# How to understand the relationship between complex baseband equivalent model and block-fading channel model?

Considering a bandpass communication system, where $$y(t)=x(t)*h(t)+n(t)$$ ($$x(t)$$ is the transmitted bandpass symbol, $$h(t)$$ is the channel impulse response, $$n(t)$$ is the AWGN, and "$$*$$" means convolution). I can write its baseband equivalent model as $$v(t)=u(t)*h_l(t)+n_l(t)$$, where $$v(t),u(t),h_l(t),n_l(t)$$ are the complex baseband counterparts. But for block-fading channel (i.e., the channel gain remains constant over many symbols), there is also a representation $$y=hx+n$$, where $$x,y,h,n$$ are all complex-valued. I am confused about how I can intuitively or rigorously deduce $$y=hx+n$$ from $$v(t)=u(t)*h_l(t)+n_l(t)$$. In other words, are there any correspondence between the signal, channel, and noise terms between these two representations? Moreover, is $$x$$ in the block-fading representation the same as "constellation point" of a transmitted symbol in essence? They both seem to be complex-valued, but I am confused about their relationship. Thank you.

$$y=hx+n$$ describes a memoryless channel, i.e., a channel with unlimited bandwidth. With such a channel, an output value $$y$$ is dependent on the current input symbol $$x$$ only.
On the other hand, $$v(t)=u(t)∗h_l(t)+n_l(t)$$ describes a more general case where the channel may have memory, i.e., the channel with limited bandwidth. With such a channel, the output value of $$v$$ is also dependent on previous and/or future symbols, in addition to the current symbol. For example, a channel with inter-symbol-interference (ISI).
For a memoryless channel, $$h_l(t)$$ in the second expression is a $$\delta (t)$$ function with a scaling. In such case, the expression $$v(t)=u(t)∗h_l(t)+n_l(t)$$ degenerates to $$y=hx+n$$, and both expressions are the same.