Spectral Analysis Of RLL Codes Modulated By FSK Or CPM

Abstract

In this paper, the method developed in [l] is generalized to the spectral analysis of RLL codes modulated by FSK or CPM. It shows that by using the formulae given in this paper, the state size involved in the spectrum computation can be largely reduced compared with the traditional method, hence the burden in the spectrum calculation can be much released. I .Introduction I t is known that RLL codes are attractive at their spectral shaping and self clocking properties. Modulated RLL codes may be the candidates for mobile communications and satellite communications. Since the one step FSSM representation for the RLL codes involves a very large size of states [2], it makes the spectral analysis formidable by using the known method [3,4,5 361. As shown in [l], a run-length diagram can be associated with RLL codes. that will largely reduce the number of states involved in the representation of the structure of the codes. By then, the base band spectrum can be calculated by a much simple way as shown in [l]. In this paper, the method proposed in [l] is generalized to the modulated RLL codes. It shows that the spectrum calculation can be simplified for modulated RLL codes as well when the run-length diagram is adopted. II . RLL Codes Modulated By FSK or CPM A RLL code [7] is a set of binary sequences that satisfy the (d,k) constraint. Any 1 binary sequence, denoted by {c,] of finite or infinite length can be parsed uniquely into a concatenation of phrase, each phrase ending in a single "I" and beginning with a string of none or one or more "OS". A binary sequence is said to satisfy a (d,k) constraint if and only if all its phrase contain no less than (d+l) digits and no more than *+I) digits. Thus "d" refers to the minimum number of 0's in a phrase and "k" refers to the maximum number of 0's in a phrase. A (d,k) constraint RLL code can be generated by a finite state sequential machine (FSSM). And a directed graph, called run-length diagram can be used to represent the FSSM [l]. Each of the directed edge connecting state sj to si has alabel . Where pi,j and are the state transition probability and the phrase length in that state transition respectively. For each binary RLL sequence, there is a input signal process [A,], which takes value in { k I), associated with it. The input sequence [A,] satisfies that: A,= 1-2b,; b,= b,-@ c,. In other words, the input sequence has same pulse length as the length of phrases in the binary (d,k) sequence {c,], and has an alternating polarity. There are many ways to modulate a RLL code. By using BPSK to modulate the input sequence is the simplest one. Since only frequency shift is involved in the BPSK modulation, the spectrum of BPSK modulated input sequence is the same as the spectrum of baseband RLL code. Hence many results can be found in literature [1,6]1. In this paper, FSK modulated input signal process will be addressed. In this modulation scheme, different frequencies is assigned to different length of phrases. To avoid the discrete component in the spectrum, the polarity of the modulated signal is alternated phrase by phrase. For example, a run-length sequence ll,lZ, ..., ln,,.., will be modulated into cos(2nfl,t)P,,(t),-co~(2nf~(t-Tll))P,(t-Tll), ...,(l)"cos(2nfh(t-T(~1+l2+ ...+ l(n-1))) P,(t-T(ll +I2 +. . .+l(n-l))),.. ., where O < t < T , other w ise . If the modulation frequency is chosen properly as: fl = f,,+l/ (21T) (3) then the modulated waveform will have continuous phase. It will be refered to 2 -