Method to Design UF-OFDM Filter and its Analysis

Transcription

1 ethod to Design UF-OFD Filter and its Analysis Hirofumi Tsuda Department of Applied athematics and Physics Graduate School of Informatics Kyoto University Kyoto, Japan Ken Umeno Department of Applied athematics and Physics Graduate School of Informatics Kyoto University Kyoto, Japan arxiv:70797v [csit] 0 Nov 07 Abstract Orthogonal Frequency Division ultiplexing OFD) systems have been widely used as a communication system In OFD systems, there are two main problems One of them is that OFD signals have high Peak-to-Average Power Ratio PAPR) The other problem is that OFD signals have large side-lobes In particular, to reduce side-lobes, Universal- Filtered OFD UF-OFD) systems have been proposed In this paper, we show criteria for designing filters for UF-OFD systems and a method to obtain the filter as a solution of an optimization problem Further, we evaluate PAPR with UF-OFD systems Our filters have smaller side-lobes and lower Bit Error Rate than the Dolph-Chebyshev filter However, the PAPR for signals with our filters and the Dolph-Chebyshev filter are higher PAPR than those with conventional OFD signals I Introduction Orthogonal Frequency Division ultiplexing OFD) systems have been widely used for broadband multicarrier communication since OFD systems have a number of advantages One of advantages is that OFD system can deal with multipath fading The reason for this is that OFD signals are generated with inverse Fourier transformation Another advantage is that OFD systems can be adopted to multipleinput multiple-output IO) channels Due to these advantages, OFD systems are used as some telecommunication standards However, it is known that there are two main problems in OFD systems One of them is that OFD systems have high Peak-to-Average Power Ratio PAPR), which is the ratio between the maximum value of signal powers and the average value of them For low values of the input powers, the output power grows approximately linear However, for input signals with large power, the growth of the output power is not linear Then, in-band distortion and out-of-band distortion are caused [] The effects of non-linear amplifiers are investigated in [] Therefore, techniques to reduce PAPR are demanded There are many methods to reduce PAPR to avoid such problems [3]- [5] Further, the distribution of the PAPR and performances of methods are investigated in [6] [7] The other main problem is that OFD signals have large side-lobes [8] This problem leads to leakage of signal powers among the bands of different users Therefore, there are some improved and proposed systems to solve this problem One of the systems is a Universal-Filtered OFD UF-OFD) system [9] In UF-OFD systems, band-bass filters are used to reduce side-lobes Therefore, methods to design efficient filters are demanded There are some investigations about improving UF-OFD systems [0] [] In this paper, we show a method to design filters for UF-OFD systems As a signal processing technique for recovering symbols, we consider Zero Forcing equalization Then, we derive the sufficient condition for increase of the Signal-to-Noise Ratio SNR) From this condition, we obtain the optimization problem to increase the SNR and reduce sidelobes Therefore, the filter for UF-OFD systems is obtained as a solution of the optimization problem Further, we evaluate the PAPR with UF-OFD systems and show the relation between PAPR and SNR Then, we show the numerical results about power spectrum density, Bit Error Rate BER) and PAPR with filters obtained from the optimization problem athematical Notation In what follows, we use the Fourier transformation as a discrete Fourier transformation F x ω)= x n jnω), ) n where x C p, x n is the n-th element of x, j is the unit imaginary number and p is a positive integer For convenience, we write the discrete Fourier transformation of x as its capital, that is, Xω)=F x ω) ) Further, we write the complex Gaussian distribution whose average and variance areµandσ asnµ,σ ) Note that their real part and imaginary part obey the Gaussian distribution whose variance isσ /, respectively II UF-OFD odel In this section, we fix our model used thorough this paper and mathematical symbols that will be used in the following sections We consider Zero Padding ZP) UF-OFD systems For more details of this model, we refer the reader to [3] Let us define A k as a symbol transmitted by the k-th carrier We assume that each A k is independent and EA i A k }=δ ik, where EX} is the average of X, z is the complex conjugate of z and i=k δ ik = 0 i k 3)

2 Then, a discrete OFD signal x n is written as x n = A k π j kn ) n ) n=0,,, ), 4) where J 0,,, } is the set of the numbers of used carriers andx) is defined as [] x)= x< 0 otherwise 5) Note that the duration of symbols is In particular, we denote K by the number of elements in the set J Equation 4) is implemented with the Inverse Fast Fourier Transformation IFFT) In UF-OFD systems, discrete OFD signals are filtered to reduce their side-lobes A filter f C N is written as f= [ f 0 f f N ], 6) where x is the transpose of x We assume that f 0 0 and f N 0 This assumption is equivalent to that the convolved signal has the length + N When the signal x n is filtered by f, the output signal y n is written as y n = N m=0 f m A k π j kn m) ) n m ) for n=0,,, +N Note that the duration of symbols becomes to +N due to the filtering Equation 7) is often written as a linear system with a Toeplitz matrix and a Discrete Fourier Transformation DFT) matrix In our model, we apply a zero padding technique to avoid intersymbol interference Let us define D as the length of zero paddings Then, the transmitted signal ỹ n is written as yn 0 n< +N ỹ n = 8) 0 +N n< +N+ D In wireless communication systems, fading effects should be considered In OFD systems, fading effects are ressed as a discrete convolution Let us define h C L as 7) h= [ h 0 h h L ], 9) where L is the length of the fading h Further, we assume that L D This assumption is equivalent to that there we have the knowledge about the length of the fading effects This assumption is often used [4] Then, the received signal r n is written as L r n = h l ỹ n l + v n n=0,,, +N+ D ), 0) l=0 where v n is additive white Gaussian noise AWGN) To estimate symbols, zero padding techniques are used Then, the length of r n is extends to Therefore, the zero padded signal r n is written as rn 0 n< +N+D r n = 0 +N+ D n< ) From Eqs 0) and ), their Fourier transformations are written as Rω) = Rω) = Fω)Hω)Xω) + Vω) ) In particular, X ) π k ) is written as X π k ) = A k 3) Therefore, Eq ) is rewritten as R π k ) = A k F π k ) H π k ) + V π k ) 4) The quantity R π ) k can be obtained with the -Fast Fourier Transformation and a down sampling technique [3] We can calculate and obtain Fω) since we know f With some techniques, for example pilot symbols, Hω) can be estimated and obtained Therefore, from Eq 4), the symbol A k can be recovered with multiplication of the inverse of Hω) Then, when the Hω) is given and fixed, the SNR about the symbol A k is written as F π k SNR k = +N+ D ) H π k ) σ n, 5) whereσ n satisfies Ev n v n}=σ n From Eq 5), SNR gets higher as F π ) k gets larger From this result, the large F π ) k is demanded for high SNR Note that UF-OFD systems are different from OFD systems in a sense of SNR In UF-OFD systems, the duration of a symbol becomes longer than original signals with zero padding techniques and filtering Then, the duration of a symbol with UF-OFD system is longer than that with conventional OFD systems As seen in Eq 5), the SNR in +N+D UF-OFD systems has the term, which is related to the length of filters N and do not appear in conventional OFD systems From this result, UF-OFD systems are different from OFD systems in a sense of SNR III Optimization Problem for Designing Filter In this section, we show the criteria for designing filters We focus on Bit Error Rate BER) and side-lobes For convenience, let us define the n-th element of g as g n = f n+m fm 6) Note that g is written as m g=[ g N+ g N+ g N ] 7) It is known that the Fourier transformation of g is written as F g ω)= N n= N+ g n jnω)= Fω) 8) From Eq 8), Fω) is ressed as the inner product of the two vectors, g and the vector whose n-th element is jnω) Further, when the vector g is given, we can obtain the original vector f with a spectral factorization method [8]

3 A Condition First, we consider the condition that filters have to satisfy We assume that the average power of signals is conserved through the filter convolution This assumption is written as n=0 E x n }= +N E y n }, 9) +N where EX} is the average of X With g, this condition is written as [] K+N )=b J g, 0) where K is the number of the elements of J and b J is the vector written as n=0 b J = [ ] b J, N+ b J, N+ b J,N, b J,n = n ) π j kn ) ) Note that Eq 0) is invariant under the following transformations: each carrier k k+ s and the n-th element of the filter f n f n π j sn ) for any real value s Therefore, when J consists of K consecutive integers, we can assume that the frequencies of carriers are symmetric around ω = 0, where ω is an angular frequency This assumption is realized with the operation k k+s Then, it is sufficient to design the filter whose power spectrum is symmetric aroundω=0 Therefore, we treat f R N, and rewrite g as g=[ g 0 g g N ] ) sinceg n } is symmetric around n=0 B Optimization Problem First, we divide the region [0,π) into three regions as follows [0,π)=Ω p Ω t Ω s, 3) whereω p,ω t andω s are the regions of the passband, the transition-band and stop-band, respectively We define the set of frequencies of carriers asω c, which is the frequency shifted set of J In Section II, we have shown that the large F ω c ) ω c Ω c ) is demanded for high SNR As criteria of designing filters, high SNR and the rapid decay of side-lobes are demanded Therefore, we obtain the following optimization problem whereλis the positive weight parameter, E Xω) } is written as E Xω) }= α ω c ω), ω c Ω c ω=nπ n Z) αω)= sin ) ) ω / sin ω otherwise and b c R N is the vector whose n-th element is defined as 4) ω b c,n = c Ω c n ) cos nω c ) n=,,, N N n=0 5) In the above equations, we have used the assumption that each A k is independent and EA i A k }=δ ik, which has been made in Section II In problem P ), the variables are g, t and t It is ected that the side-lobes get lower as the lambda becomes smaller Besides, it is ected that the SNR gets larger as the lambda becomes larger In the problem P ), the first constraints are about side-lobes The second constraints are about desired signals, which have been discussed in Section II The third constraints are about the conservation of the signal power The fourth constraints are necessary to satisfy Eq 6) [5] Note that the vector b c is the same as b J defined in Eq ) when the setω c consists of the angular frequencies πk/, where k is the number carriers In the problem P ), it is not straightforward to handle the first and fourth constraints sinceω s and [0,π] are regions To overcome these obstacles, we transform the problem One method to overcome the obstacles in the first constraints is to transform the constraints into discrete constraints [5] [6] With this method, the first constraints in the problem P ) is replaced with F ω i ) E Xω i ) } t i=,,, S, 6) whereω i i=,,, S ) is the element in the regionω s In this paper, S = 5N is chosen Note that how to choose the number S is discussed in [5] [6] [7] It is shown in [5] that the fourth constraint is satisfied if there is a symmetric matrix P which satisfies the following inequality [ ] P A PA C A PB C B PA D+ D B 0, 7) PB P ) min t λt st F ω) E Xω) } t ω Ω s ) F ω c ) t ω c Ω c ), K+N )=b c g, F g ω) 0, ω [0,π], t, t 0, For example,,, 0,,, 3} is allowed where A=, B=, C= [ ] g g g N, D= g 0 8)

4 and X 0 means that X is a positive semidefinite matrix With these methods, the problem P ) is rewritten as P) min t λt st F ω i ) E Xω i ) } t ω i Ω s, i=,,, S ) F ω c ) t ω c Ω c ), K+N )=b c [ g, P A PA C A PB C B PA D+ D B PB t, t 0, P S N, ] 0, where S n is the set of n n symmetric matrices Note that the variables in the problem P) are g, P, t and t Further, the problem P) is convex IV Peak-to-Average Power Ratio Analysis In Section III, we have shown the criteria for designing filters and the optimization problem In this section, we discuss the PAPR of UF-OFD signals and show the trade-off between PAPR and BER We make the following assumptions: ) Continuous OFD signals are regarded as a Gaussian process when the time t is fixed ) Continuous UF-OFD signals are regarded as a Gaussian process when the time t is fixed 3) Continuous OFD signals are perfectly reconstructed from their discrete signals that are sampled The first assumption is often used [6] [7] This assumption is based on the central limit theorem To analyse PAPR of UF- OFD signals in a same way for OFD signals, we make the second assumption About the last assumption, we assume that the following formula is satisfied: π j kt ) t = n= π j kn ) n ) t )) sinc π t n 9) for 0 t T, where sincx) is the sinc function, T and t are the symbol duration and the sampling time which satisfy T = t This formula is based on the sampling theorem Note that Eq 9) is not always satisfied since OFD signals are not strictly band-limited due to the rectangular pulse t) see Eq 4)) [8] Therefore, Eq 9) is an approximation The definition of the PAPRPis written as P= max 0 t T st) P av, 30) where st) is a baseband signal and P av is the average power of baseband signals Note that PAPR is a random variable since signals are regarded as random variables In this paper, it is called that the PAPR becomes high when the PAPR tends to have a large value From Eq 9) and the previously made assumption that the average power of signals are conserved through the filter convolution, the average power of UF-OFD signals is written as P av = K 3) Therefore, instead of y n, we consider the following signal ȳ n, ȳ n = y n Pav = N m=0 f m A k π j km n) ) m n K ) 3) Let us consider the continuous UF-OFD signal We fix t [TN )/, T] Note that the symbol duration of UF-OFD signals is T +N )/ From the sampling theorem, the continuous UF-OFD signal is written as ȳt)= ȳ n sinc n= π With Eq 9), Eq 33) is rewritten as ȳt)= N A k f n K n=0 = A k F K t t n )) 33) π j k t nt )) T π k ) π j kt ) 34) t for t [TN )/, T] Note that ȳt) is a random value since A k is a random value The following formulas are satisfied: Eȳt)}=0, E ȳt) }= π K F k ) 35) We denote σ by F π ) k /K In OFD signals, the covariance of signals equals Therefore, from assumptions and 3, signals of UF-OFD ȳt) and ones of OFD xt) obey N 0, σ ) and N0, ), respectively Therefore, their amplitudes obey the Rayleigh distributions whose scale parameters equal σ/ and /, respectively From this result, it is ected that the PAPR for UF-OFD signals becomes higher as the parameter σ gets larger As seen in Section II, the parameter σ relates to desired signals and large σ is necessary for high SNR Therefore, it is ected that the PAPR for UF-OFD signals becomes higher as SNR gets higher V Numerical Results In this section, we show the BER, PAPR, and the power spectrum of filters obtained from the problem P) In our simulations, the parameters, the size of IFFT =8, the length of filter N= 6, the size of the zero-padding D=6 and the length of fading channels L= are chosen Further, we assume that each element of the fading effect h n obeys N0, /L) As carrier frequencies, we set J =4, 5,, 9} With frequency shifts, we design filters on the region [0, π) We set the regions,ω p,ω t andω s as follows Ω p = [0, 7π/56),Ω t = [7π/56, 7π/64),Ω s = [7π/64,π) 36) With the above parameters, we obtain filters from the problem P) for variousλ To solve the problem P), we use the matlab package, CVX [0] The filter f is obtained from the solution of the problem P), g with spectral factorization

5 methods [7] We compare these filters with Dolph-Chebyshev filter whose side-lobe level is 45 db In each simulation, the modulation scheme is the QPSK modulation In each figure, Our filters means the filters obtained from the problem P) with various λ, D-C means the Dolph-Chebyshev filter Figure shows the power spectrum of each filter With filters obtained from the problem P), the side-lobe decreases as the weight parameter λ gets smaller In particular, with the filter whose parameter is λ = 0000, its side-lobe is nearly 90 db and lower than one of the Dolph-Chebyshev filter It is known that side-lobe reductions in order of 70 db are mandatory for broadcasting systems [] We observe that the filter whose parameter isλ=0000 is appropriate for broadcasting systems Further, the side-lobe tends to be increased by the PAPR reduction schemes [] Thus, the filter whose parameter isλ=0000 may be still appropriate after processes of PAPR reductions Power Spectrum [db] Stopband Transition band Passband OFD λ= λ=0 λ=00 λ=000 D-C45dB) λ=0000 Stopband Normalized Angular Frequency OFD Fig Power Spectrum for UF-OFD and OFD systems Figure shows the BER with various filters We assume that the receiver has the perfect knowledge about the fading function Hω) Then, the receiver can estimate the symbol A k with the knowledge of Hω) see Eq 4)) In Fig,σ s is the energy per bit before filtering andσ n is the variance of the AWGN channel, which has been defined in Eq 5) Note that the energy per bit of transmitted signals is +N σ s The BER for our filters is lower than that for the Dolph-Chebyshev filter In designing filters with the problem P), we increase the power of all desired signals However, in the Dolph-Chebyshev filter, the power of all desired signals is not equal and there are some desired signals whose power is low Therefore, it is conceivable that this difference at each power spectrum causes the difference of BER Figure 3 shows the PAPR with each filter To compare equally, there is no zero paddings in UF-OFD systems, that is, UF-OFD signals are uniformly filled in the symbol duration In calculating the PAPR, we set the = 8 points in each symbol duration and obtain each amplitude in the points Then, the maximum amplitude is regarded as the maximum value in them In OFD signals, it is conceivable that the over-sampling factor /K = 8 is sufficiently large D-C Bit Error Rate Our filters D-C λ= λ=0 λ=00 λ=000 λ=0000 D-C45dB) σ s /σ n [db] Fig Bit Error Rate of UF-OFD with various filters to measure the PAPR As the over-sampling factor, /K 4 is commonly used The discussion about the over-sampling factor is described in [9] In Section IV, we have discussed the PAPR of the filters obtained from the problem P) As seen in Fig 3, the PAPR for the signals with filters obtained from the problem P) is higher than that with the Dolph-Chebyshev filter and the OFD signals Since each BER of the filters obtained by the problem P) is nearly the same, it is conceivable that each σ, which is defined in Section IV is the same Therefore, the PAPR for the filters obtained by the problem P) is uniformly high and they are nearly the same On the other hands, the PAPR for the Dolph-Chebyshev filter is lower than that for the filters obtained by the problem P) From this result and the result about BER, there is a relation between BER and PAPR in UF-OFD systems, that is, PAPR gets higher as the SNR gets larger Complementary Cumulative Distribution Function OFD Our filters D-C λ= λ=0 λ=00 λ=000 λ=0000 D-C 45dB) OFD Peak-to-Average Power Ratio [db] Fig 3 Peak-to-Average Power Ratio for UF-OFD and OFD systems VI Conclusion In this paper, we have shown the UF-OFD model and the criteria of designing filters, which is related to side-lobes and SNR Then, we have obtained the convex optimization problem and the filters as its solutions As numerical results, we

6 have shown their BER, power spectrum and PAPR and compared them with the Dolpf-Chebyshev filter and conventional OFD signals From these results, the BER for our filters is lower than that for the Dolpf-Chebyshev filter In particular, the side-lobe of the filter whose parameterλ=0000 is the lowest This filter is appropriate for communication systems since the power of side-lobes is low However, the signals of UF-OFD systems have high PAPR than ones of conventional OFD systems It is necessary to consider this relation for communication systems [0] Grant, S Boyd, and Y Ye CVX: atlab software for disciplined convex programming 008) [] T Jiang, and Y Wu An overview: Peak-to-average power ratio reduction techniques for OFD signals IEEE Transactions on broadcasting ): Acknowledgment The one of the authors, Hirofumi Tsuda, would like to thank Dr Shin-itiro Goto for his advise References [] R van Nee, and R Prasad OFD for wireless multimedia communications Artech House, Inc, 000 [] E Costa, and S Pupolin -QA-OFD system performance in the presence of a nonlinear amplifier and phase noise IEEE transactions on Communications ): [3] J Armstrong, Peak-to-average power reduction for OFD by repeated clipping and frequency domain filtering Electronics letters ): [4] Y-C Wang, and Z-Q Luo Optimized iterative clipping and filtering for PAPR reduction of OFD signals IEEE Transactions on Communications 59 0): [5] Y Jiang, New companding transform for PAPR reduction in OFD IEEE Communications Letters 44 00) [6] H Ochiai, and H Imai On the distribution of the peak-to-average power ratio in OFD signals IEEE transactions on communications 49 00): 8-89 [7] H Ochiai, and H Imai Performance analysis of deliberately clipped OFD signals IEEE Transactions on communications 50 00): 89-0 [8] B Farhang-Boroujeny OFD versus filter bank multicarrier IEEE signal processing magazine 83 0): 9- [9] V Vakilian, T Wild, F Schaich, S ten Brink, and J F Frigon Universal-filtered multi-carrier technique for wireless systems beyond LTE Globecom Workshops GC Wkshps), 03 IEEE IEEE, 03 [0] Z Zhang, H Wang, G Yu, Y Zhang, and X Wang Universal Filtered ulti-carrier Transmission With Adaptive Active Interference Cancellation IEEE Transactions on Communications 07) [] F Tang, and B Su Filter optimization of low out-of-subband emission for universal-filtered multicarrier systems Communications Workshops ICC), 06 IEEE International Conference on IEEE, 06 [] H Schulze and C Lüders Theory and applications of OFD and CDA: Wideband wireless communications John Wiley & Sons, 005 [3] T Wild, and F Schaich A reduced complexity transmitter for UF- OFD Vehicular Technology Conference VTC Spring), 05 IEEE 8st IEEE, 05 [4] B uquet, Z Wang GB Giannakis, de Courville and P Duhamel Cyclic prefixing or zero padding for wireless multicarrier transmissions? IEEE Transactions on communications 50 00): [5] S P Wu, S Boyd, and L Vandenberghe FIR filter design via semidefinite programming and spectral factorization Decision and Control, 996, Proceedings of the 35th IEEE Conference on Vol IEEE, 996 [6] T Baran, D Wei, and A V Oppenheim Linear programming algorithms for sparse filter design IEEE Transactions on Signal Processing ): [7] D attera, F Palmierl, and S Haykin Efficient sparse FIR filter design Acoustics, Speech, and Signal Processing ICASSP), 00 IEEE International Conference on Vol IEEE, 00 [8] A Papoulis Signal analysis Vol 9 New York: cgraw-hill, 977 [9] Sharif, Gharavi-Alkhansari, and B H Khalaj On the peakto-average power of OFD signals based on oversampling IEEE Transactions on Communications 5 003): 7-78