I thought in the wrong direction, the half is needed only to normalize the channel…
The first project of the course required me to simulate a narrowband fading channel using Jake’s model. I thought that for many days but didn’t come with a solution.
Finally I understood that it is a problem about the understanding of the word “fading”. Fading is due to multi-path, and according to the previous course, multi-path communication will made the channel to be dispersive. However this is not the case for narrow band. In fact, narrow band means that the channel is approximated to be memoryless (neglectable inter-symbol interference), and different paths will “fade out” each other (by differing a phase) at that exact moment. The autocorrelation of the channel is to describe the “change of the channel itself in the time domain”, not “how the source signal are correlated with each other”. In other words, the received signal are correlated not because of inter-symbol interference, but because the channel is not time invariant. That’s why we made the assumption that the source is unmodulated.
Thus, the channel function h(t) means the change of the channel in the time domain, not the impulse response as we normally may think.