LIKELIHOOD RECEIVER FOR FH-MFSK MOBILE RADIO*

We investigate the performance of a likelihood receiver for the detection of frequency hopped multilevel frequency shift keyed signals (FH-MFSK) to a mobile user operating in a multi user environment. The analysis assumes synchronous transmission from base to mobiles operating in an isolated cell-cellular system and a simplified mobile radio channel. The likelihood receiver attempts to discriminate spurious rows of the decoded matrix of a user, which consists of samples from an exponential-mixture, from the correct row, which consists of samples from a simple exponential density, by computing the log-likelihood statistic for each row. It declares the row possessing the minimum value as the correct row, corresponding to the word transmitted to the user. An approximate analysis of the probability of bit error of this receiver by three means, viz. (i) large sample approximation (ii) simple Chernoff bound and (iii) Chernoff bound with central limit theorem, reveals that the likelihood receiver is only marginally superior to a hard limited combining receiver. INTRODUCTION The use of FH-spread spectrum modulation in mobile radio has been well-known for quite some time. In [1], a FH-MFSK system has been proposed and the performance of a hardlimited combining receiver was evaluated, assuming an approximate but realistic mobile radio environment. We analyze here a likelihood receiver operating under similar conditions, i.e. a mobile user in an isolated cell, perfect synchronization, Rayleigh fading, receiver noise and multi-user interference as the transmission impairments. A non-coherent envelope-analyzer is used to estimate the energy in the received frequency tones [1, 2]. The random address assignment and the address decoding procedure are the same as in [1]. The likelihood receiver makes a decision based on the log-likelihood statistic. Applying (i) large sample approximation (ii) simple Chernoff bound and (iii) Chernoff bound with central limit theorem, we estimate the performance of the likelihood receiver to be marginally superior to a hard-limited combiner. It is believed that the (iii) method gives the closest estimate of probability of error. In section I we discuss the receiver structure for FH-MFSK modulation scheme. In section II we make an approximate estimate of the likelihood receiver performance. We conclude with discussion of the problem associated in finding an improved performance estimation of the likelihood receiver. I. RECEIVER STRUCTURE FOR FH-MFSK MODULATION Figure 1 shows a section of the non-coherent envelope analyzer. As in [1], let J be the chip duration, K be the number of bits of information transmitted every LJ seconds, W = 20 MHz be the one way bandwidth and R be the bit rate. Then, we have 2 such sections in operation corresponding to different orthogonal tones. Let , denote the envelope squared output at the i envelope analyzer after the j chip. Corresponding to either signal plus noise or noise only case, we have , to be either exponentially distributed with mean value (1/81) or exponentially distributed with mean value (1/80), respectively. A mobile user u receives the signals from the base and creates a decoded matrix every LJ seconds. In each entry of the matrix of size 2 x L, is present the output of the envelope analyzer, X. In general a receiver chooses a row as the row corresponding to the transmitted word, based on some decision criterion. II. LIKELIHOOD RECEIVER Receiver Description We shall assume that the minimum frequency spacing between the hops in the transmitted waves is larger than the coherent bandwidth of the Rayleigh fading channel. Even though such an assumption is not truly valid with the parameter values of interest, it is made for tractable analysis. This, then, implies that Xij are independent and exponentially distributed. Among the 2 rows in the decoded matrix, only one row is the correct row, wherein all the Xij’s have a mean value (1/81). In the rest of the (2 -1) spurious rows, some elements have a mean value of (1/80). A spurious row has contributions partly from the interfering users plus noise and partly from the receiver noise. On an average, each spurious row will have a proportion, p, of Xij’s created due to interference where p is given by p = 1 (1 2)M (1) and M equals the number of users operating in the cell. Since each row can be a spurious row (hypothesis H0) or not (hypothesis H1), we have the following testing problem applied to a k row: