It is often the case when the receiver requires knowledge of the channel impulse response (CIR) - e.g., the equalizer needs this channel state information (CSI) in order to remove the intersymbol interference (ISI). This knowledge is usually provided by a separate channel estimator at the receiver. Usually the transmitter sends a known sequence of bits, referred to as training signal, and the channel estimator exploits the knowledge of this sequence and of the received signal's samples in order to estimate the CIR.

Let's say that we have a channel with CIR h and additive Gaussian noise n. The transmitted training signal is expressed as a circular matrix S. Thus, the received signal is [Note: Lower-case denotes vectors, upper-case denotes matrices]

y = Sh + n

The estimation of the received signal is

y' = Sh'

Hence, the estimation error is

e = y - y' = y - Sh'

My question is: Why do we ignore the noise in the estimation y'? Should not that be

y' = Sh' + n' ?

I searched for days in Google, I found several web-pages, lecture notes, e-books, but no one addresses this matter. It seems that the reason is obvious, but apparently it is not obvious for me!

Any help, comment, suggestion or whatever would be really appreciated ...