# Demodulation in Rayleigh Flat Fading Channels

A core part of demodulation in Flat-Fading channels is (assuming) the channel as a(random) constant complex number multiplied with your signal. This is done in order to de-rotate the symbol for decision making at the sampling time.

I'm wondering how you go about estimating what the complex number is? How do you estimate the phase change induced by the channel?

The short answer is: pilot symbols.

The long(er) answer is: The sender periodically transmits known data symbols; this sequence is known to the recipient ahead of time.

• Thanks, one other thing - the receiver doesn't actually receive a complex number, this terminology is referring to the in-phase and quadrature components right? So when we are referring to multiplying by a complex number, we are actually multiplying the in-phase and quadrature components by two separate real numbers? – FourierFlux May 20 '18 at 21:44
• @FourierFlux-- in many cases, the processing is actually done with complex numbers once the signal has been digitized. For instance, GNU Radio represents baseband signals in this way. – Robert L. May 21 '18 at 1:20

Before starting to transmit data and when the receiver uses coherent detection, a known pilot sequence is used at the beginning of each coherence time to estimate the channel.

Let the received signal over flat block fading channel be

$$y = h\,x + n$$

where $$h$$ is the channel coefficient, $$x$$ is the transmitted symbol, and $$n$$ is AWGN. In the above equation, when $$x$$ is known, the received signal signal is a noisy version of the channel coefficient. A simple way to estimate the channel is to use zero forcing. Mathematically

$$\hat{h}=\frac{y}{x}$$

More robust estimation schemes exist, but this is the idea behind channel estimation. At the beginning, the transmitting symbols are known but the channel is not, in the remaining part of the coherence time, the transmitted symbols are unknown (at the receiver), but the (estimated) channel is known.