The spatiotemporal robustness of the MIMO transmissions is exploited aiming at inducing opportunistic secondary communications on the primary radio resources. The proposed model makes use of the smallest singular values and the Extreme Value Theory for characterizing the robustness of the MIMO systems, having always the premise of preventing significant degradations in the performance of the primary communications, while at the same time an attractive number of potential opportunities for secondary are intended to be offered. So, the idea behind the suggested scheme consists in reusing the information that is already available within the primary MIMO networks aiming at cotransmitting opportunistically in a clever manner, which at the end will allow us to make a much more efficient use of both the natural and technical resources.
After reviewing the vast activity related with the cognitive radio (CoRa) technology, it is quite easy to realize that the spectrum awareness turns out to be the keystone of this technology. This is because, according to the cognitive radio cycle [
That is the reason why this research work proposes a model that is focused on making use of the advanced information that nowadays is inherently collected at the modern primary networks (e.g., multicarrier MIMO systems), aiming at establishing opportunistic secondary communications under an assisted detection scheme.
So, the LTE standard was selected as study object for the hypothesis made in this proposal, since its current releases and planned enhancements have inherently embedded an advanced network knowledge which could be reutilized in a clever manner by other technologies such as the cognitive radio.
Concretely, the information extracted from the channel state information in a single user MIMO LTE Uplink is reutilized aiming at establishing a CoRa communication over the particular time instants where the primary system is more resistant to the impairments. This is done by developing a model which pretends to identify the spatiotemporal robustness of the MIMO system by making use of the smallest singular values and the extreme value theory. So, as part of the obtained results several thresholds were determined aiming at evaluating the disruptions caused on the performance of the primary user (MIMO LTE uplink) as a function of the number of discovered opportunities for secondary purposes.
This paper is organized as follows. Section
Instead of transmitting data over a single radio channel, a multipleinputmultipleoutput (MIMO) system transmits over several radio channels aiming at increasing the data rate (spatial multiplexing) or improving the BER (transmit diversity) [
In the above equation
Mathematically speaking, the algebraic rank
The baseband block diagram of the MIMO SCFDMA transceiver chain as used by LTEA is shown in Figure
Transmitter and receiver structure of a 2 × 2 MIMO SCFDMA system.
The above figure depicts the transmitter, as well as the receiver structure of a 2 × 2 spatiallymultiplexed MIMO SCFDMA system (Closed Loop). At the transmitter, the procedure initiates in the time domain with the serial arranged symbols which in first instance are demuxed aiming at building two codewords (CW), then each of them are arranged in parallel in order to be modulated and fed to an
Once a circular convolution per channel took place (e.g., theoretically there are four virtual links in this array) the set of initial steps per layer at the receiver consists in removing the cyclic prefix, putting the information into parallel form, and utilizing the
In the above equation,
In (
Then, continuing with the description of the procedure performed at the receiver, the best estimation of the modified complex numbers is arranged in parallel (i.e., per CW) to be later on transformed back to the time domain aiming at obtaining the input needed by the demodulator, which provides the recovered symbols that are finally feed to a MUX and rearranged in serial form.
In the SCFDMA system a generic radio frame has a duration of 10 ms and consists of 20 slots. On the other hand, a subframe is composed by 2 slots each slot, being built from seven or six SCFDMA symbols depending on the cyclic prefix length configuration (i.e., normal or extended). So, based on the above, it is possible to write an equation describing the way of computing the peak bit rates of the system (i.e., per every branch of the MIMO structure) [
In the numerator, starting from left to right we have the number of bits given by the modulation in use (e.g., QPSK will lead to 2 bits), then the subcarriers are related with those that were allocated to the user in question, while the number of symbols per subframe depends on the cyclic prefix length just as it was mentioned before. For its part, the denominator refers to the required time for transmitting a subframe.
Under the context of a MIMO transmission in the LTE uplink, we refer to the particular case of having two transmit antennas and two receive antennas, just as it was described in Section
We know that after decomposing the channel matrix
For a symbol error to occur, the norm of the noise vector would have to go beyond half the minimum distance:
But (
In a MIMO transmission, the sudden arising of the largest singular values in between the smallest ones leads to the most robust fourdimensional combination (i.e., latitude, longitude, frequency, time) for facing better the errors in a communication. Therefore by using the Extreme Value Theory we have modeled the arising of such largest singular values aiming at studying the feasibility of opportunistically interfering (i.e., with a superimposed CoRa communication) a LTEMIMO transmission without significantly degrading its performance, while at the same time an attractive number of potential opportunities for a secondary transmission are being offered.
According to the Central Limit Theorem, any population or parent distribution having finite variance will lead to the following fulfillment: the probability density function (PDF) of the means coming from
So, once the Central Limit Theorem has been cited, then it is possible to introduce an analogous result which applies for the extreme values instead of the means, leading this time to a PDF having the following form [
The above equation is known as the Generalized Extreme Value (GEV) distribution and encompasses the Gumbel (Type I), Frechet (Type II), and Weibull (Type III) distributions. In general, for some values of
So, instead of estimating a unique set of MLEs for the parameters
The parameters implicitly involved into the implementation of a single user MIMO SCFDMA LTE (2 × 2 spatial multiplexing scheme) are summarized in Table
Simulation parameters of the MIMO SCFDMA (LTEA Uplink).
Variable  Value  Description 


128  Total number of subcarriers in the system 

32  Number of subcarriers per user 

4  Number of simultaneous transmissions (users) without cochannel interference 
BW  1.4  Channel bandwidth (MHz) 

0.52083  Sampling time (us) 

1850.7  Carrier frequency (MHz) 
CW  2  Number of codewords 

2  Algebraic rank (enabled layers) 


Precoding matrix (kth subcarrier) 


MMSE equalizer (frequency domain, see ( 
In order to study the sudden arising of the largest singular values in between the smallest ones, 11,200,000 samples were collected. So, aiming at extracting the maximums, blocks of data were created on persubframe basis in such a way that at the end 25,000 maximums were obtained.
The GEV distribution fitting the data set representing the largest singular values is shown in Figure
Fitted PDF of the largest singular values on a per subframe basis.
In the above figure, the histogram shown was scaled in order to make it comparable to the fitted PDF, which was obtained as described in Section
Fitted CDF of the largest singular values on a per subframe basis.
On the other hand, by fixing the value of the
Region of GEV parameters leading to a PDF modeling the data set.
So, a particular return of level
This way and by repeating the above procedure, several return levels were forecasted aiming at testing their impact on the primary communication, just as shown in Figure
BER of the interfered primary user (MIMO SCFDMA).
In general, from the above figure it can be noticed that when the primary system was interfered by the opportunistic secondary communication, the system’s performance did not move too much away from the reference curve. In fact, for the carriedout analysis the distortion never went beyond one order of magnitude since for all cases the BER was maintained around
Theoretical peak bit rates (PU and SU).
User type  Return of level  Data rate 
Data rate 

PU  —  896 Kbps  2.688 Mbps 
SU 

12.8 Kbps  38.4 Kbps 
SU 

6.4 Kbps  19.2 Kbps 
SU 

3.2 Kbps  9.6 Kbps 
SU 

2.1 Kbps  6.4 Kbps 
SU 

1.6 Kbps  4.8 Kbps 
On the other hand, if as instead of considering only
Peak data rates of a SU cotransmitting opportunistically along with a PU in an LTE system.
As it can be verified from the above set of curves, most of the obtained data rates by the SU would be better described in terms of Kbps (
On the other hand and getting back to our proposal, although the arising of the singular values of interest is well characterized, a much more realistic scenario should encompass the time varying nature of the mobile channels which could lead to a severe interference of the primary system. For that reason, in the future this proposal should be extended by considering such a small scale fading mechanism, which among other things will allow us to acquire the necessary knowledge for determining the equations corresponding to probability of collision detection and false alarm of the proposed scheme. Therefore, for the moment, this research work should be seen as a framework connecting the CoRa technology and the MIMO systems waiting to be complemented in order to increase the feasibility of its implementation.
In this research work, the extreme value theory was utilized for modeling the tracking of the most robust singular values appearing on the MIMO matrices, which represent to be the spatiotemporal conditions that are more resistant to the impairments within a MIMO system.
So the proposed model was focused on guaranteeing the nonsevere disturbance of a primary MIMO user, while an attractive number of potential opportunities for secondary purposes were also pursued. In this study, several thresholds (return levels) were obtained from the proposed model, which were tested over the MIMO SCFDMA spatial multiplexing transmission (3 GPPLTE/LTEA uplink) which played the role of the primary system. From the conduced analysis it was found that the performance degradation of the primary user never went beyond one order of magnitude with respect to an uninterfered communication used as reference, while the total number of secondary opportunities was given by the own selected threshold (i.e., by tuning
The authors would like to thank the anonymous reviewers for their helpful comments, the COST Action IC0905 TERRA (because part of this work was presented there), and to the Technical University of Catalonia (UPC) for the financial support granted.