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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.

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$y=hx+n$ 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.

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