# Multiplication and Convolution in Wireless Communications

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?

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 $$s(kT_s)=s_k$$.