# Wireless Channel Multiplicative Model

Let's assume a SISO channel in the equivalent baseband, the standard channel model (discrete) is: \begin{align} y(k) = h(k)x(k) + z(k) \tag1 \end{align} Where $$y(k)$$ is the detected signal, $$h(k)$$ denotes the channel, $$x(k)$$ is the sent signal and $$z(k)$$ is additive noise. If we consider reflections on buildings, cars, etc., why isn't there a convolutional model like in acoustics: \begin{align} y(k) = h(k)*x(k) + z(k)\text?\tag2 \end{align}

What's labeled $$(1)$$ in your question is a special case, the flat channel. It can be represented as a single coefficient.

In general, channels aren't flat, and we then need to apply $$(2)$$ instead. That's no different from acoustics.

So, your claim that $$(1)$$ generally applies to SISO channels is plain wrong.

However, when naming something "SISO", one usually means "in contrast to MIMO"; and in the context of MIMO, we basically need our channels to be flat, so that the MIMO channels can be represented as a $$m\times n$$ matrix of single coefficients between $$m$$ and $$n$$ transmitting/receiving antennas, respectively.¹

But, you say,

if the individual channel is like $$(2)$$, how can we model it like $$(1)$$?

We can't. But we can apply multi-carrier techniques (mostly, OFDM), to first divide a wideband channel with a convolutive channel impulse response into many narrow channels, which individually are flat.

¹ In theory, one can do MIMO on multipath / frequency-selective channels instead of first dividing them in many flat subchannels, but as far as I know that's not done commercially. OFDM implementations are mature and cheap, and it's much easier to come up with working math that allows us to decompose the superpositioning MIMO channel when it's flat. In fact, there's no standard textbook I'm aware of that even does MIMO on frequency-selective channels.

• Thank you for your detailed answer! – ulfgar Jan 26 at 13:05