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The problem of communicating sensor readings over a multiaccess channel for detecting a target is considered. A natural way of communication in target detection is to let sensors simultaneously transmit one of two predetermined frequency tones indicating whether the target is detected or not. Recently, this scheme has been generalized to consider non-binary sensor observations by letting sensors simultaneously transmit orthogonal waveforms depending on the value of their observations-type-based multiple access (TBMA). TBMA was shown to be asymptotically optimal in terms of detection-error probability under the idealistic assumptions that the sensor channel gains are identical, and the sensor data are conditionally independent and identically distributed (i.i.d.). In this paper, TBMA is analyzed in a more general framework by considering non-i.i.d. data and non-identical channel gains. An asymptotically optimal detector is proposed and its error-exponents for detection probabilities are characterized using tools from large deviations theory. Numerical simulations are used to demonstrate that the error exponents provide reasonably accurate estimates of the performance of TBMA.