In this paper we investigate some properties of the union bound for a basic MIMO system, assuming uncoded QAM transmission, uncorrelated Rayleigh fading, and maximum likelihood detection. The presented union bound results cover several choices of the number of transmitting and receiving antennas (up to 16). It is also illustrated how the pair-wise error probability approaches the Q( )-function as the number of receiving antennas increases

The bit error probability properties of signals consisting of convolutional coding combined with partial-response multilevel continuous phase modulation (CPM) are studied. It is assumed that the channel is an additive white Gaussian noise channel and that the receiver performs coherent maximum-likelihood sequence detection by means of the Viterbi algorithm. An upper bound on the bit error probability is derived, using the average generating function technique, and evaluated numerically for a number of coded multilevel full-response CPM schemes. Simulation results are also presented. It is concluded that the free Euclidean distance is the best one-parameter description of the error probability for the considered class of signals for high signal-to-noise ratios. However, the upper bound results show that the free distance alone yields pessimistic bit error probability behavior for some interesting cases.>

Continuous phase frequency shift keying (CPFSK) is a constant amplitude modulation method with good spectral sidelobe properties. Good error probability properties can be obtained with coherent maximum-likelihood detection. In this paper we study the Euclidean distance properties of signals formed by a conventional rate1/2convolutional encoder followed by a binary or4-level CPFSK modulator. The minimum Euclidean distance is calculated for these signal sets as a function of the modulation index and the observation interval length. The optimum detector is discussed for rational modulation index values. The best obtainable codes are found for the case of short rate1/2codes with binary or4-level CPFSK modulation. Lists of the best codes are given. Among the results are that the noncatastrophic rate1/2convolutional codes with optimum free Hamming distance do not in general give the best Euclidean distance with CPFSK.