# Wake-Up Receiver with Equal-Gain Antenna Diversity +
## 2 Background  
Deep and rapid amplitude fluctuations caused by the close surroundings of a receiver are called small-scale fading. In wireless propagation, a signal from an antenna reaches another antenna over several paths with their associated path lengths and attenuations. As a result, many copies of one signal reach the antenna after different delays, where they superimpose each other destructively or constructively [13]. If the bandwidth \(B_{s}\) of a transmitted signal is smaller than the bandwidth \(B_{C}\) over which a wireless channel has a constant gain and a linear phase response, the transmitted signal will experience flat fading [14]. A flat fading signal is a locally coherent signal. The symbol amplitudes of a transmission will change over time due to flat fading, as much as 30 dB [14]. On the receiver side, an automatic gain control (AGC) can be used to mitigate flat fading effects [15], but, more so, diversity techniques help to mitigate small-scale fading effects [16]. Assuming sufficiently separated signals in time, frequency, space, or polarization, they are independent of each other [17]. Hence, if one antenna does not receive a signal along one branch due to small-scale fading, the other uncorrelated antenna will likely still receive the signal along a diversity branch [16]. According to [17], a diversity system is a system that provides two or more similar copies of the same signal. For example, a receiver detects signal \(f_{1}(t)\) and stores it locally. Then, the receiver detects signal \(f_{2}(t)\). By combining \(f_{1}(t)\) and \(f_{2}(t)\), that is \(f(t)=f_{1}(t)+f_{2}(t)\), the signal quality can be improved. Resulting from additional noise present in the wireless channel, signal \(f_{1}(t)\) is a combination of the desired signal \(s_{1}(t)\) and a noise signal \(n_{1}(t)\), that is \(f_{1}(t)=s_{1}(t)+n_{1}(t)\), and likewise is true for signal \(f_{2}(t)\). Therefore, \(f(t)=s_{1}(t)+s_{2}(t)+n_{1}(t)+n_{2}(t)\) is a combination of the desired signal and the present noise. In a diversity system, \(s_{1}(t)\) and \(s_{2}(t)\) are similar to each other and combining \(s_{1}(t)\) and \(s_{2}(t)\) increases the desired signal amplitude. Assuming \(n_{1}(t)\) and \(n_{2}(t)\) to be independent additive white Gaussian noise signals having zero means, they partially cancel out each other. Consequently, the SNR in \(f(t)\) may increase due to diversity. More generally, a diversity system may combine \(n\) signals \(f_{i}(t)\) that have, due to the uncorrelated channel gains, different signal amplitudes \(a\) as described by Equation (1) [17]:  
\[f(t)=\sum_{i=1}^{n}a_{i}f_{i}(t). \tag{1}\]  
Combining the outputs of several antennas can increase the detected signal strength at a receiver [18] to improve communication robustness and reliability. There are existing several diversity techniques that can be utilized. The example above describes time diversity as the two signals \(f_{1}(t)\) and \(f_{2}(t)\) reach the detector at different times. Separating receiver antennas in space achieves independent signal paths between sender and receiver. This technique is called space diversity. In frequency diversity systems, signals experience uncorrelated fading resulting from separate transmit frequencies. Furthermore,in polarization diversity, signals are received either along the horizontal or the vertical polarization. Polarization diversity has the advantage that it requires no spatial separation of the antennas [16]. Combining multiple versions of the same signal \(f_{i}(t)\) according to Equation (1) is common to the above-mentioned diversity methods. However, if the multiple versions \(f_{i}(t)\) are not in phase during combination, the resulting signal amplitude could even be lower than the originally transmitted signal amplitudes due to destructive interference. Diversity systems usually take care about this by using some sort of a phase-control technique [17].