A distributed detection problem over fading Gaussian multiple-access channels is considered. Sensors observe a phenomenon and transmit their observations to a fusion center using the amplify and forward scheme. The fusion center has multiple antennas with different channel models considered between the sensors and the fusion center, and different cases of channel state information are assumed at the sensors. The performance is evaluated in terms of the error exponent for each of these cases, where the effect of multiple antennas at the fusion center is studied. It is shown that for zero-mean channels between the sensors and the fusion center when there is no channel information at the sensors, arbitrarily large gains in the error exponent can be obtained with sufficient increase in the number of antennas at the fusion center. In stark contrast, when there is channel information at the sensors, the gain in error exponent due to having multiple antennas at the fusion center is shown to be no more than a factor of 8/π for Rayleigh fading channels between the sensors and the fusion center, independent of the number of antennas at the fusion center, or correlation among noise samples across sensors. Scaling laws for such gains are also provided when both sensors and antennas are increased simultaneously. Simple practical schemes and a numerical method using semidefinite relaxation techniques are presented that utilize the limited possible gains available. Simulations are used to establish the accuracy of the results.
I. INTRODUCTIONSensors are becoming commonplace in factories, environmental, and home appliance monitoring, as well as in scientific study. In many such applications, a number of independent sensors each make a local observation, which are transmitted to a fusion center (FC) after limited initial processing at the sensors, and combined at the FC to calculate a global decision [1]. Sensors may adopt either a digital or an analog method for relaying the sensed information to the FC. The digital method consists of quantizing the sensed data and transmitting the digital data over a rate-constrained channel [2]. In these cases, the required channel bandwidth is quantified by the number of bits being transmitted between the sensors and the FC. In contrast, the analog method consists of amplifying and then forwarding the sensed data to the FC, while respecting a power constraint [3], [4]. The transmissions can be appropriately pulse-shaped and amplitude modulated to consume finite bandwidth. The channels between the sensors and the FC can be orthogonal, in which case, the transmissions from each sensor are separately received at the FC [2]. On the other hand, with multiple-access channels between the sensors and the FC, the noisy sum of all the transmissions are received at the FC to make a decision [3], [5]-[8]. The bandwidth requirements of sensor networks with orthogonal channels scale linearly with the number of sensors, whereas, when the channels are multiple-access, transmissions are simultaneous ...