# Representing data in combination of QAM and BPSK

I have a question regarding the data representation in different modulations. Assuming we are using MIMO system whith $$4$$ $$T_x$$ x $$4$$ $$R_x$$ which means four transmission antennas by four receiving antennas. My question, if three antennas are using QAM modulation, and the other antenna uses BPSK modulation. Is it possible the Receiving antennas can distinguish the data sent from each antennas ? I mean is it possible to detect the data sent from transmission antennas?

I'm asking that because in QAM modulation, data will be represented with real and imaginary parts, where in BPSK, data will be represented in real part only. So I see it logic that receiving antenna can detect the data perfectly. But I'm not sure.

I think that is possible. As you said, the three first antenna will be using QAM where the other antenna will be using BPSK. In that case, I advise you to read about Spatial modulation (SM), and Generalized Spatial modulation (GSM), Quadrature Spatial modulation (QSM) and finally the newest paper I've ever seen is F-QSM. All those papers are talking about MIMO, to be more specific, they are talking about the alternative approach of MIMO which is SM. If you want to go further, the same purpose was also implemented for OFDM resulting the OFDM-IM. It's a very good area to study and search.

In general, the new SM, is how to use the index of antennas in order to transmit more bits which will improve the spectral efficiency.

Regarding to the answer coming from Marcus !!, I think Adding up the QAM-based modulation with BPSK-based modulation will not happen as long as you are transmitting that on different antennas, "Exactly as you've said in your question". Secondly, the channels should usually be uncorrelated, so interference can be entirely avoided.

In such new articles, such that Quadrature spatial modulation, that can work as long as the signal transmitted on antennas are orthogonal. So, interference can be entirely avoided too.

You are good, just keep going on. and if you needed any help, you'd write it as a question here, many people are helping and they will try to help you too.

thank you.

• Your answer is wrong: In a MIMO channel, the transmissions of multiple antennas do add up. That's how MIMO works. And that's why you represent such channels as (non-diagonal) channel matrix, to be multiplied with the transmit signals vector. – Marcus Müller Dec 1 '18 at 16:47
• With Spatial Modulation you're bringing in a different topic. The whole beauty of spatial modulation is that you're getting effects that you'd usually only get when doing MIMO, but without having multiple simultaneously active transmit chains. If you use different modulations for different antennas, you're not only embedding bits into the antenna choice, but also the modulation choice. Not quite sure this is a smart move, because typically, you'd assume that all antennas are realizations of the same stochastic process that yields channels, so that all the things you'd decide which – Marcus Müller Dec 1 '18 at 16:51
• … constellation to pick would be the same (unless you have apriori CSI on the transmitter, which opens up a whole can of other questions about why you're using SM). So, one of these antennas would then be using an unnecessarily suboptimal constellation. I can't imagine a case where this would be desirable – if the channels are similar enough so that you need a different constellation just to tell the active antenna apart, why not do that for a single antenna and get even more bits across. Do you have a paper that quantifies the advantages of SM + different constellations? – Marcus Müller Dec 1 '18 at 16:54
• Why you always change the subject into MIMO !!! I said many times, in that case SM, GSM and others should be read. Anyway, that question might take long discussion, the New_student can search and asks others and let us know. I really can't explain more than I did. I talked about GSM, SM, and QSM. and you are still talking about SM. SM is different about GSM and QSM where just only one antenna is active, GSM and QSM one OR more antennas can be active. What I said that in his case, he should look at SM and its new versions instead of MIMO. what he said represents the case of QSM more than MIMO. – Zeyad_Zeyad Dec 2 '18 at 5:33
• I asked some professors at my university, They confirmed that in case of GSM and SM, that can be done but symbols must be transmitted in different time. they said in each instant the antennas are active, they should use one modulation. . I think I'll agree on that. – New_student Dec 2 '18 at 9:06

Let's us take the simple case $$2\times 2$$ MIMO. Let $$y_1$$, and $$y_2$$ be the received signal at the first and second receive antennas, respectively. Then we have

$$y_1 = h_{11}x_1 + h_{21}x_2 + n_1$$

and

$$y_2 = h_{12}x_1 + h_{22}x_2 + n_2$$

where $$h_{ij}$$ is the channel coefficient from transmit antenna $$i$$ to receive antenna $$j$$, $$x_i$$ is the transmitted signal from transmit antenna $$i$$, and $$n_i$$ is the AWGN at receive antenna $$i$$.

We can write the above signals in matrix-vector form as

$$\mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{n}$$

Apparently, there is interference here. So, we need to do some sort of equalization. Let's use zero forcing (ZF) for simplicity. In this case you get

$$\mathbf{H}^{-1}\mathbf{y} = \mathbf{\tilde{x}}=\left[\begin{matrix} \tilde{x}_1\\\tilde{x}_2\end{matrix}\right]$$

where $$\tilde{x}_i = x_i + \tilde{n}_i$$

for $$i = 1,\,2$$.

Now, suppose that $$x_1$$ is 4-QAM, and $$x_2$$ is BPSK, both have the same symbol time. Will that be a problem? and why? They seem separable to me, and can be detected given that the receiver knows in advance that $$x_1$$ is drawn from a 4-QAM constellation, while $$x_2$$ from a BPSK constellation. Actually we do this in OFDM. It's called bit loading, where different subcarriers are modulated with different modulation sizes, and schemes when the CSI is know at the transmitter. Of course in OFDM the symbols are separable because of the orthogonality of the subcarriers, but we can somehow separate the signals in spatial multiplexing, although not perfectly.

• Thank you so much .. I also asked professor, he replied me by the same thing. he said we can do it as long as the antenna are different, and he advised be to read about Dual-OFDM. he said it's new article, but I didn't get it :( .. you mean you agree with answer coming fomr Zeyad_Zeyad ?? – New_student Dec 2 '18 at 5:23
• @New_student I don't know about spatial modulation, which is different than spatial multiplexing by the way. My answer is different I believe. I am not sure how interference can be avoided completely in spatial modulation. For example I don't agree with this statement, at least as it is "I think Adding up the QAM-based modulation with BPSK-based modulation will not happen as long as you are transmitting that on different antennas, ..". As long as we transmit different signals on different antennas at the same time, interference is inevitable. He needs to elaborate on his answer more, I think – BlackMath Dec 2 '18 at 5:35
• Fully agreeing with you! Zeyad's answer as it's written is wrong, and the fact that (Q)SM is proposed doesn't remedy that at all. – Marcus Müller Dec 2 '18 at 9:53
• @MarcusMüller You commented "if you have multiple antennas transmitting at the same time, they should have the same modulation", why is that again? Why cannot you have different modulations on different antennas? They will add up, but you can separate them by equalization, and then detect each symbol based on the modulation scheme it used. I think my derivation shows that, doesn't it? – BlackMath Dec 2 '18 at 17:27
• @Zeyad_Zeyad If we transmit different symbols over different times, then that has nothing to do with MIMO. The question was about spatial multiplexing MIMO where we transmit different symbols (possibly from different constellations) over different transmit antennas at the same time. My answer is yes, given that the receiver knows the modulation scheme from each transmit antenna, and that all symbols have the same symbol time. – BlackMath Dec 3 '18 at 6:24

No, they can't simply tell things apart. How should they?

On the air, the transmitted QAM-based signal adds up with the BPSK signal, and even in the exact symbol times, you get numbers with real and imaginary part.

Also, you forget that signals still are pulse shaped; a sensible pulse shaper can have a complex part when going from one BPSK symbol to the other.

What you describe has nothing to do with how MIMO works! MIMO works because for uncorrelated channels, you can find a matrix decomposition of your $$\mathbb C^{N_{TX}\times N_{RX}}$$ (flat channel case) matrix, not because the signals are different. With that matrix decomposition, you can find "separable" channels, even though the signals add up "on the air" and you just receive a mishmash of all TX signals at each RX antenna (though the amount of the TX signals in that mishmash has to be different for each RX antenna).

Because Zeyad_Zeyad introduces SM but doesn't agree on whether in a MIMO channel, signals from different antennas sum in the air:

The typical MIMO channel is modeled like this:

With each RX antenna seeing a linear combination of all TX signals; hence, the representation as channel matrix $$\mathbf H$$

$$\mathbf H=\begin{pmatrix} h_{11} & h_{12} & h_{13}\\ h_{21} & h_{22} & h_{23}\\ h_{31} & h_{32} & h_{33} \end{pmatrix}$$

so that you can get the first receive signal as the first entry, the second as the second entry, … of

$$\vec r =\mathbf H \vec t\text,$$

where $$\vec t$$ is the transmit signal vector (i.e. the first entry is the symbol at TX antenna 1, the second at TX antenna 2 …).

As you can see, if say $$t_1$$ is real but $$t_2$$ or $$t_3$$ have an imaginary part, $$r_1$$ can very well be complex.

So, as you can see, for MIMO to work, you don't need the signals to have different characteristics – you need the channel coefficients $$h_{i,j}$$ to be independent enough!

Please allow me a personal note from the perspective of someone who helps students survive exams in "Digital Communications Basics" courses at university, @Eng. Badr:

In your previous questions, you were wildly jumping from extremely involved topics like FBMC to more entry-level topics like CDMA then down to the very, very basic of QAM vs PSK (which you had a great misunderstanding about!). Now you're back to a topic that is relatively complex (MIMO) and requires solid understanding of the things like constellations.

You sound like a student who's either preparing for an exam for which they are missing the basic course, or like someone doing a thesis on a topic where they have to acquire basic knowledge by themselves. In either case, jumping into topics, asking a single question, and then jumping to the next topic will not enable you to understand much, and cost very much of your time, without enabling you to do your own deductions. Especially in the thesis case, that's a hazard, because at some point, someone will point at a blaring mistake in what you're doing and say "ok, every person who works on {topic} should know that this can't work, it's really basic".

So, do yourself the favor of getting a digital communications textbook that takes you from the basics of complex baseband over constellations, pulse shaping, a tiny bit of channel coding, over to multi-access schemes like FDMA and CDMA and finally MIMO. I promise that in the medium run, reading that book from start to finish will save you time and effort and, if my hypothesis on the situation you're in is right, will definitely improve the grade you'll be getting.

• They are not going to be added up since the transmission is on different antennas. Clarify your answer . !! – Zeyad_Zeyad Dec 1 '18 at 15:56
• But they are. That's the beauty of MIMO. – Marcus Müller Dec 1 '18 at 16:46
• Yes, after those modifications you have done, It seems little be ok. I might agree with you. but excuse me to tell you that your answer before that modification is completely wrong. – Zeyad_Zeyad Dec 2 '18 at 9:09
• What. No. That's not true, sorry. You can go into the edit history and see that I didn't change anything. I just copy/pasted the definition of the MIMO channel here. – Marcus Müller Dec 2 '18 at 9:17