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In Wireless Communication, sometimes the received signal is expressed (in time-domain) as follows: $$r=hx+n$$ where $h$ is called a fading coefficient and sometimes it is expressed as: $$r=h*x+n$$ where * is the convolution operator and $h$ is called the channel impulse response.

My question is: Are the fading coefficient in the first equation and the impulse response in the second equation the same thing? If so, why we use the multiplication instead of convolution?

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The short answer is that in the first case the channel delay spread spans one symbol time duration or less (which is called frequency flat fading channel), and can be approximated by a complex number according to the central limit theorem, while in the second, the channel delay spread spans several symbol duration (which is called ISI or frequency selective channel).

EDIT1: Mathematically, the flat fading channel can be represented as $$h(t)=h_k,\,\,\,\text{ for } kT_s\leq t<(k+1)T_s$$ while in the frequency selective case it can be represented as $$h(t)=\sum_{n=0}^{N-1}h_n\,\delta(t-nT_s)$$ where $N$ is the number of symbols the delay spread of the channel spans, and $T_s$ is the symbol time duration.

Now suppose the transmitted signal is $s(t)$. The noise-free received signal for the first channel is $$r(t)=h(t)\star s(t)$$ which when sampled at time $kT_s$ for the $k$th symbol equals to $$r_k=h_k\,s_k$$ while in the second case it's $$r(t)=\sum_{n=0}^{N-1}h_n\,s(t-nT_s)$$ which when sampled at $t=kT_s$ equals to $$r_k=\sum_{n=0}^{N-1}h_n s_{k-n}$$ where for brevity I used the notation $s(kT_s)=s_k$.

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