Large-scale propagation models
Measurements-based propagation models have showed that average received sig-nal power decreases logarithmically with distance separating a transmitter-receiver (Tx-Rx) pair. These models are used to estimate the received signal power as a function of distance and are called path loss models. The average path loss for an arbitrary Tx-Rx pair with distance d is expressed as 
P¯L(d) ∝ d , or P¯LdB(d) = P¯LdB(d0) + 10n log10 d , (2.1) d0 d0.
where n indicates the path loss exponent and d0 the close-in reference distance. n depends on the propagation environment, e.g. in a free space environment n is equal to 2. The model in (2.1) considers that any distance d separating a Tx-Rx pair results in a constant path loss. However, two diﬀerent Tx-Rx pairs at two diﬀerent locations with the same separation distance may have the surrounding environment totally diﬀerent. This implies that measured signals can be far away from the average value predicted in (2.1). Measurements have been derived in [11, 12], and have showed that at any distance d, the path loss P L at a particular location is random and log-normally distributed, i.e. Gaussian distributed when measured in dB . That is, (d) = P¯L d P L (d ) + 10n log 10 + X , (2.2) dB dB 0 σ d0.
where Xσ is a Gaussian distributed random variable with zero mean and variance σ 2 (in dB). The log-normally distributed random variable given in (2.2) describes the random shadowing eﬀects. The close-in reference distance d0, the path loss exponent n and the standard deviation σ are the parameters that characterize the path model, which can be used to estimate the received power levels at a random location for communication system analysis purpose .
Small-scale propagation models
Small-scale fading, or simply fading, is a term used to describe the fluctuations of the parameters (phase, amplitude, frequency) of a radio signal over a small transmission distance. The most important eﬀects in a fading channel are the fluctuation of the signal power, the fluctuation of the frequency modulation due to Doppler eﬀect, and the time dispersion caused by the multipath propagation delays. For mobile wireless communications, fading eﬀect usually occurs in urban areas where the mobile is surrounded by obstacles that prevent from a line-of-sight connection between the base station and the mobile terminal.
In a mobile radio channel, small-scale fading is influenced by the following major factors:
• Multipath propagation: the presence of reflecting objects and scatters in the channel disperses the signal parameters, and creates multiple modified versions of the transmitted signal displaced with respect to one another in time and phase.
• Terminal mobile motion: the movement of the destination causes a random modulation due to diﬀerence Doppler shifts on each of the multipath com-ponents. The frequency variation depends on the speed of the destination movement.
• Obstacles motion: when the obstacles of the propagation environment move, the multipath components undergo a time variation.
• Bandwidth of the transmitted signal: the small-scale fading strength depends on the transmitted signal bandwidth.
The fading channel may be referred to as fast fading or slow fading channel depending on how fast the channel variations are compared to the transmitted baseband signal variations. If the coherence time Tc of the channel is smaller than the symbol period, i.e. the channel impulse response changes rapidly within the symbol duration, the channel is said fast fading. The coherence is used to characterize the time varying nature of the frequency variations of the channel in the time domain. It is inversely proportional to the Doppler spread Ds, i.e.
Tc ≈ 1 . If the symbol duration is much slower than the channel coherence time, Ds i.e. T ≪ Tc, the channel is supposed to be slow fading. In this case, the channel may be assumed to be static over one or several reciprocal bandwidthes.
Frequency fading is mainly related to time dispersion due to multipath prop-agation. When the fading channel has a constant gain and linear phase response over a coherence bandwidth Bc that is greater than the bandwidth of the trans-mitted signal Bs, the radio channel is said flat fading channel. In a flat fading, the channel can be approximated with no excess delay, since the reciprocal bandwidth (symbol period) of the transmitted signal T is much larger than the channel time delay spread τmax, i.e. Bs ≪ Bc, and T ≫ τmax. (2.3).
Flat fading channels are also referred to as narrowband channels for which the transmission bandwidth is considered narrow when compared to the flat fading coherence bandwidth. On the other hand, if the fading channel has a constant gain and linear phase response over a bandwidth smaller than the transmitted signal bandwidth, the radio channel is called frequency selective fading channel. In such a case, the channel response has a time delay spread greater than the symbol period, i.e. τmax > T . This results in multipath eﬀects, and multiple versions of the transmitted symbols are received attenuated and time-delayed which induces inter symbol interference. In the frequency domain, the transmitted components at diﬀerent frequencies undergo diﬀerent attenuation and phase shifts. Frequency selective fading channels are also referred to as wideband channels for which the signal bandwidth is supposed wide when compared to the coherence bandwidth.
The frequency selective fading channels can be transformed into adjacent flat fading subchannels by applying techniques such as orthogonal division frequency multiplexing (OFDM) that divides the wideband into many sub-bands much nar-rower than the coherence bandwidth. In our work, an OFDM scheme in a Rayleigh distributed channel. This assumption is commonly used in mobile radio channels to describe the envelope of an individual multipath component.
Interference mitigation in mobile wireless communications
In wireless communications, the ideal would be to allow the users in the same area to send information simultaneously in the same bandwidth to their intended receivers. Sending information at the same time in the same bandwidth will cause interference at the receiver side that, if dealt with as noise, enhances the noise strength. Therefore, a wise management of interference is a challenging task so the users can share the same wireless medium. Multiplexing techniques used so far allocate the available resources in an orthogonal way. The three major multi-plexing techniques used to share the available resources in wireless communication systems are: frequency division multiplexing (FDM) (orthogonal bandwidth allo-cation thanks to parallel sub-bands), time division multiplexing (TDM) (orthog-onal time share thanks to successive time slots), and code division multiplexing (CDM) (orthogonal user signatures). Such techniques are applied for both nar-rowband and wideband systems.
Table of contents :
List of Figures
1 Introduction and motivations
1.3 Summary of the PhD contributions
2 State of the art on interference mitigation in wireless communications
2.2 Radio propagation
2.2.1 Large-scale propagation models
2.2.2 Small-scale propagation models
2.3 Interference mitigation in mobile wireless communications .
2.3.1 Multiplexing techniques
2.3.2 Achievable rate
2.4 Multi-user channel categories
2.5 Advanced techniques for data rate improvement under Gaussian input assumption
2.5.1 Broadcast channels
2.5.2 Multiple access channels
2.5.3 Interference channels
2.6 Optimality of the IA technique with respect to the input alphabet .
3 Interference alignment for a multi-user SISO interference channel
3.2 System model
3.3 IA design in a SISO interference channel
3.3.1 Precoding design
3.3.2 Linear decoding design
3.4 IA precoding subspaces optimization
3.4.1 MMSE-based decoder – Iterative solution
3.4.2 ZF-based decoder – Closed-form solution
3.4.3 Complexity and sum-rate performance
3.5 Precoding vectors design within IA subspaces
3.5.1 MMSE-based decoder
3.5.2 ZF-based decoder
3.5.3 Complexity and sum-rate performance
3.6 Convergence rate of the iterative solutions
3.7 Comparison of the proposed optimized designs to the state of art schemes
4 Linear detectors for downlink transmission with interference alignment
4.2 Context and transmission network
4.3 System Model
4.4 Spatial IA design in a K-user MIMO IC
4.5 Traditional linear decoding in a spatial IA scheme
4.6 Desired signal extraction in a spatial IA scheme using high-cumulants order
4.6.1 Desired signal Extraction
4.6.2 Second-order information: Whitening
4.6.3 Joint Approximate diagonalization of Eigenmatrices .
4.6.4 Semi-Blind separation
4.7 Simulation Results
5 Low complexity detectors based on sparse decomposition for uplink transmission
5.2 System model
5.3 Joint decoding of interference and desired signal
5.4 Sphere decoding
5.5 Sparse decomposition
5.6 Iterative detection of sparse transformed MIMO via ℓ1-minimization
5.6.1 Noiseless MIMO channel
5.6.2 Noisy MIMO channel
5.7 Iterative detection of sparse transformed MIMO via minimum distance minimization
5.8 Complexity and bit error rate performance
5.9 Turbo detection of a sparse detected signal
5.9.1 Turbo detection concept
5.9.2 Turbo detection scheme
5.9.3 New detection criterion
5.9.4 Bit error rate performance
6 Conclusion and perspectives
6.2 Future works
A Mutual information in the MIMO interference channels
B Projected gradient method
C Sum-rate gradient with respect to the combination matrix
D Sphere Radius for the ℓ1-minimization problem constraint