Columbia International Publishing Journal of Advaned Eletrial and Computer Engineering Researh Artile Effet of Pulse Shaping on Autoorrelation Funtion of Barker and Frank Phase Codes Praveen Ranganath 1* and Sathyanarayan Rao 1 Reeived 12 May 2014; Published online 28 June 2014 The author(s) 2014. Published with open aess at www.usip.us Abstrat Phase oded pulse-ompression is a ommonly used tehnique in a variety of appliations; for example to improve resolution in range-finding appliations. In this paper, we onsider two phase odes- binary Barker ode of length 13 and polyphase Frank ode of length 9. The effets of using four different pulse shapes (Retangular, Triangular, Gaussian, and Exponential) on these two phase odes are assessed using the autoorrelation funtion. The performane parameters (Peak Sidelobe Level (PSL), Integrated Sidelobe Level (ISL), and 3dB mainlobe widths) derived from the autoorrelation funtion are ompared and disussed. Possible appliations are disussed based on the exellent properties demonstrated by exponential pulse shape for these two phase odes. Keywords: Phase odes, Pulse shape, Autoorrelation, 3dB width, Peak and Integrated sidelobe levels 1. Introdution Pulse ompression is a phase-oded modulation tehnique employed to improve the range resolution in radar signals. In this tehnique, eah hip within a pulse is modulated in aordane to the sequene of phases defined by the ode. The odes are either real binary or omplex poly phaseodes. One suh set of ommonly used binary odes in phase modulation is the well known lassial Barker odes (Barker, 1953). Barker odes have been under study sine 1950s, due to their low autoorrelation binary sequenes. Similar to binary odes, polyphase odes are also used. One suh popular polyphase ode is the Frank ode (Frank, 1963). Pulse shaping is an important aspet to be onsidered while designing wireless systems suh as radar and ommuniation systems. Pulse shaping enhanes the performane of a ommuniation system by reduing intersymbol interferene and dispersion (Shafhuber et al., 2002). The *Corresponding e-mail: pr0007@alumni.uah.edu 1 Member, Alumni Assoiation, The University of Alabama in Huntsville, USA 22
harateristis of a modulated pulse, or a waveform in general, an be desribed by its autoorrelation funtion. Some of the ommon derived quantities of autoorrelation funtion used in desribing the performane of a waveform are the peak sidelobe level (PSL), integrated sidelobe level (ISL), energy, and 3dB resolution widths. For example, in the ase of range-finding systems, the PSL is related to rejeting unwanted returning signals from point soures around the region of interest where the range-finding is performed. The ISL is assoiated with the suppressing the distributed returns near the area of interest (Nunn & Coxson, 2008). Due to exellent bandwidth offered by optial fibers, pulse shaping of laser beams has beome an important researh area in the reent years (Li, 2009; Wiener, 2011). It is also an important parameter to be onsidered in radar signal proessing, for example, pulse shape of the transmitted signal determines the range resolution, minimum and maximum range of a range-finding devie. One reent study by Yurhak (2012) desribes the pulse shaping effet on eho waveform of radar altimeter. In this work we have investigated the effet of four different pulse shapes on autoorrelation properties of the lassial Barker ode and Frank ode. The set of pulse shapes implemented in this paper onsists of Retangular, Triangular, Gaussian and Exponential pulses on the Barker-13, and Frank-9 phase odes. Barker-13 ode is a binary ode with 13 hips, while Frank-9 ode is a omplex polyphase ode with 9 hips. While onfining the disussion only to two phase odes and four pulse shapes, this paper illustrates the basi idea of effet of pulse shaping on autoorrelation parameters suh as ISL, PSL and 3dB widths. This study ould be of great relevane to the areas where pulse shaping and phase odes are routinely used. The remaining setions of this paper are organized as follows. In setion 2, the mathematial notations of the performane parameters are disussed. In Setion 3, the above mentioned pulse shapes and the two phase odes are implemented and the autoorrelation funtions are omputed. The qualities of eah phase-ode modulated pulse are derived and desribed in Setion 4. Finally, based on the observations, we onlude our work in Setion 5 with a disussion on the exponential pulse. 2. Performane Parameters 2.1 Autoorrelation When two signals under omparison are idential, then the ross orrelation operation is alled autoorrelation. Mathematially, the autoorrelation funtion of a disrete finite sequene x[ n],( n 1,2,..., N) is given by the following equation N t * x( ) [ ] [ ] i 1 R t x i x i t (1) where t 0,1,..., N 1 is the disrete time-index, the asterisk on the seond term in the produt of the right side of Eq. 1 denotes the omplex onjugate. The negative half sequene of the autoorrelation funtion is the mirrored omplex onjugate of the orresponding positive sequene sequene given by R t R t (2) * x( ) x( ) 23
The signal xn [ ] ould be a real signal as in lassial Barker ode, or a omplex signal suh as in the Frank ode. Sine autoorrelation funtion measures similarity of a signal with itself, there will be a maximum peak value when the two sequenes under onsideration are aligned. In other words, the funtion Rx () t will have a peak att 0. In loations other thant 0, the magnitude of the autoorrelation Rx () t will have lower values. Usually in radar signal proessing, the reeived pulse is orrelated with a opy of transmitted pulse and the resulting funtion gives information about the range of the target. In strit sense, sine reeived signal is different from the transmitted pulse, it is ross orrelation operation. But however, using the autoorrelation funtion, we an desribe three ommonly used performane parameters and it applies to senario where ross orrelation is performed suh as radar signal proessing. 2.2 Peak Sidelobe Level For illustration purposes, we have onsidered a simple retangular pulse shape and Fig. 1 shows its autoorrelation funtion whih is omputed using Eq. 1. In this figure, the entral dark peak between time lag R x(0) and is alled the mainlobe and the lighter shaded peaks between time lag ( 1) to N are referred as sidelobes. The absolute maximum value among the seondary peaks in the sidelobes is the peak sidelobe level (PSL) or the first sidelobe level (Rihards et. al., 2010), whih is expressed as PSL max{ R ( t)}. (3) t 0 x This quantity is more often expressed as a ratio relative to the mainlobe level in deibels, defined as the peak sidelobe level ratio (PSLR). The PSLR in deibels is represented as PSL PSLR[ db] 20log 10 R (0). (4) x Fig. 1. Illustrating the autoorrelation parameters 24
2.3 Energy The sequenes in the autoorrelation funtion ontribute individually to the total energy ontent of the signal. The energy of a finite sequene x[ n],( n 1, 2,..., N), using the expression for the autoorrelation funtion, is omputed as N 1 2 2 x ( ) t 1 E R t. (5) 2.4 Integrated Sidelobe Level Apart from the PSL in desribing the performane of the sidelobes, another important performane parameter of the autoorrelation funtion is the Integrated Sidelobe-level (ISL). The Integrated sidelobe level is defined (Rihards et. al., 2010) as Energy in Sidelobes (lighter region) ISL. (6) Energy in Mainlobe (dark region) Mathematially, the ISL is omputed by the expression E ISL. (7) R 2 (0) x Similar to the PSL, the ISL is more onveniently expressed in deibels by the integrated sidelobe level ratio ( ISLR ), using the logarithmi expression ISLR[ db] 10log ( ISL). (8) 10 3. Pulse Shaped Phase Coding and Implementation In a phase-oded pulse modulation, a single pulse is subdivided into N number of smaller hips of width. The phase ϕ n of eah hip is modulated aording to the sequenes in the ode to yield a final modulated sequene given by x[ n] exp( j ), n 1,2,..., N. The N 13 Barker ode has only two phases, 0 o and180 o, that orrespond to the amplitude of eah hip to be either 1or 1. The Barker-13 ode has the sequene of [+1 +1 +1 +1 +1-1 -1 +1 +1-1 +1-1+1]. n 25
The polyphase Frank ode has a length of N M 2 N, where N is a perfet square of some integer M, i.e.. The phase of eah hip in the Frank ode is omputed by the following expression (Rihards et. al., 2010) 2 n pq, n 1,..., N, (9) M where p, q 0,...,( M 1). There are M different modulo 2 phases in this ode. In the binary odes, the magnitude of the modulated sequenes is always unity, while the polyphase Frank odes an have multiple values from 0 to 1. In this work, we have used M =3 to yield N 9 for the Frank ode. The different pulse shapes along with their mathematial forms are shown in Table 1. These pulse shapes are applied to individual sub pulse of the respetive phase odes. Resulting pulse shape modulated phase ode is shown in Fig. 2 for Barker-13 and Fig. 3 for Frank-9. Table 1: Pulse shapes used and their mathematial forms 26
Fig. 2. Barker-13 phase oded pulse for retangular, triangular, Gaussian, Exponential pulse shapes. Fig. 3. Frank-9 phase oded pulse for retangular, triangular, Gaussian, Exponential pulse shapes. 4. Results and Disussions The normalized autoorrelation funtion is omputed for different pulse shape phase ode ombination and is shown in Fig. 4 for Barker-13 and Fig. 5 for frank-9. The performane 27
parameters of the autoorrelation funtions of the Barker and Frank ode for different pulse shapes is also omputed and summarized in Table 2. In the following sub-setions, we disuss the important observations, related to normalized autoorrelation funtions of the Barker and Frank modulated pulse based on Figs. 4 & 5, and Table 2. 4.1 Similarities between autoorrelation funtions of Frank-9 and Barker-13 modulated pulses. With the retangular pulse as the referene pulse shape, we notie that the non-retangular pulse shapes have a narrower main lobe width. In both the ases, the PSLR is 1/N and as the pulse shape gets narrower from retangular to exponential, the main lobe of the autoorrelation funtion also gets narrower. The autoorrelation funtion of retangular pulse ats as an envelope to the autoorrelation funtions obtained using the non-retangular pulse shapes. 4.2 Differenes between autoorrelation funtion of Frank-9 and Barker-13 modulated pulses. While the main lobe and side lobes are learly separated in the autoorrelation of Barker ode for all the pulse shapes, there is a lear separation only with the use of exponential pulse for the autoorrelation of Frank ode. Sine the Barker ode with 13 hips, it yields better sidelobe suppression than the Frank ode with 9 hips. In the ase of Frank ode in Fig. 5 for non-retangular pulse shapes, we observe that for M=3 phases there are 2 peaks within the envelope of sidelobe obtained from the retangular pulse. This observation is valid for a larger number of phases M. 4.3 Effet of pulse shaping on the autoorrelation funtions From Fig. 4 and Fig.5, it is learly seen that the non-retangular pulse shapes yield a narrower main lobe and sidelobe for both Barker and Frank odes. Among all the four pulse shapes that we have onsidered, the exponential pulse yields the narrowest mainlobe, narrowest sidelobe, and also it provides a lear distintion from the main lobe and the first sidelobe. Also, the exponential pulse shape yields a higher ISLR. Table 2 Autoorrelation properties of different ode-pulse ombinations PHASE CODE PULSE SHAPES RELATIVE 3-dB MAINLOBE WIDTHS PSLR (db) ISLR (db) BARKER-13 FRANK-9 Retangular 1.0-22.278-8.969 Triangular 0.7663-22.278-7.526 Gaussian 0.6769-22.278-6.977 Exponential 0.2405-22.278-3.484 Retangular 1.0-19.085-9.539 Triangular 0.7216-19.085-7.851 Gaussian 0.6375-19.085-7.286 Exponential 0.2265-19.085-3.786 28
Fig. 4. Normalized autoorrelation funtion for Barker-13 modulated pulse in Fig. 2. Fig. 5. Normalized autoorrelation funtion for Frank-9 modulated pulse in Fig. 3. 5. Conlusions In this paper, we have reviewed the autoorrelation properties that desribe the performane of a phase-ode modulated pulse. We disussed the ommonly used phase odes: binary Barker ode 29
with 13 hips, and polyphase Frank ode with 9 hips. The effet of using a narrower pulse shape on the phase oded modulation has been demonstrated. The use of pulse shape affets the 3dB resolution, energy and the ISLR, but the PSLR is unaffeted by the pulse shape. 5.1 Effet of using different Phase odes Irrespetive of any pulse shape used, the phase-modulated odes used in this work provide a fixed PSLR of 1/N. The Barker-13 ode gives a PSLR of 1/13 (-22.278 db), while the Frank-9 ode yields 1/9 (-19.085 db). A phase ode with more number of hips provides a better sidelobe suppression performane. 5.2 Effet of using different pulse shapes In the autoorrelation funtion of Frank ode, by using the retangular pulse shape, eah sidelobe is wide and single domed. But in the ase of non-retangular pulse shapes, multiple peaks are obtained within a sidelobe. The number of peaks is found to be M-1 when M different phases are used in the Frank ode. Sine the energy ontent of a pulse is dependent on the pulse shape for a given pulse length, the integrated sidelobe ratio depends on the pulse shape. Narrower pulse provides a larger ISLR. The Frank-9 ode provides lower ISLR than the Barker-13 ode for a given pulse shape. For a given phase ode and a fixed number of hips, the 3-dB range resolution of the exponential shaped pulse yields the lowest value. Among the four pulse shapes that we have used in this work, the exponential pulse shape yields the narrowest main lobe, and sharper peaks in sidelobes for both the phase odes. Also in the ase of polyphase Frank ode, the exponential waveform performed very well in separating the mainlobe and its immediate sidelobe, whih was not possible with other pulse shapes that we used. The exellent properties of exponential pulses suggest its potential uses in situations where attenuation of return signal is not prominent. A tehnique for realization of exponential pulse related to radiation measurements was provided by Jordanov et al. (1994). Some of the possible areas of its appliability inlude ultrasound imaging, nearby range-finding appliations suh as ollision and obstale avoidane system in automated roboti mahines (Jorg and Berg, 1998), guiding system for visually impaired, eye-safe lidar systems for surveying et. Exponential signals have potential of realization with the developments in ultrafast iruits. Further investigation on using the double exponential pulse shape in phase odes ould reveal muh more interesting features in its autoorrelation properties. Referenes Barker, R.H., (1953). Group synhronizing of binary digital systems, Communiation Theory (W. Jakson, ed.), Aademi Press, New York, pp. 273 287 Frank, R. L., (1963). Polyphase odes with good nonperiodi orrelation properties, IEEE Transations on Information Theory, vol.9, no.1, pp. 43-45. http://dx.doi.org/10.1109/tit.1963.1057798 30
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