Parks-McClellan algorisması ile sınırlı zamanlı Nyquist işareti sentezi
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Milcroişlemciler yardımı ile tasarlanan veri haberleşme sistemlerinde en önemli problemlerden biri, iletim bandmdaki işaret enerjisinin tüm enerjiye ora nını maximum yapan sınırlı zamanlı işaret elemanlarının ^tasarımını gerçekleştirmektir. Ayrıca peryodik örnek leme anlarında üretilen işaret elemanları için simgeler arası girişim sıfır olmalıdır. Bu çalışmada simgeler arası girişimi sıfır olan ve verilen Q band genişliğinde spektral enerjiyi maksi mum yapan M baud zaman aralığında Nyquist tipi işaret lerin optimal tasarımı göz önüne alınmıştır. Simgeler arası girişim kısıtlamasının olmaması durumunda bu problem Ville ve Bouzitat tarafından çözülmüş Slepian Pollack ve daha sonra Landau ve Pollack tarafından genelleştirilmiştir. Son zamanlarda Halpern varyasyo- nel tekniklerle, simgeler arası girişimi göz önüne alarak yukarıdaki problemi incelemiştir. Problemdeki bu ek sınırlamaya girmek için işaretlerin ve farklı gecikmiş durumlarının birbirine dik olduğu düşüncesi ile hareket etmiş bu durumda optimum işaretin, homojen bir integral denkleminin birkaç sınırlama ile elde edilmiş bir çözümü olduğunu göstermiştir. Bununla beraber sadece küçük M iler için bu integral denklemin nümerik çözümüne ulaşılabilmiş, ayrıca önerilen algo ritmanın yakınsaklığı ispat edilememiştir.
Synthesis of The Optimal Nyquist-Type Signals One at the most important problems in data transmission systems which are designed with micro processors is to obtain a band limited signal that has maximum ratio of transmission band energy to total energy. There a constraint is taken into account: no intersymbol interference at the periodic sampling instants is permitted. In this thesis, optimal design of Nyquist-type signals which generates no intersymbol interference at the periodic sampling instants and maximizes spectral energy in a given W bandwidth is obtained. Without intersymbol interference constraint, this problem was solved by Ville and Buzitat and generalized by Slepian and Pollack, and then by Landau and Pollack, the prolate spherodial wave functions. Recently, Panayirci and Tuğbay, have investigated a different version of this problem. They considered the optimal design of Nyquist-type signals of finite duration, which maximizes spectral energy inside a given bandwith, and which generates no intersymbol interference. The optimal signal was a solution of a homogeneous linear equation with constraint. They formulated their new problem and and the constraint for included into the problem. It was found that the optimal signal was a solution of an inhomogeneous Also a technique for analytically solving that equation was given. The formulation of this approach is given shortly below: The main concern is to design a Nyquist-type and a bandlimited signal of bandwidth W, s(t), t«(-oo,+oo), which maximizes its energy in the time interval (-o-TjO'T), given by =. -J or E. = f s"(t)dt -ffT Cl> under the constraint the total signal energy CO E = I s2(t)dt C23 :J o -co is costant. Let S(f) denote' the Fourier transform of s(t). Then the Nyquist constraint, i.e. s(kT)=ü for k=±l,±2,... can be inserted into the problem as follows. It is well known that the necessary and sufficient condution for zero interference is the first Nyquist criterion: £S( f + ~- ) = A C3} k=-co where A is any positive real constant. Furthermore, since the consideration is limited to bandlimited signals of band-width W, we express the bandwidth in terms of a rolloff factor y which is the relative amount of bandwidth in excess of Nyquist band, 1/2T W - r = T72T C4> The most interesting case is the small excess bandwidth ( y < 1). With this restriction, it must be cosidered only the k=-l,0,l terms of the sums in equation 3. It is convenient to write S(f) in terms of a normalised rolloff function B(f) S(f )=A1 PCf > + B(f -4-) + B*<- f- ^ ) C5D -[ 2T 2T where B(f) = 0 for |f| > r/2T and P(f) is the required shape for the case y - 0 P(f ) r i. |f |< i/ ~ \ 0 otherwi < 1/2T se The Nyquist constraint is equivalent to B(f) = - B ( -f ), Moreover, it can be shown that S(f) is a real and even function. Hence, B(f) is taken to be real and odd. The problem of selecting the optimal signal s(t) under above constraints reduces to finding the rolloff vi function B(.) and the A which maximize the functional J= J[BC.),A] defined by J = E - \E C6D The constant X is called a Lagrange multiplier. An efficient method for the solution of this optimisation problem is to approximate the frequency spectrum of the signal S(f) by a sequence of rectangular pulses equally spaced in frequency. Recall that the spectrum of the designed signal shape is constant for |f j < (.1-y) /2T. Let this constant is denoted by A. Then, the remaining part of the spectrum for Cl-jO/aT < |f| < (l+^)/2T can be divided into (2M+1) equal intervals, each of length A = y/T(2M-KL>, and the function may be evaluated at the midpoints f.= (l-^)/2T + as shown symmetric s. (i=l,2,., A [n (-x1] + n [-a1] ] } where, n (t/T) - < 1, |t| < t /2 0, otherwise (3 = /2T Vll Then, taking inverse Fourier transform of S(f }, the following is obtained for s(t) : ?[ s(t) = A f + £ E s. f. + E cos(2nf.t)) M+l ** t 2M+1. E c< l = M+2 and f.(t) = 2Asinc(At ) (cos(anf. t )-cosC2rcf.t) x. l 2M+2-1 ( i =1, 2..... M ) sinc(x) = sin(Trx)/rrx the energy of the signal s(t) in the interval (-oT.o'T) can be expressed as a function of s. and A, M M E. = A' l [M M *I -I c + 2 e a- s. + E Ebss i=ı i = i j =i JJ, crx cQ = | r. (t )dt -at f (t )f O u (t )dt Jar f. (t )f. (t )dt J i, j =1. 2 M On the other hand, the total signal energy is E = 2A o K M 2A E s + 2A ı=ı t = i J C7!> vıı where, c = (3 + CM + O. 25)A ı The problem of selecting the optimal signal s(t) which maximizes its energy in the interval (-cT.o'T) subject to the constraint that the total energy is constant, reduces to finding the coefficients s »s,..., s and A which maximize 12 M I =I(s,s s ;A) =E. - XE 12 M x. O setting a i = 0 i = 1,2 M, t and a s «i = o, a a yields the set of linear equations M (2a + 2 r b. s > - 4XA(2s - 1) = 0 C83 j=i lJ J i=l.2 M M M ft* c + 2 £ a. s. + T\ Tb.ss.., <. i ?. ?. ". J L J u=l l=lj=l M 2X| c - 2A £ s.d - s ) 1 = L i = i -I C93 By solving the above equations simultaneously, the o o o optimal coefficients s,s,...s and the optimal energy ratio X = (E./E ) are obtained. o t o op t A= [Eo4C*-2\?tSt(1 -S?0 ] 1/2 In this thesis, Parks-McClellan's digital filter designe algorithm is used to obtain Nyquist-type IX signal. Algorithm's formulation is changed according to approximation used in Panayirc i -Tuğbay and a new design procedure is given. The alterations made, are briefly explained below : Suppose that each rectangular pulse shown in Fig. 4.1 denotes a unique frequency band in the filter characteristic. In that case, each cut off frequency used in the algorithm has to be determined, but the number of rectangular pulses is too great. It isn't possible to use the edge frequency of each pulse is cut off frequencies, because transition region in the filter transfer function does not appear in this spectrum. Therefore, the midpoint frequency of each pulse was used as the input frequency to the algorithm, instead of the edge the edge frequencies. In this program, instead of solving (4.11) equetions directly, only X was analytically calculated at each step and the s. coefficients of the curve that approaches the characteristic shown in Fig. 4.1. were calculated until X approached maximum value. s. coefficients were exchanged with the extrema of the previous curve and iterated until the calculated X equals the previous one. The X thus obtained has an optimal energy ratio ; X = (E./ E ). The second step is to calculate the amplitude A, using the optimal coefficients already obtained. The amplitudes of 2M+1 rectangular pulses are then calculated using A and s. coefficients. Lastly, s(t) is obtained by calculating the inverse Discrete Fourier Transform of the optimal s coefficients. The program listing of this algorithm is given in Appendix A and the optimal Nyquist signal samples obtained using this algorithm are given in Appendix B. The energy concentration was found to be greater than 95 %.
Synthesis of The Optimal Nyquist-Type Signals One at the most important problems in data transmission systems which are designed with micro processors is to obtain a band limited signal that has maximum ratio of transmission band energy to total energy. There a constraint is taken into account: no intersymbol interference at the periodic sampling instants is permitted. In this thesis, optimal design of Nyquist-type signals which generates no intersymbol interference at the periodic sampling instants and maximizes spectral energy in a given W bandwidth is obtained. Without intersymbol interference constraint, this problem was solved by Ville and Buzitat and generalized by Slepian and Pollack, and then by Landau and Pollack, the prolate spherodial wave functions. Recently, Panayirci and Tuğbay, have investigated a different version of this problem. They considered the optimal design of Nyquist-type signals of finite duration, which maximizes spectral energy inside a given bandwith, and which generates no intersymbol interference. The optimal signal was a solution of a homogeneous linear equation with constraint. They formulated their new problem and and the constraint for included into the problem. It was found that the optimal signal was a solution of an inhomogeneous Also a technique for analytically solving that equation was given. The formulation of this approach is given shortly below: The main concern is to design a Nyquist-type and a bandlimited signal of bandwidth W, s(t), t«(-oo,+oo), which maximizes its energy in the time interval (-o-TjO'T), given by =. -J or E. = f s"(t)dt -ffT Cl> under the constraint the total signal energy CO E = I s2(t)dt C23 :J o -co is costant. Let S(f) denote' the Fourier transform of s(t). Then the Nyquist constraint, i.e. s(kT)=ü for k=±l,±2,... can be inserted into the problem as follows. It is well known that the necessary and sufficient condution for zero interference is the first Nyquist criterion: £S( f + ~- ) = A C3} k=-co where A is any positive real constant. Furthermore, since the consideration is limited to bandlimited signals of band-width W, we express the bandwidth in terms of a rolloff factor y which is the relative amount of bandwidth in excess of Nyquist band, 1/2T W - r = T72T C4> The most interesting case is the small excess bandwidth ( y < 1). With this restriction, it must be cosidered only the k=-l,0,l terms of the sums in equation 3. It is convenient to write S(f) in terms of a normalised rolloff function B(f) S(f )=A1 PCf > + B(f -4-) + B*<- f- ^ ) C5D -[ 2T 2T where B(f) = 0 for |f| > r/2T and P(f) is the required shape for the case y - 0 P(f ) r i. |f |< i/ ~ \ 0 otherwi < 1/2T se The Nyquist constraint is equivalent to B(f) = - B ( -f ), Moreover, it can be shown that S(f) is a real and even function. Hence, B(f) is taken to be real and odd. The problem of selecting the optimal signal s(t) under above constraints reduces to finding the rolloff vi function B(.) and the A which maximize the functional J= J[BC.),A] defined by J = E - \E C6D The constant X is called a Lagrange multiplier. An efficient method for the solution of this optimisation problem is to approximate the frequency spectrum of the signal S(f) by a sequence of rectangular pulses equally spaced in frequency. Recall that the spectrum of the designed signal shape is constant for |f j < (.1-y) /2T. Let this constant is denoted by A. Then, the remaining part of the spectrum for Cl-jO/aT < |f| < (l+^)/2T can be divided into (2M+1) equal intervals, each of length A = y/T(2M-KL>, and the function may be evaluated at the midpoints f.= (l-^)/2T + as shown symmetric s. (i=l,2,., A [n (-x1] + n [-a1] ] } where, n (t/T) - < 1, |t| < t /2 0, otherwise (3 = /2T Vll Then, taking inverse Fourier transform of S(f }, the following is obtained for s(t) : ?[ s(t) = A f + £ E s. f. + E cos(2nf.t)) M+l ** t 2M+1. E c< l = M+2 and f.(t) = 2Asinc(At ) (cos(anf. t )-cosC2rcf.t) x. l 2M+2-1 ( i =1, 2..... M ) sinc(x) = sin(Trx)/rrx the energy of the signal s(t) in the interval (-oT.o'T) can be expressed as a function of s. and A, M M E. = A' l [M M *I -I c + 2 e a- s. + E Ebss i=ı i = i j =i JJ, crx cQ = | r. (t )dt -at f (t )f O u (t )dt Jar f. (t )f. (t )dt J i, j =1. 2 M On the other hand, the total signal energy is E = 2A o K M 2A E s + 2A ı=ı t = i J C7!> vıı where, c = (3 + CM + O. 25)A ı The problem of selecting the optimal signal s(t) which maximizes its energy in the interval (-cT.o'T) subject to the constraint that the total energy is constant, reduces to finding the coefficients s »s,..., s and A which maximize 12 M I =I(s,s s ;A) =E. - XE 12 M x. O setting a i = 0 i = 1,2 M, t and a s «i = o, a a yields the set of linear equations M (2a + 2 r b. s > - 4XA(2s - 1) = 0 C83 j=i lJ J i=l.2 M M M ft* c + 2 £ a. s. + T\ Tb.ss.., <. i ?. ?. ". J L J u=l l=lj=l M 2X| c - 2A £ s.d - s ) 1 = L i = i -I C93 By solving the above equations simultaneously, the o o o optimal coefficients s,s,...s and the optimal energy ratio X = (E./E ) are obtained. o t o op t A= [Eo4C*-2\?tSt(1 -S?0 ] 1/2 In this thesis, Parks-McClellan's digital filter designe algorithm is used to obtain Nyquist-type IX signal. Algorithm's formulation is changed according to approximation used in Panayirc i -Tuğbay and a new design procedure is given. The alterations made, are briefly explained below : Suppose that each rectangular pulse shown in Fig. 4.1 denotes a unique frequency band in the filter characteristic. In that case, each cut off frequency used in the algorithm has to be determined, but the number of rectangular pulses is too great. It isn't possible to use the edge frequency of each pulse is cut off frequencies, because transition region in the filter transfer function does not appear in this spectrum. Therefore, the midpoint frequency of each pulse was used as the input frequency to the algorithm, instead of the edge the edge frequencies. In this program, instead of solving (4.11) equetions directly, only X was analytically calculated at each step and the s. coefficients of the curve that approaches the characteristic shown in Fig. 4.1. were calculated until X approached maximum value. s. coefficients were exchanged with the extrema of the previous curve and iterated until the calculated X equals the previous one. The X thus obtained has an optimal energy ratio ; X = (E./ E ). The second step is to calculate the amplitude A, using the optimal coefficients already obtained. The amplitudes of 2M+1 rectangular pulses are then calculated using A and s. coefficients. Lastly, s(t) is obtained by calculating the inverse Discrete Fourier Transform of the optimal s coefficients. The program listing of this algorithm is given in Appendix A and the optimal Nyquist signal samples obtained using this algorithm are given in Appendix B. The energy concentration was found to be greater than 95 %.
Açıklama
Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1992
Konusu
FIR filtre, Nyquist işareti, Parks-Mcclellan algoritması, FIR filter, Nyquist signal, Parks-Mcclellan algorithm