I am giving a simple reasoning to understand the number of antennas at Tx and Rx. It is well known that MIMO system is used either for spatially multiplexing (different symbols from each antenna) or for diversity (same symbol from each antenna) or a combination of both (giving rise to diversity multiplexing trade of).

The processing of signal can be done both at transmit side as well as receive side.

Let us denote M transmit antennas and K receive antennas.

Denote transmit symbol vector by [X1, X2, ..XM] and received signal vector by [Y1, Y2,...YK].

Received signal at antennas:

Y1 = H11X1+H12X2+H13X3....H1MXM,

Y2 = H22X2+H21X1+H23X3....H2MXM,

Y3= H33X3+H31X1+H32X2....H3MXM, and finally

YK= HKKXK+HK1X1+HK2X2....HKMXM,

We arrive at simple linear equations and we know everything about it from school algebra. It is noted that we have to decode X1, X2,...XM to avoid interference from other transmit antenna for different X1, X2,...XM.

1. Given a certain number of transmitting antennas what is the minimum number of receiving antennas needed in a MIMO system for correct signal detection?

Ans:

a) Multiplexing scenario:

It means all Tx antenna transmit different symbols.

Given a certain number of transmit antennas means number of variables X.

To solve a simultaneous linear equation, number of equations must be greater than or equal to number of variables i.e K>=M. Hence, minimum is equal to number of transmit antenna.

b) Diversity scenario:

Diversity means same symbol X from all Tx, hence X1= X2.=XM (only one unknown variable). In this case a single receive antenna is enough.

c)Diversity multiplexing scenario:

It means some antenna (Say M1) of Tx chain will transmit same symbol while some others (M-M1) transmit different symbols. Number of unknown variables in this case would be M-M1+1.

In essence number of receive antenna denotes the number of different variables sent from transmit side.

2. How can I derive the number mathematically (or by some other means)?

Ans:The above description seems sufficient.

3. Why can't the number of receiving antennas be less than the number of transmitting antennas? Any proof? Can someone direct me to some papers that discuss any of these issues?

Ans: See Scenario b and c described above.

Now the above description was related to the receive processing. What about transmit processing? Transmit processing is very similar to the above.

Here we fix number K.

a) Multiplexing scenario:

Here we pre-code ie. linear combine symbols [X1, X2, X3...XM] before sending.

Why we precode? It is done to minimize the interference at the receiver. It is obvious that minimum transmit antenna should be equal to K since each antenna of transmit side corresponds to each antenna at the receive side.. Moreover, to avoid interference at receive antenna 1, the transmit symbols from all other antenna intended for K-1 antennas should be linear combined. Hence number of antenna at transmit M>=K

a) Diversity scenario:

Since same symbols from each antenna is required, single transmit antenna is enough for receive only diversity. However, more antennas at transmit side are beneficial for diversity as well as capacity enhancement (capacity is related to minimum of number of antennas).

c) Diversity multiplexing scenario:

Tx antenna should be equal to one greater than number of different variables used in [X1, X2, X3...XM].

In essence, if we realize that MIMO gives rise to simultaneous linear equations, it is very easy to understand the process as well as number of antennas.

Just write the equations and see.

Noted that number of different symbols are variables while each antenna (either at transmit or receive side) acts as equations. Thanks for reading the whole !!!

Dear Kwasi Adu-Boahen Opare, The article entitled "Theoretical Results for Fast Determination of the Number of Antennas in MIMO Transmission Systems" is accessible via link provided below. Hope it will be of your use.

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.73.9820&rep=rep1&type=pdf

You can have 1000 transmit antennas and 2 receive antennas, and that would still be a MIMO system. You can only spatially multiplex two signals in this case, but you can use the large number of transmit antennas to achieve a huge array/beamforming gain.

Perhaps you can clarify your problem formulation, because in general you would pose the problem in a different way: Given a certain number of transmit antennas and receive antennas, how should you design the transmission to maximize the data rate. An answer to this question is given in the seminal work by E. Telatar:

Capacity of Multi-antenna Gaussian Channels

http://mars.bell-labs.com/papers/proof/

For MIMO the term antenna is not adequate. We have to consider complete transmitter module(TX) and complete receiver module (RX). The number of antenna describe just spatial diversity which is different from MIMO concept. The number of antenna must be at leat 2 at the emitter and 2 at the receiver. The optimisation of the number of antenna is not the same if you use MIMO system in the spatio-temporal coding scheme (like Alamouti codes) or for spatial multiplexing scheme with or without a knowledge of the propagating channel.

Thank you all for your answers. I was interested in the MIMO concept (as applied to multiple antennas or spatial diversity). @Emmanuel, your answer helped me to know that MIMO concept is widely applicable in other contexts, beyond spatial diversity and multiple antennas. thanks.

Afaq and Emil, thanks for the links. I realized that, capacity increases proportional to the minimum of Nr (number of receive antennas) and Nt (number of transmit antennas) . That means, if I transmit with 3 antennas, and I have two receiving antennas, my capacity gain will not be more than a factor of 2. Does this fact mean that only two of the three transmitted signals will be detected, as mentioned by Neeraj?

@Neeraj, what if the channel state information is perfectly known at the receiver? Can I then detect the signals even if the number of receiving antennas are fewer?

Thanks.

The receiver needs to know the channel (perhaps not perfectly but partially) to benefit from having multiple antennas. This is one of the assumptions in the work by Telatar and most other research papers.

You are right that the capacity grows roughly proportional to M = min( Nt, Nr ), which is known as the prelog factor, multiplexing gain or degrees of freedom. This means that if C_1 is the capacity of a single-input single-output (SISO) link, then the MIMO capacity behaves roughly as M*C_1 + A. The additive parameter A increases with N_t and N_r (and not only M), thus having N_r > N_t improves A and but not M.

If you have 3 transmit antennas and 2 receive antennas, then it all depends on the channel knowledge available at the transmitter. If the transmitter has perfect CSI, it spatially multiplex two streams of data and the receiver can jointly decode both of them. If the transmitter has no CSI, the capacity-achieving strategy is to send three data streams but only decode two of them. However, nothing will be lost because the information is coded such that only 2/3 of it is needed for decoding.