculty of Engineering in 2004.In 2005 he obtained a full-time PhD scholarship from the Tertiary Education Commission of Mauritius and was awarded his PhD degree in Electrical and Electronic Engineering in 2010 by the University of Mauritius.He joined the University of Mauritius as an academic in June 2009 and is presently an Associate Professor at the Department of Electrical and Electronic Engineering of the University of Mauritius.His research interests include Communications Theory, Multimedia Communications, Mobile and Wireless Communications, Networking, Security and Internet of Things.He has published over 60 papers in these areas and is actively involved in research supervision, reviewing of papers and also organizing on international conferences.Louvi Doorganah graduated with a B.Eng (Hons) in Electronics and Communications Engineering from the University of Mauritius in 2018.He was then selected by Huawei Technologies Ltd to attend the Seeds for the Future program that was held in the Huawei Headquarters in China.He is presently working as an Assistant Telecommunication Consultant at Huawei Technologies Ltd.His research interests include Communication Theory, Modulation and Pulse Shaping Filters.

Table 1 .
Coefficients used to determine best tradeoff for MPEXP.

Table 2 .
Coefficients used to determine best tradeoff for BTMFEXP.

Table 4 .
Number of RMs and RAs in each equation part.

Table 5 .
The number of computations for the different combined equation parts.