problem and solvable by non-linear tree search approaches such as K-Best.
However, for fading channels approximations have to be made in order to enable a convex problem formulation that can be solved by state-of-the-art solvers.
Section 5.5 contains the numerical evaluation via simulations. For the AWGN channel, we will show that MAP estimation allows for reliable multiuser energy estimation with M = √
N observations only, which is a remarkable result. However, especially in fading channels, the MAP approach sacrifices optimality due to approximations made during the derivation.
Surprisingly, the MUSIC algorithm performs well in fading and in AWGN channels yielding a good trade-off between robustness and optimality. The introduced MMP suffers optimality in the fading channel and can only be applied for AWGN channels. Further, it can be shown that the SNR enhancing effect outperforms state-of-the-art algorithms such as the GOMP.
The results and schemes for the AWGN channel introduced in this chapter have been published in [MBD14]. Augmenting these concepts to the fading channel is a novel contribution solely contained in this thesis.
Pre-Filter (·)2 T1
-T (·) ≷PresentNot-Present
y(t)
Threshold Figure 5.1: Block diagram of energy detection.
5.2.1 Energy Detection
The main idea of energy detection originates back to the task of detecting the presence of an unknown signal in a noisy environment [Urk67, DAS07]. Based on the received energy, the energy detector performs a binary hypothesis test that estimates whether the signal is present or not. The received energy over a certain time period is thereby used as input statistic of the hypothesis test. The basic principle of energy detection is relatively simple and shown in Fig. 5.1. Based on some knowledge about the signal of interest the energy detector consists of a filter that filters out the band of interest followed by a squaring device1 that squares the input signal which is then fed into an integrator. Depending on the time the signal is integrated over the output corresponds to the received energy in that time interval2. If the signal of interest is present, the received energy is composed of the energy of that particular signal plus the accumulated noise energy. If the signal is not present, the output contains the noise energy only. The goal is to define a threshold that performs an estimation whether the signal was present or not. Depending on this threshold the corresponding false alarm and missed detection probabilities of the energy detector are defined.
Due to the recent research in the field of Cognitive Radio, where spectral holes have to be identified, energy detection experienced an upswing. Here energy detection concepts were reinvented under the name radiometry to detect the spectral occupation of a band [SF92, HSCK06, ZKL08]. Beyond that sub-Nyquist sampled energy detection has been investigated as a strategy to estimate the spectral occupation in Cognitive Radio, combining energy detection concepts with the idea of sub-Nyquist sampling from the field of Compressed Sensing [ME11, CE16, CE14]. This idea is picked up in Subsection 5.4.3 for MAP multiuser energy estimation.
Furthermore, energie detection was also used in the field of communications.
Here concepts were used to estimate the set of active users in CDMA systems.
1Sometimes also called square law device
2Depending on whether the signal is normalized to the time period where the integration is carried out the power is considered instead of energy. Since both only differ in a scaling factor the statistical properties do not change and the term energy detector is kept.
The concepts used there range from subspace methods via MUSIC [WC98]
to Kalman filtering and random set theoretic approaches [VM13]. Especially sub-space based concepts are of major interest as the underlying estimation problem is based on the receive covariance matrix. Thus, the MUSIC approach considered in Subsection 5.4.2 is based on state-of-the-art concepts.
The major difference to the state-of-the-art concepts is to consider the application of MUSIC in overloaded systems.
5.2.2 Multiple Measurement Vector Compressed Sens-ing
The energy detection task via the receive covariance matrix has not been directly considered in the Compressed Sensing literature. In Compressed Sensing the problem of estimating variables with a common support as (5.2) refers to the so-called Multiple Measurement Vector Compressed Sensing (MMV-CS) problem. Solving the MMV-CS model (2.5) is a commonly known problem [CREKD05]. However, instead of only estimating the frame support SX as pursued here, MMV-CS algorithms aim at estimating the whole matrix X, which is a quite more complex task than the estimation˜ of the frame support only. Among the numerous algorithms existing for MMV-CS problems we employ the GOMP subsequently as a benchmark to assess the performance of multiuser energy detection. The application of the GOMP in CS-MUD has been extensively studied in the past [SD12] where it was used to jointly detect activity and data. The GOMP algorithm is shown in Listing 5.1 and is a straightforward extension of the single measurement OMP algorithm to the MMV-CS case.
Akin to the OMP, the GOMP is an iterative and greedy algorithm that identifies active users by correlating the signature matrix with the resid-ual, corresponding to the received signal in the first iteration. The node corresponding to the column of A with the highest correlation is assumed to be active and the data of that particular node is estimated in a least-squares sense. In the next step, the residual is updated by subtracting the contribution of the nodes that have been estimated as active. The algorithm proceeds by correlating the signature matrix with the residual. Clearly, this approach is heuristic and greedy as an activity decision is never revised during the iterative algorithm. The GOMP produces an estimate for the whole matrix X. Instead of processing the matrix˜ X˜ we only consider the estimated support set SXˆ from the GOMP.
Throughout this chapter we assume genie knowledge of Nact at the base station. This assumption is clearly not feasible in general and requires future research. With this knowledge, the GOMP stops after v =Nact iterations,
Algorithm 5.1 Group Orthogonal Matching Pursuit (GOMP) R0 = Y, SX(0)ˆ = ∅, v = 0,Xˆ = 0N×LF
repeat
v = v + 1
#Find Column with highest correlation to residual#
n = arg max
n
LF
l=1|AH(:,n)R(:,l)|
#Update estimated support set#
SX(v)ˆ ∪n
#Estimate data of active nodes#
Xˆ(S(v)
Xˆ ,:) = A†
(:,SX(v)ˆ )Y
#Update residual#
Rv = Y −A ˆX(S(v) Xˆ ,:)
until v = Nact
#Output#
Xˆ = Xˆ(S(v)
Xˆ ,:), SXˆ = SX(v)ˆ
making the dimensions of the partial system matrix to be M ×Nact in the last iteration. Thus, the complexity of the matrix inversion can be upper-bounded by O
Nact3
[GVL12]. Incorporating Nact iterations, we have a complexity that scales as Nact· O
Nact3 .