Chapter 9

This chapter is an introduction to Multiple Input Multiple Output radio systems. It presents the underlying fundamentals of MIMO, with some theoretical aspects as well as some practical applications in modern wireless standards making use of MIMO.

9.1 Introduction

MIMO techniques are the most important advance in recent wireless systems; they are a critical part of important standards such as LTE (and LTE Advanced), HSPA+, WiMax, 802.20, 802.11n/ac/ad; the purpose of MIMO is mainly:

9.1.1 History

Bell Labs research started research on MIMO in the 80’s, but one of the turning point might have been the major attention given by the media to BLAST, the Bell Labs Space Time which demonstrated impressive throughput rates. 1

9.1.2 Techniques

At least five multi-antenna techniques have been defined to improve radio link performance:

  1. Receive diversity : classic selection, MRC, the method has been used at the base station since the early days of cellular telephony. Of course the concept of combining multiple receivers to fight fading is even older, and widely used such as in microwave links. More recently it is seeing renewed interest on a smaller scale in multiple antennas in mobile handsets.
  2. Transmit diversity using Space-Frequency Block Coding (SFBC) at the base station.
  3. Beam Steering (towards a specific device) and null forming (for interference cancellation). This is the classic approach for an antenna array designer.
  4. MIMO spatial multiplexing at the base station, for multiple user access.
  5. Cyclic Delay Diversity (CDD), used with spatial multiplexing (an OFDM technique).

9.1.3 Capacity

The main theoretical aspect of MIMO is one of channel capacity. The Shanon’s capacity theorem for a simple RF channel is:

C = B log2(1+ S ∕N)

where C= capacity (bits/s), B=bandwidth (Hz), S∕N= signal to noise ratio.

That capacity equation 2 is widely used and refers to a system with one transmitter, and one receiver (with possibly added diversity, but ultimately combined into one receiver); now we consider a system of N × M antennas: N transmitters, and M receivers.


Figure 9.1: MIMO channel is represented by a N×M matrix of throughput correlation parameters (amplitude and phase) from each transmitted signal to each received signal.

The H-matrix is a matrix [Hij] defines complex throughput correlation parameters (with amplitude and phase) from each transmit antenna i to each receive antenna j. The new capacity equation for MIMO systems is

     ∑n       (    Si  2   )
C  =    B log2  1+ N- σi(H)

where n is the number of independent transmit/receive channels (which is no greater than min(N,M)), and reflects the number of sufficiently uncorrelated paths (It is the rank of the matrix H, and in LTE it is referred to as the rank of the channel), Si are the signal power in channel i, N the noise power, and σi2(H) are singular values of the H matrix.

This channel capacity equation shows that capacity increases linearly with n, which optimally approaches the number of antennas min(N,M). But remember that the channel itself (meaning the propagation media) has a rank, and has a capacity, no matter how many antennas are in the system.

That linear variation is where the value of MIMO lies. In equation (9.1), capacity increases in log 2(1+SNR), which is nearly a linear increase in SNR for small values of SNR (since log(1+x) x for x 0), however modern wireless standards tend to aim at higher modulation rates, which require higher SNR, for which log 2(1+SNR) log 2(SNR), which is a much slower increase in capacity.

9.2 Fundamental Principles

MIMO systems include considerations around signal preconditioning, as well as channel predictions, as illustrated in figure (9.2).


Figure 9.2: This great diagram (from [130]) illustrates preconditioning, channel variations, and processing. Each transmit signal is mapped into layers (streams), which are sent on each antenna. The receiving side retrieves the initial signal from the multiple received layers.

9.2.1 Preconditioning

Preconditioning consists of combining multiple signals over multiple streams, each transmitted over a specific antenna, in a manner convenient to be decoded and recovered on the receiving side. Alamouti preconditioning is a specific choice of combinations that aims at producing well conditioned matrices.

Figure 9.3: Alamouti preconditioning consists of combining multiple signals over multiple transmit streams.

r0 = h00s0 + h01s1
r1 = h10s0 + h11s1

or in matrix form: [r] = [H][s] On the receiving side, the original signal is retrieved (at least formally) by [s] = [H]-1[r]. Practically that matrix may be difficult to invert for too reasons.

  1. The receiver has to estimate the matrix [H], which is not intuitive since that matrix characterizes the channel, changes constantly (recall coherence time – 4.3.2), and therefore may deviate from the actual instantaneous propagation environment.
  2. The matrix condition number has to be high enough for the matrix to be well conditioned to be accurately invertible.

The condition number K(H) of the system is defined as follows: first we define G = HH (where indicate transposed complex conjugate), and its eigenvalues of G: λmin to λmax. The conditioning number is then defined as

        ∘ -----
K (H ) =  λmax-

Further, we need to take into consideration the noise from the channel [r] = [H][s] + [n], which makes the condition number of the matrix even more important. Small variations (like added noise) will cause the inverted estimation of an ill-condition matrix to be inaccurate, which will induce errors in retrieving the original symbols si.

Preconditioning is very important to MIMO, without it, the multiple transmit stream will not be properly isolated, and will interfere with one another on the receiving side. To illustrate the importance of preconditioning, let us look at the case of two transmit signals, and two receiving antennas, where both sides are in LOS:

       [      ]
H  = 1-  1  1
     2   1  1

trying to invert the matrix is useless, it is so extremely ill conditioned (determinant =0) that it is not invertible. But if we pre condition the transmitter (transmitting the opposite on one antenna)

       [        ]
H  = 1-    1  1
     2   - 1  1

the matrix determinant is now δ=1, H-1 is easy to compute.

9.2.2 Space-Time Coding (STC)

Figure 9.4: Alamouti 2x1 Space-Time Coding (STC) block diagram.

Let us examine a simple example of 2x1 Alamouti space-time code.

r  = h s +  h s
 0    0 0*    1 1 *
r1 = h1s0 + h0(- s1)

s0 = A (h*0r0 + h1r*0)
s1 = A (h*1r*0 + h0(r*1))

where A = 1(|h0|2 + |h1|2)

In OFDM the parameters h0, h1 are estimated with pilot signals (known signals to estimate the channel). Alamouti 2x1 can improve overall performance given a certain SNR, but does not improve capacity.

Let us now move on to a 2x2 Alamouti example:

r0 = h00s0 + h01s1
r  = h  s + h  s
 1    10 0   11 1

        *       *
s0 = A (h0*r0* + h1r0)*
s1 = A (h1r0 + h0(r1))

where B = 1(h00h11 - h01h10) or in matrix form [s] = [H]-1[r] This setup can provide two independent streams; the physical channel can be pictured as having a number of layers (corresponding to the rank of the matrix).

9.2.3 Performance and Condition Number

There are empirical relationships between the MIMO condition number and the relative SNR required to achieve the same level of performance as SISO systems (figure 9.5).

Figure 9.5: MIMO channel condition number required to achieve the same level of performance as SISO systems.

The above capacity considerations give an insight in the best conditions for MIMO, and in some of its limitations. The ability of the MIMO modem to converge depends on three factors: phase orthogonality, gain isolation, and SNR. Main parameters to increase phase orthogonality are: delay spread, frequency, and distance between transmit and receive antennas. The main parameters to create gain isolation are: polarization isolation between antennas, antenna pattern isolation (directive beam, or different propagation modes). Parameters to effect the available SNR are: distance, interferers, frequency, antenna gain and efficiency.




Close to Cell, high SNR, rich multipath


High peak data rates, higher device complexity

Edge of Cell, low SNR


Increased reach

Many scattered users, high SNR, rich multipath


Device separation, low device complexity

Table 9.1: Summary of MIMO benefits in different scenarios, from good SNR close to cell to poor conditions at the edge.

9.2.4 Channel Estimation

Channel estimation is an important part of MIMO performance. It is important to measure time delays, attenuation, and phase (as a function of frequency) for each path between antennas in order to maximize signal recovery and hence performance.

To improve estimation, some MIMO systems use closed loop MIMO estimation for the channel and channel states. Some systems are called open-loop MIMO, in which the receiver side regularly measures channel characteristics (from some known transmit sequence), but the transmit side has no knowledge of the propagation channel. Open-loop MIMO systems typically use orthogonal coding such as space-time coding.

Other systems use closed-loop MIMO, where the receiver regularly report channel information back to the transmitter; there are different types of closed Loop MIMO such as beam forming or precoding.

Closed loop MIMO systems cannot waste bandwidth in transmitting a complete description of the channel state; instead they use codewords describing important channel parameters (such as rank). Different standards use different codewords to report channel state. LTE for instance simply a use a table of codewords to describe the channel: a 4-bit word describing one of 16 channel states. 802.11n supports Channel Sounding, where a receiver can send Channel State Information (CSI) to a transmitter. 802.11n also supports ‘Implicit TxBF’, where the transmitter assumes reciprocity of the channel and calculates the steering matrix using training symbols.

Of course in a mobile environment these characteristics vary, and must be estimated often (depending on coherence time of the channel – see 4.3.2).

9.3 Classic MIMO Systems

Classic MIMO systems are illustrated in in figures 9.6, 9.7; they include:

Figure 9.6: Space-Time Transmit Diversity (STTD) illustrative diagram.

Figure 9.7: Space-Frequency Transmit Diversity (SFTD) illustrative diagram.

antennas are fractions of a wavelength apart, multiple elements phase and amplitude act as an antenna array, and steer the beam.
replaces [H] by a better conditioned [P][H][P] matrix. Each transmit signal is preconditioned before transmission. Antennas are several wavelengths apart.
Spatial Multiplexing:
multiple streams are combined in a true MIMO method as much as the rank will allow

9.4 Homework

  1. Compare SISO capacity (Shannon calculation), and 2x2 MIMO capacity. Use required SNR of 26 for 64QAM, 13 for QPSK. List any other assumption.
  2. There are concerns that higher order MIMO may not be very practical in many cases. Search the Internet for test reports on various LTE throughout tests (in lab, indoors, or drive test reports). Try to find the rank of the LTE channel. Can you find enough reports on the rank to assess how often a LTE system will make efficient use of 2x2 MIMO vs. 4x4 MIMO, or 8x8 MIMO? (Specify indoor, outdoor, etc).