Transient probability response of cycle slip occurrence ... - IEEE Xplore

Transient probability response of cycle slip occurrence in the PSK-TDMA carrier recovery T. Fujino, B.S., M.S., Y. Umeda, B.S., and E. Yamazaki, B.S., M.S., Dr.Eng. Indexing terms:

Telecommunication, Radiocommunication

Abstract: The class of carrier recovery which is often employed in PSK-TDMA systems is 'frequency multiplier/narrow-band filter/frequency divider.' Cycle slips occur when a signal extracted from the narrowband filter is embedded in some noise. In the PSK-TDMA systems, a transmitted signal S is of the burst form while received noise N is in the steady state. A preceding burst signal / is often allocated immediately before the burst signal S. In this paper, the transient occurrence probability of the cycle slips is analytically derived, assuming that the products of the multiplier, which are a burst signal (S x S) and a preceding burst signal (/ x /) as well as a steady-state noise (N x N), a burst noise (S x N) and a preceding burst noise (/ x A/), are coexistent at the narrow-band filter output. Also the transient cycle slip occurrence probability is numerically calculated using the results of the analysis.

Introduction

1

The class of carrier recovery which is often employed in PSK signalling systems is 'frequency multiplier/narrowband filter/frequency divider', as shown in Fig. 1 [1, 2, 3], narrow-band frequency - divider filter FnQr

frequency multiplier

receive filter Fnr

\ \ I

r

L

tl ) !

A

I (S): burst signal of interest I (SxS):burst signal of interest

jI (l):preceding burst signal) | (N): steady-state noise

Fig. 1

steady-state noise (N x N), a burst noise (S x N) and a preceding burst noise (/ x N) are produced by the multiplier and then supplied to the narrow-band filter F n a r . Here the burst noise (S x N) and the preceding burst noise (/ x N) are naturally associated with the burst signal of interest (S x S) and the preceding burst signal (/ x /), respectively. A typical illustration of the set of these burst signals and the classes of noise is presented in Fig. 2.

( I x l ):preceding burst signal (SxN):burst noise ( I x N ) : preceding burst noise i (Nx N):steady-state noise

preceding burst signal (1)

steady-state noise(N)

Typical carrier recovery system

where the multiplier consists of a memoryless nonlinear device. Cycle slips occur when a signal extracted from the narrow-band filter Fnar is embedded in some noise. The phenomenon of the slip is essentially the same as that of the FM click. Rice has already presented its occurrence probability in the steady state [5]. In TDM A systems, a transmitted signal S is of the burst form while received noise N is in the steady state. When they enter the frequency multiplier in Fig. 1, there appear at its output a burst signal (S x S), a burst noise (S x N) and a steady-state noise (JV x N) [1, 6]. In case a preceding burst signal / is allocated immediately before the burst signal S, the multiplier produces at its output a preceding burst signal (/ x /) and a preceding burst noise (/ x N) in addition. A set of these burst signals and the classes of noise, when narrowbanded by the Fnar, are superposed in the vicinity of the burst edge, and the burst signal of interest (S x 5) is interfered by the preceding burst signal (/ x /). With respect to such a burst operation, it is important to express the occurrence probability of the cycle slips in the form of the transient response. In this paper we will analytically derive the transient occurrence probability of the cycle slips. The probability will be expressed as a function of the time counted from the burst head, assuming that a burst signal of interest (S x 5) and a preceding burst signal (/ x /) as well as a Paper 3490F (E8), first received 17th October and in revised form 15th May 1984 The authors are with the Information Systems & Electronics Development Laboratory, Mitsubishi Electric Corporation, 325 Kami-Machiya, Kamakura City 247, Japan

IEE PROCEEDINGS,

Vol. 131, Pt. F, No. 7, DECEMBER 1984

burst signal of interest (S)

preceding burst signal (Ixl)

preceding burst noise (ixN)

burst signal of interest (SxS)

burst noise (SxN)

steady-state noise(NxN)

(Ixl)

(IxN)

(SxN) (NxN)

Fig. 2

Illustration of a typical set of signals and classes of noise

a At the multiplier input point b At the multiplier output point (narrow-band filter F M r input point) c At the narrow-band filter F M r output point

729

Also, some numerical calculations for the transient cycle slip occurrence probability are made using the results of the analysis. The analysis results obtained should be useful for determining the preamble length of bursts and the Fnar's bandwidth for modems in those PSK-TDMA systems that restrict the cycle slipping rate performance for the recovered carrier in the continuous mode [10]. 2

Analysis

Fig. 3 shows the analysis model. Let F(jco) be the equivalent low-frequency transfer function of the narrow-band P u(t)

I

j

= P,{1 - Ec(t)} cos K t + ,).

(8)

2.3 Steady-state noise (N x N) Let nA and nB be the steady-state noise observed at the F,,flr input and output points, respectively: nA(t) = xA(t) cos coct — yA(t) sin wc t

(9)

nB(t) = xB(t) cos (oc t — yB(t) sin coc t

(10)

where xA, yA and xB, yB are equivalent low-frequency noise. The autocorrelation function for xB or yB is expressed [7] as

P E c (t) narrow-band filter Fnnr

where Pt denotes the amplitude of the carrier, and A( = (f)j — 0O) distributes uniformly on [ — n, 7t]. The outgoing signal is expressed as

xB .y B

dt

RB{t,, t2) = E\_xB{ti)xB{t2)] h(t)

= ElyB{ty)yB{t2)'\

6"(t)

= RA(h,t2)*h(tl)*h(t2) h'(t) Fig. 3

where RA(tu t2) is the autocorrelation function for xA or

Analysis model

filter F n a r . Assume that both the amplitude and the group delay performances of the transfer function F(jco) are even functions of the angular frequency co. Then the impulse response h(t) for F(jco) is a real function with zero value for t < 0; and the following relationship holds: hln\t) =

(1)

where at least the first order derivative of h(t) is assumed to exist (n = 0, 1). The one-sided noise bandwidth BNo and the RMS bandwidth Brms for the Fnflr are defined as (2)

B

"° ~ 2 I.

(11)

(3)

Assume that Bm ^> BNo (where Bm = one-sided noise bandwidth of Frec). Then, x^, yA can be regarded as white noise, and hence the autocorrelation function RA(tlf t2) can be expressed as

where Ns is the power spectral density for xA or yA. Substituting eqn. 12 into eqn. 11, we have ± RB(x) = Ns

h(v)h(v + T) dv =>