JOURNAL OF TELECOMMUNICATIONS, VOLUME 21, ISSUE 1, JULY 2013

1

Eigenvalue Based Selection of Prolate Spheroidal Wave Functions for Pulse Shape

Modulation D. Adhikari and C. Bhattacharya

Abstract— Proper selection of ultra wideband (UWB) waveforms has become an interesting challenge such that the spectral

limits of the frequency band approved by Federal Communications Commission (FCC) can be efficiently utilized. Pulse shapes

based on prolate spheroidal wave function (PSWF) are a suitable choice for UWB communication systems because of their

property of dual orthogonality. In this paper, we propose a set of PSWF pulses by appropriate selection of the order of pulse (n)

and the time-bandwidth product (c) based on the eigenvalues of both integral and differential forms of PSWF. The selected set

of pulses are analysed in terms of power spectral density (PSD), and autocorrelation function (ACF) for their applicability as

basis functions in an efficient M-ary pulse shape modulation (PSM) scheme.

Index Terms—Prolate spheroidal wave function, pulse shape modulation, time-bandwidth product, order of pulse, power

spectral density, autocorrelation function.

—————————— ——————————

1 INTRODUCTION

LTRA wideband (UWB) signal waveforms are ex-tremely short duration pulses designed for transmis-

sion without any carrier modulation and are immune to multipath fading. Baseband transmission of such subna-noseconds pulse-shapes results in a simple transceiver architecture with a high data rate of transmission. How-ever the wide spectral bandwidth is likely to interfere with existing systems such as WLAN (2.45 and 5.0 GHz), fixed wireless (5.9 - 8.6 GHz), and GPS systems (1.177 - 1.58 GHz). The Federal Communications Commission (FCC) [1] recommends transmission within a spectral mask that restricts UWB communication systems within a (3.1 fl -10.6 fh) GHz bandwidth and emitted power spectral density (PSD) to -41.3 dBm/MHz. Compliance to the FCC recommended spectral mask would warrant a proper pulse shape design that should optimally utilize the al-lowable limits of power spectrum.

A number of design approaches such as derivatives of Gaussian pulses and modified Hermite pulses (MHP) are proposed as UWB waveforms [2]. Gaussian pulses are not found to meet the regulations of FCC spectral mask in satisfactory manner as they are required to be modified and filtered [3]. Although carrier modulation of MHP pulses improves efficiency, the receiver design becomes complex than for otherwise carrierless systems [4].

Prolate spheroidal wave functions (PSWF) are an at-tractive option for UWB pulse waveforms due to their double orthogonality in time-frequency representations. They are special functions offering a solution to the ener-

gy concentration problem in a finite duration pulse. M-ary pulse shape modulation (PSM) is introduced

towards improving the performance of multiple access, time-hopping UWB communications schemes as in [5]. An M-ary PSM scheme based on the orthogonal proper-ties of Hermite pulses is discussed in [6] whereas in [7] such a scheme utilizing PSWF is shown to achieve high data rate in a severe multipath fading environment. A 4-ary PSM scheme based on PSWF acts as a physical layer for broadband satellite communications in W-band show-ing distinct advantages over raised cosine filtered quadra-ture amplitude modulation (QAM) [8]. However, the au-thors in [8] have utilized PSWF pulses of first and second order only.

In Section 2 of this paper, we analyse the performance criterion for an M-ary PSM scheme in a dense multiple access scenario in terms of signal to noise plus interfe-rence ratio (SNIR) and bit error probability (BEP). In Sec-tion 3 we select a set of PSWF waveshapes based on the eigenvalues of the differential and integral form of ex-pression of PSWF in the normalized time interval [-1, 1] of Tp. We analyse the above selected PSWF pulses as set of basis functions for an M-ary PSM scheme with respect to autocorrelation function (ACF) and power spectral densi-ty (PSD). In the process we develop a closed form expres-sion for ACF of PSWF.

2 M-ARY PSM SCHEME WITH ),( tcn AS BASIS

FUNCTIONS

The underlying idea in a PSM scheme is to represent the M-ary symbols (M = 2N) with a set of N orthogonal pulses as basis functions [9]. The advantage of M-ary PSM scheme is the requirement of less timing precision, better immunity to multipath and independence of the received signal polarity during detection [10]. We analyse the PSM

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D Adhikari is with the Defence Institute of Advanced Technology, Pune – 25, India.

C Bhattacharya is with the Electronics Engg Department, Defence Insti-tute of Advanced Technology, Pune – 25, India

U

2

scheme for a dense multiple access scenario, which results in a multiple access interference (MAI).

Let ),( tcn , 0 ≤ n ≤ N-1 be a set of pulses such that

(1) form a set of orthonormal basis functions. Let the set of symbols transmitted be si, 0 ≤ i ≤ M-1. An N-tuple representation of M-ary symbol set is

(2)

The transmitted signal for the ith symbol of the jth user is shown in Fig.1 and is

(3)

where ai,n is the nth bit of the ith symbol, Tf is the frame time. The transmitted energy is Et,

)( jic is the pseudoran-

dom (PN) code to avoid catastrophic collision due to probable simultaneous arrival of received pulses, and Tc is the slot time. The total number of slots per frame is Nc where Tf = Nc Tc.

The composite received signal at the input of the kth receiver for the ith symbol with additive channel interfe-rence 𝝎(t) is shown in Fig.2 and is

(4)

The SNIR for the ith symbol of the desired lth (j≠l) user in a multiple access PSM scenario is derived as

(5)

Here 2MAI is the variance due to MAI and )(, ll is the

autocorrelation function (ACF) between the template and received signal of the desired (lth)j≠l user.

(6) Depending upon the value of )( j

ic the template would occupy one of the slots out of [1, 2, ..., Nc] for each frame of the lth user. Based upon the magnitude of the random delay , an overlap of pulses for other users (j ≠ l) with the template of the lth user would exist only for those slots, resulting into MAI. The MAI for users’ j≠l is given by

(7)

The variance of MAIZ gives variance due to MAI ( 2MAI ),

that would result in the evaluation of SNIR from (5). The symbol error probability is given as Pe = Q(SNIR) where Q(.) is the Marcum Q-function.

Each of Nu users in a PSM scheme utilize different sets of orthogonal basis functions 𝜓n(t) so that )(, jl ≈ 0 in (5). Also it is found that large values of Nc in (5) results in smaller 2

MAI [11]. The average BER for a wireless com-munications system being inversely proportional to the SNIR, a large value of )(, ll alongwith a smaller value of

2MAI would yield a smaller average BER. Consequently,

correct estimation of the ith symbol is governed by the ACF )(, ll as well as the PSD of the selected pulse-shape.

In the next section we select a set of PSWF waveshapes as basis functions for an M-ary PSM scheme. We analyse the suitability of such waveshapes in terms of ACF and PSD.

3 CHOICE OF PSWF PULSE-SHAPES BASED ON

PARAMETERS C AND n

PSWF of the form ),( tcn are real, continuous functions of time t for c ≥ 0, where c = ΩTp/2 is the time-bandwidth product (TBW) constant, Ω is the bandwidth, and n represents the order of the pulse. The conventional ap-proach is to approximate ),( tcn as numerical solutions for the eigenfunction of a multiband spectral mask [12],[13][14]. However, such numerical solutions do not

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lead to analytical expressions for PSD and ACF of PSWF. To evaluate the efficiency of the PSWF pulses in the FCC designated spectral masks it becomes necessary to devel-op analytical tools so as to properly choose the parame-ters c and n. Also, comparison of performance in terms of spectral efficiency with those of MHP is in order.

The PSWF may be defined [15] as the eigenfunctions of the integral equation

(8)

or, the eigenfunctions of the differential equation (9) The eigenvalues )(cn of (1) are a set of real positive-numbers 1 > )(0 c > )(1 c > …> )(cn those determine the concentration of energy of the pulse within the nor-malized time interval Tp [-1, 1]. On the other hand, the values of n those admit solutions of (2) are the discrete positive real eigenvalues for real c such that 0 < 0 < 1 < …< n .

The normalized expression of pulse shapes for PSWF is given by [13]

(10)

The functions ),( tcn are derived from kth degree ortho gonal Legendre polynomials Pk(t), k = 0,1,2,... The series expansion coefficients associated with Pk(t) are n

kd (c).

3.1 Selection of parameters c, n

The time domain behaviour of ),( tcn generated from (10) are shown in Fig. 3 and Fig. 4 for constant c and n respectively. As shown in Fig. 3, ),( tcn are characterized by exactly n zero-crossings in the interval Tp. The wave-shapes in the figure show that as n increases the time in-dices of the zero crossings shift towards origin. It is seen from Fig.4 that as c increases the waveshapes turn asymptotic beyond the bounds of Tp making them unrea-lizable. In such cases ),( tcn in (10) cannot be obtained as functions of orthogonal Legendre polynomials. We there-fore restrict variation of c from 0.5 to 12 for the pulse shapes of ),( tcn .

The values of )(cn in (8) should be close to unity within the time interval Tp as they represent attenuation in power spectrum within the allowable spectral limits for different n [15]. The nature of variation of )(cn and

n [16][17] with successively increasing values of c, n are shown in Table I , Fig. 5 and 6 and are enumerated below:

)(cn decreases with increase in n for a particular value of c, whereas n increases with increasing n.

The values of )(cn fall off rapidly with increasing n once n > ( 2c/𝜋) as shown in Fig. 5.

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TABLE 1 VARIATION OF EIGENVALUES WITH n AND c

On the other hand, Fig. 6 shows that for lower orders of the pulse the energy concentration )(cn remains close to unity for a wider choice of c. A set of ),( tcn pulses based on proper selection of c,

n parameters would yield higher energy content in the normalized time interval Tp, and sharper ACF with better decorrelation property between PSWF pulses of different orders. As seen from Table I and in Fig. 5-6, we obtain the maximum energy concentration (90% for )(cn ) of the pulse within the bounds of Tp upto n = 6 for c = 12 where both forms of eigenvalues are desirably large. We can therefore select a set of pulses for an M-ary PSM scheme that that would provide values of )(cn ≈ 1 and sharper ACF. We now analyse the PSD and ACF for such a set of selected pulses. 3.2 Evaluation of PSD and ACF

The power spectral density (PSD) is given by the square of the Fourier transform of (10) as

(11)

and the autocorrelation function (ACF) for timing asyn-chronisation as (12) Here )(kR is the ACF of of kth order Legendre polyno-mials, and )(klR is the cross correlation for k ≠ l given as

(13) The Legendre polynomials are expressed by Rodrigue's formula as

(14)

Fourier transform of the above expression of Legendre polynomial is (15) where (16)

is the Bessel function of fractional order (k+1/2) in the Fourier domain. From (15) and (16) (17) The contribution of higher order ACF for k > 2 in (12) is negligible, and the cross-correlation terms are minimal.

)(kR for k = 0, 2 (even order PSWF ) is evaluated as

(18)

(19) where (*) stands for convolution of the time function. The time functions fn (t) may be recursively derived as, (20)

c = 2 c = 6

n )(cn n )(cn n

0 0.88056 1.127734 0.999901 5.208269

1 0.35564 4.287129 0.996062 16.000443

2 0.035868 8.225713 0.940173 25.356479

3 1.1522 E-3 14.100204 0.646792 33.204199

4 1.8882 E-5 22.054830 0.207349 40.720194

5 1.9359 E-7 32.035263 2.73817 E-2 49.773712

6 1.3661 E-9 44.024748 1.95501 E-3 61.180757

7 7.0489 E-12 58.018371 9.48488 E-5 74.852867

8 2.7768 E-14 74.014194 3.43678 E-6 90.651161

c = 8 c = 12

n )(cn n )(cn n

0 0.99999787 7.221579 1.00000000 11.232421

1 0.99987898 22.092154 0.99999992 34.157136

2 0.99700462 35.706417 0.9999967 55.953547

3 0.96054568 47.757099 0.99991663 76.505853

4 0.74790284 58.016771 0.99858732 95.639741

5 0.32027663 67.364750 0.98366430 113.05467

6 6.07844 E-2 77.825226 0.88175663 128.36084

7 6.12629 E-3 90.691432 0.55736081 141.74502

8 4.18252 E-4 106.01179 0.18342927 155.07470

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and for n ≥ 2 (21) Here H (.) is the Heaviside function for -2 < t < 2. From the values of )(kR and )(klR we evaluate ACF ( )(n ) of PSWF, the plot of which is shown in Fig. 10. 3.3 Analysis of PSD and ACF

The scaled PSD of PSWF pulses shown in Fig. 7 and Fig. 8 are compared for sucessively increasing values of c and n. For a particular order n the PSD increases in Fig. 7 with increasing values of c signifying higher concentra-tion of power in the limited bandwidth. The PSD goes down in Fig. 8 with increasing values of n for a constant c as is expected from (11). The PSD demonstrate distinct bandpass characteristics of PSWF in both the figures. This multiband property of PSWF is certainly useful in choos-ing the band of operation in presence of existing licensed communication such as WLAN and GPS. The lower PSD for higher orders of pulses in Fig. 8 ensures applications those require low probability of detection.

The in-band energy concentration of PSWF is com-pared with the PSD of modified Hermite pulses (MHP) in

Fig.9. The PSD of PSWF is found to have better spectral concentration in the limited bandwidth recommended by FCC than that of MHP [18]. It is observed in Fig. 9 that the PSD for MHP spreads over to the adjacent bands for n ≥ 5. The option of choosing the parameter c in case of PSWF provides an additional degree of freedom for selection of orthogonal pulses for an M-ary PSM scheme as compared to MHP.

Results of normalized ACF of PSWF ( )(n ) for even values of n = 0 to 6 are shown in Fig. 10. These results are obtained for c = 12 utilizing the expression for ACF in (12). The normalized ACF plots shown in Fig. 10 become sharper as n increases. These results show that for higher values of n the decorrelation property among ),( tcn is more pronounced. Therefore, M-ary PSM schemes would yield better detection in multiple access scenarios for val-ues of n more than 2.

4 CONCLUSION

In this paper, we have analysed the influence of both forms of eigenvalues of PSWF on the parameters c and n to obtain an optimal set of orthogonal pulses those can be efficiently utilized in an M-ary PSM scheme. For a certain choice of c and n we have shown the analytical behaviour of ACF and PSD of ),( tcn .We have demonstrated that a trade-off between the values of c and n may lead to effi-cient utilization of the allowable spectral bounds go-verned by FCC. Finally we have justified the suitability of the selected PSWF pulseshapes in terms of ACF and PSD for an M-ary PSM scheme yielding better receiver per-formance.

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)(

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D. Adhikari is a BTech from the Institute of Radio Physics and Elec-tronics from the University of Calcutta (1990) and ME from University of Pune (2000) and is presently pursuing his PhD from the Defence Institute of Advanced Technology (DIAT), Pune, India. He has been working as a Faculty in DIAT since last five years. His current re-search interest is in the area of Ultrawideband communications sys-tems. C. Bhattacharya is a PhD (Engg) from Jadavpur University, Kolkata, India. He is presently the Head of Department, Electronics Engg, DIAT. He is a Senior Member, IEEE.