# SIMO systems diversity and matched filter response

Let's assume we have a SIMO system with multiple transmitter antennas (e.g. qty 4), and a single receiver or 4x1. Let's suppose we're talking about the SSR steady-state response, Gaussian noise, and standard LTI system as in mobile communications. Let's suppose the receiver knows precisely the channel responses for each 4 MIMO paths, e.g. $$h_1$$, $$h_2$$, $$h_3$$, $$h_4$$. Let's assume the transmitter does NOT use pre-coding and does not know the channel parameters; only the receiver does, and there is no way to send CSI to the transmitter.

It is my understanding that in this scenario it is NOT possible to obtain diversity benefits. Specifically, if the diversity rank is K (or 4), it is not possible to obtain BER ~ SNR ^ (-K) in this scenario.

I just wanted to confirm the reason. I saw a discussion that the receiver is not able to form a matched filter response independent for each path even knowing $$h_1$$, $$h_2$$, $$h_3$$, $$h_4$$., but need an intuitive explanation. I assume it has something to do with only receiving a combined signal as opposed to a matrix, but wanted to double check.

• Confusingly, you seem to be describing a MISO system and then call it SIMO. Then even more confusingly, it becomes a MIMO system! Are you sure you're using the right terms here? Commented Feb 12 at 2:40
• I ask because "diversity rank" is not a usual term. Either we speak of "diversity gain", or we speak of "diversity order", or we speak of the "channel matrix's rank", and all three are different things. I'm taking it from the context of your question that you ask about diversity order, but I feel like I might be taking to a beginner who thinks similarly sounding concepts are the same. If that's the case: nope, they aren't! (And nothing here has rank 4!) Commented Feb 12 at 2:52
• Yes I agree that there is no diversity rank 4 here, because min(M,N) is 1 , not 4. Commented Feb 12 at 7:31
• Yes, you're right, it's a MISO system. I later said MIMO to refer to paths, although it's not a MIMO system. Sorry for the confusion on the rank, diversity gain, and order. Essentially, I'm looking for an intuitive explanation why we don't get diversity gain in this case, even though the receiver knows h1, h2, h3, h4. I'm intuitively understanding that it has something to do with not having an independent matched filter for each, so we don't get to multiply probabilities hence we don't get the Prob-error ~ (SNR) ^ -K , but still looking for a bit more detail. Commented Feb 12 at 7:38
• @sinabala You seem to know what you're talking about :) I would appreciate any comments on the MISO system I described. It is one of the examples featured in a DSP communications book, and I'm trying to understand better. Any help would be much appreciated. Commented Feb 15 at 20:15