BER Enhancement of OFDM-Based Systems Using

BER Enhancement of OFDM-Based Systems Using the Improved Parametric Linear Combination Pulse Cesar A. Azurdia-Meza

S. Kamal, Kyesan Lee

Department of Electrical Engineering Universidad de Chile Santiago, Chile {cazurdia}@ing.uchile.cl

Department of Electronics and Radio Engineering Kyung Hee University Yongin-Si, Republic of Korea {shka, kyesan}@khu.ac.kr

Abstract—In this work, the performance of the sub-optimum improved parametric linear combination pulse (IPLCP) in orthogonal frequency division multiplexing (OFDM) based systems is evaluated. An OFDM-based system, in the presence of carrier phase noise and carrier frequency offset, is evaluated in terms of BER. The IPLCP is characterized by having three additional design parameters, adding extra degrees of freedom to reduce the BER in OFDM-based systems. Two of the constants were fixed, whereas the other one was optimized via numerical simulations for BER reduction. The sub-optimum IPLCP outperforms the other evaluated pulses in terms of BER. Keywords—Bit error rate (BER), improved parametric linear combination pulse (IPLCP), Nyquist firt criterion, orthogonal frequency division multiplexing (OFDM).

I.

I NTRODUCTION

Orthogonal frequency division multiplexing (OFDM) is a bandwidth-efficient communication technique that has been widely used in several wireless communication standards; such as Long Term Evolution (LTE) Advanced, Wireless Fidelity (WiFi), Wireless Personal Area Network (WPAN), digital terrestrial TV broadcasting (DVB-T), Worldwide Interoperability for Microwave Access (WiMAX), broadband powerline communication (BPLC), amont others [1]–[5]. Further, OFDM-based systems are being studied and proposed as one of the key technologies to be implemented at the physical layer of 5G systems combined with multiple-input multiple-output (MIMO) techniques [6]–[9]. The extensive use of OFDMbased systems is due to various advantages; such as the use of guard intervals to deal with the effect of delay spread, high data rate transmission capability, and robustness to multi-path fading. Despite of all of the benefits offered by OFDM-based systems, there are certain technical issues that need to be addressed. OFDM-based systems are very sensitive to frequency offset errors caused by frequency differences between the local oscillators in the transmitter and the receiver, Doppler spread, and distortions within the channel, among others, causing intercarrier interference (ICI) [3], [5], [10], [11]. Further, OFDMbased systems are characterized by having high PAPR values [3], [5], [12]. High PAPR values and large frequency offsets (ICI) increase the BER in OFDM-based systems [13]. The use of Nyquist-I pulses to reduce the BER due to the effects of carrier phase noise and carrier frequency offset in OFDM-based systems has been studied, proposed, and implemented by several scholars [10], [11], [14]. Pulses such

as the raised cosine (RC) pulse, ‘better than’ raised cosine (BTRC) pulse [15], sinc power (SP) pulse [16], improved sinc power (ISP) pulse [17], new windowing function (NW) [18], and the phase modified sinc pulse (PM) [19] have been proposed, among others. The ISP pulse is characterized by having two design parameters, whereas the PM and NW have four. At the moment, the NW and PM are arguably the pulses that best diminish the effects of frequency offset in OFDMbased systems. The main goal of this manuscript is to evaluate the performance of the improved parametric linear combination pulse (IPLCP) in OFDM-based systems in terms of the BER as a function of the carrier frequency offset and carrier phase noise. The IPLCP pulse was originally derived in [20] to reduce ICI power in OFDM-based systems, and later evaluated in [13] for various key OFDM parameters. II.

G ENERAL OFDM S YSTEM M ODEL

An OFDM symbol is formed by the sum of N different digital data symbols (M-PSK, M-QAM), each transmitted on a different orthogonal subcarrier. The complex envelope of the transmitted OFDM symbol with pulse shaping is expressed as [13], [19], [20] { } N −1 ∑ j2πfc t j2πfk t s(t) = Re e p(t)dk e , (1) k=0

where dk represents the k-th data symbol (which should not be confused with the OFDM symbol), N is the number of subcarriers implemented in the system, fk is the k-th subcarrier frequency, fc is the central carrier frequency, and p(t) is the pulse-shaping function that limits or narrows each data symbol in a certain interval of time. It assumed that the input data symbols are uncorrelated, where {dk } is a zero-mean independent sequence of unit variance and normalized symbol energy. Frequency offset, ∆f , and carrier phase errors, θ, are introduced in OFDM-based systems because of channel distortion, Doppler spread, and desynchronization between the oscillator of the transmitter and receiver. Therefore, the received signal is given by [15] r(t) = ej(2π∆f t+θ)

N −1 ∑ k=0

dk p(t)ej2πfk t

(2)

Hence, the BER of the system increases proportionally to the phase error and frequency offset.

Because each data symbol of an OFDM-based system is transmitted on a different subcarrier, the analysis of the signals cannot be made in the time domain. This is because the envelope given by (1) is the result of the sum of N orthogonal signals. Therefore, the analysis of the signals of an OFDM-based system is usually done in the frequencydomain. To achieve zero ICI, the pulse-shaping function P (f ) has to behave as a Nyquist-I pulse. Nyquist first criterion in the frequency-domain is defined as follows [13], [15] { 1, if f = 0 (3) P (f ) = 0, if f = ±1/T, ±2/T, . . ., which indicates that that the pulse P (f ) should have spectral null points at the frequencies ±1/T, ±2/T, . . . to ensure subcarrier orthogonality. The explicit frequency-domain expression of the IPLCP is given as follows [20] PIP LCP [

2 2 (f ) = e(−επ (f T ) ) ×

sin(πf T ) πf T

2

(παf T /2) × [ 4(1−µ)sin + π 2 α2 (f T )2

παµf T sin(παf T ) ] π 2 α2 (f T )2

]γ ,

(4)

where µ, γ and ε are constants defined for all real numbers, and the roll-off factor α is defined for 0 ≤ α ≤ 1. The constants µ, γ and ε add extra degrees of freedom to reduce the BER in OFDM-based systems. It was proven in [13] and [20] that the IPLCP fulfills the criterion defined in (3). In [13] it was found that the constant γ is the one that has the largest impact on reducing the BER in OFDM-based systems. Therefore, the value of the constants µ and ε will be the ones used in [13] (µ = 1.6 and ε = 0.1), whereas the sub-optimum γ will be determined via numerical simulations. In general there is an optimum µ, ε, and γ for every rolloff factor and transmission scheme, although they might not be unique. To reduce the BER of OFDM-based systems, in the presence of carrier phase noise and carrier frequency offset, the sidelobes of the pulse shaping filter P (f ) should be diminished. Therefore, a Nyquist pulse that decays rapidly is required to diminish the effects to carrier frequency offset. The frequency response of the IPLCP is plotted in Fig. (1) with α = 0.22, µ = 1.6, ε = 0.1, and different values of γ. For comparison purposes, the frequency response of the RC pulse is also plotted. It can bee seen in Fig. (1) that the sidelobes of the IPLCP are diminished as the value of ε increases. IV.

RC IPLCP (γ=5) IPLCP (γ=4) IPLCP (γ=3) IPLCP (γ=2) IPLCP (γ=1)

0.8

I MPROVED PARAMETRIC L INEAR C OMBINATION P ULSES

P ERFORMANCE E VALUATION

Yielding small ICI power and diminishing PAPR are not the only evaluation metrics to be considered in the design of an optimal pulse shaping filter in OFDM-based systems. The most important evaluation metric in any digital communication system is the BER, as it plays an important role in assessing the performance and effectiveness of an OFDM-based communication system. To evaluate the BER of the IPLCP and the other pulses, we used the theoretical expression derived in [10]. This theoretical expression is used to compare the BER of different pulse shaping functions assuming an additive white

0.6 P(f)

III.

1

0.4 0.2 0 −0.2 0

1

2

3

fT Fig. 1. Frequency response of the RC and IPLCP pulse for µ = 1.6, ε = 0.1, α = 0.22, and different values of γ.

Gaussian noise (AWGN) channel, and BPSK modulation. The average BER is given as a function of the carrier phase noise θ, average ICI power (PICI ), carrier frequency offset ∆f , and the pulse shaping function P (f ). The average BER is given as follows [10] BEROF DM = 1 − (1 − OF DMsymbol )N ,

(5)

where N represents the number of subcarriers or symbols per modulated block. Further, the BER of an BPSK-OFDM symbol is given as follows [10] ( { [ ]√ } √ BERsymbol = 12 Q cos θ P (−∆f ) + PICI 2γb { [ ] √ }) √ (6) +Q cos θ P (−∆f ) − PICI 2γb , where γb = Eb /N0 . If the spectral sidelobes of the pulse shaping functions are very small in comparison to its main lobe, as it is the case with the IPLCP and the other pulses evaluated, then (6) can be approximated as [10] [ √ ] BERsymbol ≈ Q cos (θ) P (−∆f ) 2γb . (7) Therefore, in this manuscript the BER of the proposed system is evaluated using the expression given in (7). The BER of the proposed system is evaluated using a frequency offset ∆f equal to 0.3, a carrier phase noise of θ = 10o and θ = 30o , and α = 0.22. For comparison purposes, the design parameters of the SP [18], ISP [19], NW [18], and PW [18] pulses used in this manuscript are the ones that provided the best performance in OFDM-based systems. The sub-optimum IPLCP pulse was obtained via extensive numerical simulations, and it was determined for γ = 0.03. In Fig. 2 is plotted the BER of the BPSK-OFDM system with θ = 10o , whereas in Fig. 3 the BER is plotted with θ = 30o . In general, a lower BER was achieved for a smaller carrier phase noise angle, in accordance with the expression given in (7). Overall, the sub-optimum IPLCP outperforms the other pulses evaluated in the system. This is because the frequency response of the sub-optimum IPLCP possesses smaller sidelobes, and its central lobe does not behave as a delta Dirac function as the value of γ is reduced, as depicted in Fig. 1. This is

extremely important because a pulse with a narrow central lobe and no sidelobes increases the BER of the system because the synchronization between the oscillator of the transmitter and receiver need to be almost perfect to recover the transmitted signal [10], [13] 0

10

[2]

[3] [4]

−1

Bit Error Rate (BER)

R EFERENCES [1]

10

[5] −2

10

[6] −3

10

−4

10

−5

10

SP (a=0.5) ISP (a=0.5, n=2) IPLCP (µ=1.6, ε=0.1, γ=0.03) NW (a=2, n=2, β=1, γ=1) PM (a=0.5, b=0.5, c=2, n=2)

[7]

[8]

0

2

4

6

8

10

Eb/N0 (dB)

[9]

Fig. 2. BER evaluation of different pulse shaping functions using a BPSKOFDM system with ∆f = 0.3, θ = 10o , and α = 0.22. [10] 0

10

[11]

Bit Error Rate (BER)

−1

10

[12] −2

10

[13] −3

10

−4

10

−5

10

0

SP (a=0.5) ISP (a=0.5, n=2) IPLCP (µ=1.6, ε=0.1, γ=0.03) NW (a=2, n=2, β=1, γ=1) PM (a=0.5, b=0.5, c=2, n=2) 2

4

6

8

[14]

[15]

10

12

Eb/N0 (dB) Fig. 3. BER evaluation of different pulse shaping functions using a BPSKOFDM system with ∆f = 0.3, θ = 30o , and α = 0.22.

V.

C ONCLUSION

In this work the performance of the sub-optimum IPLCP in OFDM-based systems, in the presence of carrier phase noise and carrier frequency offset, was evaluated in terms of BER. The proposed sub-optimum IPLCP pulse achieved smaller BER values compared to those of the other evaluated pulses. The performance of the IPLCP pulse reported in this paper could potentially be improved by using optimization techniques.

[16]

[17]

[18]

[19]

[20]

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