A New Method of Measurement Matrix Optimization for Compressed Sensing Based on Alternating Minimization

In this paper, a new method of measurement matrix optimization for compressed sensing based on alternating minimization is introduced. The optimal measurement matrix is formulated in terms of minimizing the Frobenius norm of the difference between the Gram matrix of sensing matrix and the target one. The method considers the simultaneous minimization of the mutual coherence indexes including maximum mutual coherence μmax, t-averaged mutual coherence μave and global mutual coherence μall, and solves the problem that minimizing a single index usually results in the deterioration of the others. Firstly, the threshold of the shrinkage function is raised to be higher than the Welch bound and the relaxed Equiangular Tight Frame obtained by applying the new function to the Gram matrix is taken as the initial target Gram matrix, which reduces μave and solves the problem that μmax would be larger caused by the lower threshold in the known shrinkage function. Then a new target Gram matrix is obtained by sequentially applying rank reduction and eigenvalue averaging to the initial one, leading to lower. The analytical solutions of measurement matrix are derived by SVD and an alternating scheme is adopted in the method. Simulation results show that the proposed method simultaneously reduces the above three indexes and outperforms the known algorithms in terms of reconstruction performance.

Introduction
Compressed sensing (CS) [1] can sample the sparse or compressible signals at a sub-Nyquist rate, which brings great convenience for data storage, transmission, and processing. By adopting the reconstruction algorithms, the signal can be exactly reconstructed from the sampled data. As a marvel way of signal processing, CS is applied in different fields such as image encryption [2], wideband spectrum sensing [3], wireless sensor network data processing [4], etc.
The original signal x ∈ R N×1 is assumed to have a sparse representation in a known domain Ψ ∈ R N×L N (N ≤ L) as x = Ψs where Ψ is the dictionary matrix and s is a K-sparse signal. The incomplete measurement y ∈ R M×1 is obtained through the linear model y = Φx = Ds (1) where Φ ∈ R M×N (M < N) is called the measurement matrix and D = ΦΨ is the sensing matrix. Some specific properties of Φ have great impacts on the reconstruction performance. In [5,6], spark and restricted isometric property (RIP) are respectively proposed as the sufficient conditions on Φ to recovery guarantee. However, computing the spark of a matrix has combinatorial complexity and certifying RIP for a matrix requires combinatorial search, that is to say, these tasks are NP-hard and difficult to accomplish. To a large extent, the coherence between Φ and Ψ reflects the performance of meeting the above conditions. Actually, the coherence is equivalent to the mutual coherence of D. Since the mutual coherence can be easily manipulated to provide recovery guarantees, it is commonly used to measure the performance of Φ. The frequently-used mutual coherence indexes include the maximum mutual coherence µ max [7], t-averaged mutual coherence µ ave [8] and global mutual coherence µ all [9], which respectively represent the maximum, the average and the sum of squares of the correlation between any distinct pair of columns in D.
The first attempt to consider the optimal design of Φ is given in [8]. The simulation results carried out in [8] show that the optimized Φ leads to smaller µ ave and a substantially better CS reconstruction performance is obtained. Then the optimization of Φ becomes an important issue in CS. The recent works try to find the optimal Φ which has excellent performance in reducing the mutual correlation by minimizing the Frobenius norm of the difference between the Gram matrix G = D D and the target Gram matrix G t . The main work focuses on designing G t and finding the best Φ.
In [8], G t is obtained by shrinking the off-diagonal entries in G. The shrinkage technique reduces µ ave but it is time-consuming. Furthermore, µ max is still large, which ruins the worst-case guarantees of the reconstruction algorithms. A suitable point between the current solution and the one obtained using a new shrinkage function is chosen to design the G t in [10]. It is of very strong competitiveness in µ ave and µ max . However, the optimal point is hard to determine and the unsuitable point may seriously degrade the algorithm's performance. In [9], G t is obtained by averaging the eigenvalues of G. Simulation results show that µ all is reduced effectively but reducing µ ave and µ max is hard to be guaranteed, which means µ ave and µ max may maintain large values. Duarte-Carajalino and Sapiro [11] set G t as an identity matrix. Since D is overcomplete and G cannot be an identity matrix, simply minimizing the difference between G and G t does not imply low µ max [12]. In [13][14][15][16][17], G t is chosen from a set of relaxed Equiangular Tight Frames (ETF) [18]. The set can be formulated as where µ welch denotes the Welch bound [19] and G t (i, j) denotes the (i, j)th entry of G t . However, the maximum absolute value of off-diagonal entries in G is almost always greater than µ welch . In this case, the optimization usually implies a solution D with low µ ave but high µ max . In summary, the target Gram matrices mentioned above only focus on a certain mutual coherence index, and fail to take into account µ ave , µ max and µ all simultaneously. When a certain index is targeted, the other indexes may not decrease significantly or even increase. Therefore, Φ is not 'good' enough and the reconstruction performance is well below par.
After designing G t , the next step is to find the 'best' Φ by approaching G to G t . In [8][9][10][11]17], G is obtained by applying SVD to G t primarily and then the square root D of G is built asD D = G. At last, Φ is obtained by Φ =DΨ † where † denotes the Moore Penrose pseudoinverse. This kind of method is intuitive, but the generalized pseudoinverse poses problems of calculation accuracy and robustness [15]. In [14,15], gradient algorithm and quasi-Newtonian algorithm are respectively utilized to attain Φ. Firstly, the cost function F(Φ) with Φ as the variable is constructed. Then the search direction is determined by the derivative of F(Φ). Finally, Φ is obtained with a fixed step size. However, choosing a suitable step size which has a great influence on the accuracy of the solution requires a lot of comparison work. Moreover, the gradient algorithm and quasi-Newtonian algorithm cannot converge until a certain number of iterations is accomplished, resulting in high computational cost. In [11,16], the method for designing Φ shares the same concept as K-SVD [20], that is to update a matrix row by row. Eigenvalue decomposition is required to find the square root of the maximum eigenvalue for each row, which results in a significant increase in the calculation. For solving this problem, Hong et al. [16] utilize the power method instead of eigenvalue decomposition. However, the eigenvalue obtained by power method is the one with the largest absolute value. When the eigenvalue is negative, eigenvalue decomposition is still necessary.
The primary contributions of this paper are threefold:

•
The new target Gram matrix G t targets µ ave , µ max , and µ all of D simultaneously is designed. Firstly, a new shrinkage function whose threshold exceeds µ welch is utilized to determine the initial target Gram matrix. Then G t is obtained by sequentially applying rank reduction and eigenvalue averaging to the initial matrix. • Analytical solutions of the measurement matrix Φ to minimize the difference between G = Ψ Φ ΦΨ and G t are derived by SVD. • Based on alternating minimization, an iterative method is proposed to optimize the measurement matrix. The simulation results confirm the effectiveness of the proposed method in decreasing the mutual coherence indexes and improving reconstruction performance.
The remainder of this paper is organized as follows. Some basic definitions related to mutual coherence indexes and frames are described in the next section. The main results are presented in Section 3, where the solutions to the G t design are characterized and a class of the solutions to the optimal Φ is derived in detail. The procedure of our method and the discussion can be also found in Section 3. In Section 4, simulations are carried out to confirm the effectiveness of the proposed method. In the end, the conclusion is drawn.

Mutual Coherence Indexes
entry at the position of row i and column j in G, where i, j = 1, 2 · · · L. Here, we quote the definitions of the mutual coherence indexes as that presented by Donoho [7], Elad [8], and Zhao [9]. Definition 1. For a matrix D, the maximum mutual coherence µ max is defined as the largest absolute and normalized inner product between all columns in D that can be described as Definition 2. For a matrix D, the t-averaged mutual coherence µ ave is defined as the average of all absolute and normalized inner products between different columns in D that are above t and can be described as Definition 3. For a matrix D, the global mutual coherence µ all is defined as the sum of squares of normalized inner products between all columns in D that can be described as As shown in [5], the original signal can be exactly reconstructed as long as K < (1 + 1/µ max )/2. The conclusion is true from a worst-case standpoint which means that µ max does not do justice to the actual behavior of sparse representations. Therefore, Elad considers that an "average" measure of mutual coherence, namely µ ave , is more likely to describe its true behavior. Different from the previous two indexes, µ all reflects the overall property of D.
In fact, the purpose of reducing the mutual coherence indexes of D is to attain G that meets the following requirements: (1) The maximum absolute value of off-diagonal entries in G is sufficiently small; (2) The number of off-diagonal entries with large absolute value is minimized; (3) The average of off-diagonal entries with large absolute value is as small as possible. However, when a certain mutual coherence index is targeted solely, we cannot guarantee that the obtained G will fully meet the requirements. Therefore, the decrease of a certain index does not always mean better Φ and improved reconstruction performance. When the three indexes are reduced simultaneously, the requirements are better satisfied and better performance is obtained.

ETFs
It is shown in [19] The bound is achievable for ETF. Here, we recall the definition of ETF [18].

Definition 4.
Let F be a M × L matrix whose columns are f 1 , f 2 · · · f L . The matrix F is called an equiangular tight frame if it satisfies three conditions (1) Each column has a unit norm: (2) The columns are equiangular. For some nonnegative θ, we have f i f j = θ when i, j = 1, 2 · · · L and i = j. Sustik et al. [18] show that a real and (L − M)(L − 1)/M must be odd integers when L = 2M, M, and 2M − 1 must be an odd number and the sum of two squares respectively when L = 2M. Fickus et al. [21] surveys some known construction of ETFs and tabulates existence for sufficiently small dimensions. The above studies show that M and L must meet some exacting requirements when an ETF is available for D. However, it is really difficult to meet the requirements in practice, which means the maximum absolute value of the off-diagonal entries in G is usually significantly larger than µ welch .

The Proposed Method
The off-diagonal entries in G actually are the inner products between different columns in D. Reducing those entries is likely to lead to lower mutual coherence indexes and better performance. The most straightforward approach is to replace large off-diagonal values with small ones. However, it is impossible to solve Φ from a certain G because of the inequality of rank between Ψ Φ ΦΨ and G when the approach is adopted. Therefore, a feasible approach is to minimize the difference between G and G t that can be formulated as where G = Ψ Φ ΦΨ. This problem can be solved by alternating minimization strategy [14,16], which iteratively minimizes (5) to find the desired Φ. The idea is to update G t and Φ alternatively and repeat this proceeding until a stop criterion is reached. In this section, we design G t firstly and then derive the analytical solutions of Φ. Finally, an iterative method is proposed to optimize the measurement matrix based on alternating minimization.

The Design of G t
It can be seen from (5) that G t plays an important role in measurement matrix optimization. In recent works, G t is frequently set as the relaxed ETF matrix, which is obtained by applying the following shrinkage function where ς = µ welch for i, j = 1, 2 · · · L and i = j. Such a scheme in designing G t guarantees that the off-diagonal entries with large value of G will be intensively constrained, which means lower µ ave and µ max . Recall from Section 2.2 that the Welch bound is not achievable for G in most cases. As shown in [14,16], different ς yields different results and µ welch is not the optimal value. Li [22] et al. found that a smaller µ max is available when ς is slightly larger than µ welch . Inspired by [22], we propose an improved shrinkage function which divides the entries in G into three segments through two thresholds. One of the thresholds is µ welch and the other is larger than µ welch . The function is as follows where Thr = µ welch + c and 0 < c < µ welch . As can be seen from Equation (7), the maximum absolute value of off-diagonal entries in G t is raised from µ welch to Thr. According to the previous analysis, the new function is likely to lead to a further reduction in µ max while maintaining the advantage of Equation (6) with respect to µ ave . After shrinkage, G t becomes full rank generally [8], that is Rank(G t ) = L. However, the rank of G is identically equal to M. Thus, we consider mending this by forcing a rank M. A new target Gram matrix, denoted as G t_M , is obtained by solving The solutions to this problem are given by Theorem 1 below.
be the matrix obtained by applying the shrinkage operation shown as Equation (7) to G and G t = PΛP be the eigendecomposition of G t . P is orthonormal with dimension L and Λ = diag(λ 1 , λ 2 · · · λ L ) with |λ 1 | ≥ |λ 2 | ≥ · · · ≥ |λ L |. The solutions of the minimization problem defined by (8) are characterized by where U X ∈ R M×M and V X ∈ R L×L are unitary. Then G t_M can be rewritten as Denote f = G t − G t_M 2 F . By substituting G t_M with G t_M = X X, can be rewritten as a function of matrix X. Let ∂ f /∂X be the derivative of f with respect to X. The optimal X should satisfy ∂ f /∂X = 0. Equivalently, we have It then follows from Substituting Equation (12) into Equation (10), we obtain It turns out from the unitary invariance with Equation (13) that With a few manipulations, we conclude that the solution of (8) is equivalent to solving where tr() denotes the matrix trace operation. Noting that A = A A and G t = PΛP , the problem in (15) is equivalent to Denote B = V X P and rewrite B as B = [b 1 , b 2 · · · b L ] where b i ∈ R L×1 for i = 1, 2 · · · L. Rewrite A as A = [e 1 , e 2 · · · e M , 0, 0 · · · 0] , where e j ∈ R L×1 denotes a unit vector with the ith entry is equal to 1 for j = 1, 2 · · · M. Then, it is easy to obtain that Let b ji = e j b i be the jth entry of b i . It can be shown with some manipulations that the problem in (16) is equivalent to With |λ 1 | ≥ |λ 2 | ≥ · · · ≥ |λ L | and In this case, As can be seen, the rank of G t_M is equal to M. The proof is then completed. After the operation of rank reduction, the rank of G t_M is equal to that of G. Additionally, G t_M is most similar to G t in terms of Frobenius norm. Inspired by [9], we reduce the sum of squares of all off-diagonal values of G t_M , namelyμ all , by eigenvalue averaging. When minimizing the difference between G and G t_M , a smallerμ all is more likely to lead to a smaller µ all .μ all can be formulated aŝ whereĝ ij denotes the (i, j)th entry of G t_M andĝ ii = 1 holds for i = 1, 2 · · · L.
Noting that Assuming that with M equal all non-zero eigenvalues is given by Recall that G t_M is most similar to G t in terms of Frobenius norm, it means that G t_M is of good competitiveness in µ ave and µ max . Furthermore, as a variant of G t_M , G t_opt reduces the sum of squares of all off-diagonal values of G t_M , leading to a better performance in minimizing µ all . Therefore, G t_opt is more likely to be an ideal solution of target Gram matrix which leads to better µ ave , µ max , and µ all simultaneously.

The Analytical Solutions of Φ
After obtaining the target Gram matrix G t_opt , the next step of the optimization is to find the best Φ. To handle the problem, we try to find the optimal solution by minimizing the difference between Ψ Φ ΦΨ and G t_opt as min G t_opt − Ψ Φ ΦΨ  (21) and Λ M ∈ R M×M be the Mth principal submatrix ofΛ. Then the solutions of the minimization problem defined by (22)

Theorem 2. Let G t_opt be the matrix shown as Equation
where U Z ∈ R M×M is an arbitrary unitary matrix.
Proof. Assume that the off-diagonal values of Σ D and Σ Ψ are non-zero, Φ can be written as By substituting Φ in (22) with Equation (24), it can be shown with some manipulations that the solutions of the problem in (22) are equivalent to the solutions of Let Z = V D PΛP V D and z i be the ith diagonal entry of Z. Denote Λ Z = diag(z 1 , z 2 · · · z M ). With further manipulations, we simplify (25) to Obviously, the minima are achievable only if Λ Z = Σ 2 D holds and Λ Z 2 F takes the maximum value. Let U = V D P and u ij be the (i, j)th entry of U where i, j = 1, 2 · · · L. Noting that the top M diagonal entries ofΛ are all equal toλ, we have It is worth noting that Λ Z where U 1 ∈ R M×M . Since z i =λ holds for i = 1, 2 · · · M, we have u 2 i1 + u 2 i2 + · · · + u 2 iM = 1 and U 2 = 0, U 3 = 0 accordingly. As U Z is a unitary matrix, it is easy to verify that U 1 and U 4 are both unitary matrices. Then it follows that Substituting Equation (28) into Equation (24), the optimal solution is obtained by The proof is then completed.

Comments
According to Sections 3.1 and 3.2, the procedure for measurement matrix optimization has been summarized in Algorithm 1.

Algorithm 1. The proposed optimization method.
Input: Dictionary matrix Ψ ∈ R N×L N which has an SVD form of Ψ = U Ψ Σ Ψ 0 V Ψ , number of iterations Iter, constant c, Welch bound µ welch . Output: Measurement matrix Φ opt . Initialization: Initialize Φ 0 ∈ R M×N to a random matrix, initial U Z ∈ R M×M to a unitary matrix. For l = 1 to Iter do 1.
Compute the sensing matrix D = Φ l Ψ and normalize the columns in D.

3.
Shrink G and obtain G t by Apply eigenvalue decomposition to G t and obtain G t = PΛP .

5.
Compute the average of the top M diagonal entries in Λ, denoted asλ. 6. Construct

end return Φ Iter
Noting that G t plays an important role in measurement matrix optimization, Algorithm 1 takes µ ave , µ max and µ all into consideration simultaneously when designing the G t . By minimizing (5), the Gram matrix is most similar to G t in terms of Frobenius norm, leading to maintain the advantage of G t in reducing the mutual coherence indexes. Therefore, Algorithm 1 is effective in reducing µ ave , µ max , and µ all .
In the shrinkage function, a different threshold yields different results. Inspired by [22], we propose a shrinkage function shown as (7) which has a new threshold µ welch + c. We have not derived the optimal value of c in theory, but setting c to a proper value can also lead to a moderate result.
After averaging the eigenvalues of G t_M , the first term on the right part of (20) is minimized. However, the diagonal entries of G t_M change accordingly. Hence, we can't assure thatμ all reaches the minima, that is to say, G t_opt may not be the optimal solution in terms of µ all . Fortunately, we find that the change of ii in (20), which means our approach is effective in reducing µ all . The proposed algorithm is an iterative one. The main complexity of Algorithm 1 for each iteration is located at steps 1, 2, 4, and 7. For those steps, the flops required are O(MNL), O ML 2 , O L 3 , and O L 3 respectively. Hence, the complexity of Algorithm 1 is approximate to be O IterL 3 . Since the complexity for similar algorithms in [8,16,17] which apply eigenvalue decomposition or SVD is no less than O IterL 3 , the proposed algorithm has not increased the complexity significantly.

Simulation Results and Discussion
In this section, we conduct simulations to predetermine a suitable c firstly. Then, we examine the mutual coherence indexes and reconstruction performance of the proposed method and compare them with the well-established similar algorithms given in [8,16,17] by presenting the empirical results. Last, we verify the effectiveness of our method with various measurement matrices and dictionary matrices. The iteration number Iter is set to 100 and t is set to µ welch . For a given dictionary matrix Ψ ∈ R 80×120 , x ∈ R 120×1 has a sparse representation as x = Ψs where s is K-sparse and each non-zero entry is randomly positioned with a Gaussian distribution of i.i.d. zero-mean and unit variance. Orthogonal Matching Pursuit (OMP) [23] algorithm is employed in signal reconstruction. Denote ε = x e − x 2 / x 2 the reconstruction error where x e is the reconstructed signal. The reconstruction is identified as a success, called exact reconstruction, provided ε ≤ 10 −6 . Denote P suc the percentage of successful reconstruction. In Sections 4.1-4.3, Φ 0 and Ψ are both Gaussian random matrices.

The Choice of c
Since the analytical solution of c is extremely difficult, here, we conduct a serious of simulations to find a suitable c. Figure 1 illustrates the change tendency of mutual coherence indexes and P suc with argument c. We fix the row number of M to 28, the sparsity to 8, and varies c from 0 to 0.16. The experiment is performed for 1000 random sparse ensembles and the results are recorded.
When c = 0, the shrinkage function shown as (7) is the same as (6). As can be seen from the graphs, when c increases, µ ave increases, µ max and µ all decrease first and then increase, P suc increases firstly and then decreases. µ max and µ all reach the minima when c = 0.02 and c = 0.03 respectively. It is worth noting that appropriate increase of c leads to decrease of µ max and µ all but increase of µ ave . When c = 0.01, better µ max and µ all are obtained and the loss in µ ave is tolerable. Moreover, P suc reaches the maxima. Therefore, 0.01 may be a moderate value for c and c is set to 0.01 in the simulations in Sections 4.2-4.4.

Comparing the Mutual Coherence Indexes
This section presents a series of simulations to compare our method with algorithms given in [8,16,17] on the three mutual coherence indexes of D obtained by D = Φ opt Ψ where Φ opt is the optimized measurement matrix. For convenience, each method is denoted as Propose, Elad, Hong, and Entezari. The down-scaling factor for Elad is set to 0.95. The inner iteration number for Hong is set to 2, which means K-SVD is applied twice in every updating of Φ. The point is set to 0.5 to update the G t in Entezari. Figure 2 illustrates the change tendency of mutual coherence indexes with iteration number for M = 28. As can be seen from the figure, the indexes corresponding to different algorithms all change monotonously with the iteration number. When µ max and µ ave converge, the number of iterations required by our method is almost equal to that of Hong and significantly less than that of Elad. When µ all converges, the number of iterations required by our method is equivalent to that of Entezari and significantly less than that of Hong and Elad. Figure 3 presents the histogram of the absolute off-diagonal values of Φ opt Ψ Φ opt Ψ for M = 28. It is seen from the figure that Elad and Entezari have long tails, showing that the number of off-diagonal values that exceed 0.34 is relatively large. The tail of Hong is shorter than that of Elad and Entezari, and reaches the maximum of 0.34. Compared with Hong, our method has a shorter tail which reaches the maximum of 0.32 and has more off-diagonal values below the µ welch (0.1662).
In conclusion, while effectively reducing µ max and µ all , our method can maintain a small µ ave at the same time. Additionally, the number of iterations required for the convergence of each index of our method is significantly less than that of Elad. Therefore, from the view of mutual coherence indexes, the measurement matrix obtained by our method has better properties than the other three methods. This coincides with the theoretical results obtained in the Section 3.

Comparing the Reconstruction Performance
Case 1. Comparison of the P suc in the noiseless case.
In this case, we conduct two separate CS experiments, first by fixing K = 8 and varying M from 12 to 44 and second by fixing M = 28 and varying K from 4 to 20. Each experiment is performed for 1000 random sparse ensembles and the number of exact reconstruction is recorded. Figures 4 and 5 reveal that the P suc of our method is the highest, which indicates its superiority over the other three methods.      To show the robustness of the proposed method in noisy cases we consider the noisy model y = Φx + v where v is the vector of additive Gaussian noise with zero means. We conduct the experiment by fixing M = 28, K = 8, and varying SNR from 10 to 50 dB. The experiment is performed for 1000 random sparse ensembles and the average reconstruction error is recorded. From Figure 6, we can see that the reconstruction errors decrease with the increase of SNR, and the error of the proposed method is smaller than that of the others.  Table 3 presents that µ all of the Entezari is slightly larger than that of our method. It is interesting to note that the number of off-diagonal entries with smaller absolute values in the Entezari is significantly larger than that of our method from Figure 3. Moreover, it can be seen from Table 2 that µ ave of Hong is slightly lower than that of our method. However, the simulation results show that our method outperforms the others in terms of reconstruction performance. It is also worthy noting that our method reduces µ ave , µ max , and µ all simultaneously, leading to better reconstruction performance in CS. This implies that a single mutual coherence index cannot accurately reflect the actual performance of the methods, and verifies the necessity of using multiple indexes simultaneously in measurement matrix optimization.

Different Kinds of Φ and Ψ Optimized by the Proposed Methods
To analyze the performance of our method with various measurement matrices and dictionary matrices, a serious of simulations are carried out in this section. We choose the measurement matrix as a Gaussian random matrix and a Bernoulli random matrix, and choose the dictionary matrix as a Gaussian random matrix and the DCT matrix, respectively. We compare the mutual coherence indexes and the reconstruction performance before and after optimization. When Ψ is the Gaussian random matrix, Φ belongs to R M×80 and Ψ belongs to R 80×120 . When Ψ is the DCT matrix, Φ belongs to R M×120 and Ψ belongs to R 120×120 . Each experiment is performed for 1000 random sparse ensembles.
The mutual coherence indexes of different measurement matrices Φ with different dictionary matrices Ψ are shown in Figure 7. As seen from the simulations, all the optimized measurement matrices produce smaller µ max , µ ave , and µ all than the random ones.  Figures 8 and 9 present the reconstruction performance of OMP with the optimized measurement matrices and the random ones. It is seen from the graphs in these figures that all the optimized matrices outperform the random ones in terms of the percentage of exact reconstruction.

Conclusions
This paper focused on the optimization of measurement matrix for compressed sensing. To decrease µ max , µ ave , and µ all simultaneously, we designed a new target Gram matrix which was obtained by applying a new shrinkage function to the Gram matrix and updated by performing rank reduction and eigenvalue averaging. Then, we characterized the analytical solutions of the measurement matrix by SVD. Based on alternating minimization, we proposed an iterative method to optimize the measurement matrix. The simulation results show that the proposed method reduces µ max , µ ave , and µ all simultaneously and outperforms the existing algorithms in terms of reconstruction performance. In addition, the proposed method is computationally less expensive than some existing algorithms in the literature.
As detailed, we gave the optimal value of c under a fixed matrix scale through simulation. When the scale changes, the value of c in Section 4.1 may no longer be applicable. Therefore, it is meaningful to find the theoretical 'optimal value' of c. Furthermore, noting that lower mutual coherence indexes mean potentially higher reconstruction performance, further efforts are needed to decrease the indexes simultaneously.

Data Availability Statement:
No new data were created or analyzed in this study. Data sharing is not applicable to this article.

Conflicts of Interest:
The authors declare no conflict of interest.


On the Energy Performance of Iridium Satellite IoT Technology

Most Internet of Things (IoT) communication technologies rely on terrestrial network infrastructure. When such infrastructure is not available or does not provide sufficient coverage, satellite communication offers an alternative IoT connectivity solution. Satellite-enabled IoT devices are typically powered by a limited energy source. However, as of this writing, and to our best knowledge, the energy performance of satellite IoT technology has not been investigated. In this paper, we model and evaluate the energy performance of Iridium satellite technology for IoT devices. Our work is based on real hardware measurements. We provide average current consumption, device lifetime, and energy cost of data delivery results as a function of different parameters. Results show, among others, that an Iridium-enabled IoT device, running on a 2400 mAh battery and sending a 100-byte message every 100 min, may achieve a lifetime of 0.95 years. However, Iridium device energy performance decreases significantly with message rate.

Introduction
In the last two decades, a wide range of communication technologies have been designed or adapted for Internet of Things (IoT) applications. In this space, devices (e.g., sensors or actuators) are typically inexpensive machines with significant resource constraints and relaxed bandwidth requirements [1].
Most IoT technologies rely on some form of terrestrial infrastructure, with a coverage range that may span from a few meters up to several kilometers, depending on the technology [2]. However, when such infrastructure is not available, satellite communication technologies provide an alternative connectivity solution for IoT devices. In fact, satellite communication technologies have recently attracted the attention of the IoT community, since such technologies are experiencing a renaissance [3][4][5][6].
New satellite deployments are planned in the near future, while the number of satellite IoT connections is expected to grow by a factor of~4 up to 2025 [6]. While early satellite systems made use of Geostationary Earth Orbit (GEO), most current and future deployments aim to leverage the lower altitude of Low Earth Orbit (LEO) systems, which offer lower cost, higher bandwidth, lower latency, and lower power consumption. Examples of current main LEO satellite systems include Iridium, Orbcomm, Globalstar, Argos, or Starlink [3].
A fundamental aspect of an IoT technology is its energy performance. This is especially critical for satellite IoT devices, which are typically not powered by the electrical grid. However, to the best of our knowledge, no previous work has focused on the energy performance of a satellite IoT technology. In this paper, we investigate the energy performance of Iridium satellite technology for IoT devices. We derive a current consumption model from measurements carried out on a real Iridium hardware module. We provide the average current consumption, theoretical device lifetime, and energy cost of data delivery 2 of 14 results as a function of different communication protocol and link parameters. Our study shows that an Iridium-enabled IoT device, running on a 2400 mAh battery and sending a 100-byte message every 100 min, may achieve a theoretical lifetime of 0.95 years, whereas the maximum achievable lifetime for the same device (i.e., the asymptotic theoretical device lifetime as the message sending period increases) is 3.55 years. However, the device energy performance decreases significantly as the message rate increases compared to long-range IoT technologies such as LoRaWAN and Sigfox.
The remainder of the paper is organized as follows. Section 2 reviews the related work. Section 3 overviews the Iridium satellite system, along with the communications system used by Iridium IoT devices. Section 4 provides a current consumption model of Iridium IoT satellite technology. In Section 5, we use the model derived in the previous section to evaluate the energy performance of an Iridium IoT device. Section 6 compares the energy performance of Iridium with that of LoRaWAN and Sigfox. Section 7 concludes the paper with the main remarks from this work.

Satellite IoT
A few published works have surveyed satellite IoT systems [3][4][5][6][7]. Centenaro et al. provided a comprehensive survey of the state-of-the-art in the area of satellite IoT, which focuses on current solutions, research, and industrial/commercial initiatives, and future research opportunities, among others [6]. De Sanctis et al. explored the use cases for satellite IoT, and pointed out open issues, especially in terms of connectivity and interoperability between sensors, satellites, and the Internet [7]. The authors created a taxonomy of satellite IoT systems, whereby end devices (e.g., sensors) use satellite connectivity directly or indirectly (in the latter, a terrestrial gateway covering a sensor network uses a satellite link for Internet connectivity). Fraire et al. analyzed the characteristics and open issues of direct-to-satellite systems [3]. Qu et al. focused on the benefits of LEO satellite systems for IoT and provided a holistic LEO satellite architecture overview [5]. Ferrer et al. reviewed the panoply of medium access control (MAC) protocols for satellite IoT systems, with a focus on low-cost nanosatellite systems [4].
Maine et al. presented the main concepts, architecture, and services of the Iridium system [8]. Pratt et al. describe the system fundamentals in more detail, and provide a network performance analysis in terms of communication latency [9]. McMahon et al. performed an experimental latency characterization of Iridium [10].
Iridium has been used as the communication means in many scenarios for both voice and data communications [9,12]. Among the latter, Iridium has enabled many IoT use cases in a diversity of remote scenarios including polar, marine, desert, forest, and UAV-based, among others [11][12][13][14][15]. Iridium has also been exploited for positioning applications [16].

Energy Performance of IoT Technologies
Since many IoT devices are powered by limited energy sources, energy performance is a crucial aspect of an IoT technology. Many studies have focused on modeling, evaluating, and/or improving the energy performance of a wide range of IoT technologies such as IEEE 802.15.4 and ZigBee [17], Bluetooth Low Energy (BLE) [18,19], LoRaWAN [20], Sigfox [21], or NB-IoT [22], among others. However, to our best knowledge, no work in the literature has evaluated the energy performance of a satellite technology for IoT.

Iridium Overview
The Iridium satellite system was originally conceived to comprise 77 satellites, which inspired the name for the system, since the atomic number of the iridium chemical element is 77. The Iridium system finally consisted of 66 satellites.
While most previously existing satellite communication systems used GEO satellites, Iridium was designed based on LEO satellites. This feature was a consequence of the need to offer worldwide voice and data communications coverage for lightweight handheld devices. For such use cases, LEO satellites require lower transmission power and offer lower propagation delay. The altitude of Iridium satellites was around 780 kms. The system became commercially available in 1998. In 2019, the launch of a new generation of Iridium satellites, called Iridium Next, was completed. There are currently 75 Iridium Next satellites in orbit, comprising 66 operational satellites, and nine on-orbit spare satellites. Note that Iridium Next satellites are fully compatible with the initial Iridium system. The altitude of 70 such satellites is 625 kms, whereas the altitude of the remaining five satellites is 720 kms [16]. Figure 1 illustrates the architecture of the Iridium satellite system. An Iridium subscriber unit (i.e., the Iridium modem of the user device) communicates via a radio link with the Iridium satellite constellation. Data from the user device may be routed through several satellites until they are transmitted in the downlink to an Iridium terrestrial gateway or to another user. An Iridium gateway is connected to the Internet, where data can reach an application server.  [17], Bluetooth Low Energy (BLE) [18,19], LoRaWAN [20], Sigfox [21], or NB-IoT [22], among others. However, to our best knowledge, no work in the literature has evaluated the energy performance of a satellite technology for IoT.

Iridium Overview
The Iridium satellite system was originally conceived to comprise 77 satellites, which inspired the name for the system, since the atomic number of the iridium chemical element is 77. The Iridium system finally consisted of 66 satellites.
While most previously existing satellite communication systems used GEO satellites, Iridium was designed based on LEO satellites. This feature was a consequence of the need to offer worldwide voice and data communications coverage for lightweight handheld devices. For such use cases, LEO satellites require lower transmission power and offer lower propagation delay. The altitude of Iridium satellites was around 780 kms. The system became commercially available in 1998. In 2019, the launch of a new generation of Iridium satellites, called Iridium Next, was completed. There are currently 75 Iridium Next satellites in orbit, comprising 66 operational satellites, and nine on-orbit spare satellites. Note that Iridium Next satellites are fully compatible with the initial Iridium system. The altitude of 70 such satellites is 625 kms, whereas the altitude of the remaining five satellites is 720 kms [16]. Figure 1 illustrates the architecture of the Iridium satellite system. An Iridium subscriber unit (i.e., the Iridium modem of the user device) communicates via a radio link with the Iridium satellite constellation. Data from the user device may be routed through several satellites until they are transmitted in the downlink to an Iridium terrestrial gateway or to another user. An Iridium gateway is connected to the Internet, where data can reach an application server. Each satellite supports three antenna arrays supporting 16 beams each, providing a total of 48 spot beams on the Earth's surface, effectively defining 48 covered cells. Note that Iridium satellites are not geostationary (i.e., they do not remain in the same location). Iridium satellites travel at 26,804 km/h, resulting in a complete orbital period of 100.13 min [9]. As a consequence, from a given ground location, a satellite is visible for seven minutes. When a satellite becomes unreachable for a user, an attempt is performed to hand communication to another satellite. In order to enable intersatellite links, satellites are equipped with additional antennas. Each satellite supports three antenna arrays supporting 16 beams each, providing a total of 48 spot beams on the Earth's surface, effectively defining 48 covered cells. Note that Iridium satellites are not geostationary (i.e., they do not remain in the same location). Iridium satellites travel at 26,804 km/h, resulting in a complete orbital period of 100.13 min [9]. As a consequence, from a given ground location, a satellite is visible for seven minutes. When a satellite becomes unreachable for a user, an attempt is performed to hand communication to another satellite. In order to enable intersatellite links, satellites are equipped with additional antennas.
Iridium uses a portion of L-band spectrum, ranging from 1616 MHz to 1626. 5  used for simplex downlink signaling. The duplex band is divided by means of frequency division multiple access (FDMA) into 240 frequency channels of 41.67 kHz bandwidth each, with an operation bandwidth of 31.5 kHz and guard bandwidth of 10.17 kHz. A frequency channel may in turn be shared among different users by means of a time division multiple access (TDMA) scheme. A TDMA frame is defined with a total duration of 90 ms, which comprises an initial signaling simplex header, four uplink timeslots, and four downlink timeslots. The simplex header has a duration of 20.32 ms, whereas each uplink or downlink timeslot has a duration of 8.28 ms. The remaining time up to the total duration of 90 ms corresponds to guard times before and after the simplex header and each timeslot [9]. A frame supports duplex communication for four simultaneous users by using time division duplex (TDD). Each timeslot contains a burst composed of three parts: an unmodulated tone, a unique word, and the data field. The unique word is transmitted by using binary phase shift keying (BPSK) modulation, whereas the data field is transmitted by means of quadrature phase shift keying (QPSK) modulation [16]. One frame carries a total of 4500 bits, leading to a bit rate of 50 kbit/s. A user duplex timeslot carries a total of 414 bits.
Iridium offers different types of communication services including dial-up, direct Internet access, paging, router-based unstructured digital inter-working connectivity solution (RUDICS), and short burst data (SBD). For IoT purposes, that is, infrequent transmission of short payloads (e.g., corresponding to sensor readings) with relaxed latency requirements (e.g., in the order of tens of seconds), SBD is the most suitable service.
SBD is a packet-switched-oriented service that uses forward error correction (FEC) encoding, along with a selective automatic repeat request (ARQ) mechanism. In SBD, the user duplex timeslot comprises 160 SBD information bits, along with 20 SBD header bits, and 234 bits that correspond to other overheads. The 160 SBD information bits are protected by BCH(31,20) FEC encoding [23]. The theoretical user data capacity of a SBD channel (i.e., 160 user data bits are carried every 90 ms) is 1.78 kbit/s. However, the actual user capacity defined for a SBD channel is 125 byte/s (i.e., 1 kbit/s). The maximum size of a message sent by a user (e.g., a sensor device equipped with Iridium satellite connectivity) is 1960 bytes. However, commercial devices may support a smaller maximum message size (e.g., 340 bytes in the RockBLOCK Mk2 model [24]). In fact, IoT applications typically handle small user payloads.
An Iridium IoT device can send and receive data by means of the aforementioned SBD duplex timeslots. When a message is sent to the device, it is stored in a queue at the gateway. If the gateway receives a message from the Iridium IoT device, the first message in the queue is then sent to the device. Therefore, downlink data transmission requires a previous uplink transmission performed by the Iridium IoT device.

Current Consumption Model for Iridium Satellite IoT
This section presents a current consumption model of a device that periodically transmits a message (e.g., a sensor reading) with payload size L, once every time interval, T, by means of the Iridium SBD service. We assume that the device is by default in sleep mode in order to save energy, except when it executes the operations required for message transmission. Based on the current consumption model, the section also shows how the device battery lifetime and energy efficiency of data delivery can be calculated.
In order to provide a realistic model, we performed current consumption measurements on a real device hardware platform, called RockBLOCK Mk2 (Rock Seven Mobile Services, Whiteley, UK). This platform uses the Iridium 9602 module, and supports the Iridium SBD service, with a maximum transmitted message size of 340 bytes. The transmit power used by the device is 32 dBm. Note that the device includes a supercapacitor, which acts as an energy reservoir and allows to buffer the highly pulsed nature of the device internal circuits. Figure 2 illustrates the experimental setup used for the measurements. The Rock-BLOCK device (left) was controlled via AT commands from a computer, while an Agilent Technologies N6705A power analyzer (right) was used to measure the current consumption of the device.
Ssors 2021, 21,7235 pacitor, which acts as an energy reservoir and allows to buffer the highly pulsed of the device internal circuits. Figure 2 illustrates the experimental setup used for the measuremen RockBLOCK device (left) was controlled via AT commands from a computer, w Agilent Technologies N6705A power analyzer (right) was used to measure the consumption of the device. Measurements were performed in an outdoor scenario, in line-of-sight con (i.e., without obstacles between the RockBLOCK device and the Iridium satellites) 3 shows the location of the RockBLOCK device, and the rest of the experimental s the mentioned outdoor scenario. We next provide the current consumption pattern and main characteristics considered Iridium IoT device for a successful message transmission, for L = 1 b Figure 4). The device is initially in a sleep state, then it becomes active to perform cessful message transmission, and finally, it returns to the sleep state. Measurements were performed in an outdoor scenario, in line-of-sight conditions (i.e., without obstacles between the RockBLOCK device and the Iridium satellites). Figure 3 shows the location of the RockBLOCK device, and the rest of the experimental setup, in the mentioned outdoor scenario.
pacitor, which acts as an energy reservoir and allows to buffer the highly pulsed nature of the device internal circuits. Figure 2 illustrates the experimental setup used for the measurements. The RockBLOCK device (left) was controlled via AT commands from a computer, while an Agilent Technologies N6705A power analyzer (right) was used to measure the current consumption of the device. Measurements were performed in an outdoor scenario, in line-of-sight conditions (i.e., without obstacles between the RockBLOCK device and the Iridium satellites). Figure  3 shows the location of the RockBLOCK device, and the rest of the experimental setup, in the mentioned outdoor scenario. We next provide the current consumption pattern and main characteristics for the considered Iridium IoT device for a successful message transmission, for L = 1 byte (see Figure 4). The device is initially in a sleep state, then it becomes active to perform a successful message transmission, and finally, it returns to the sleep state. We next provide the current consumption pattern and main characteristics for the considered Iridium IoT device for a successful message transmission, for L = 1 byte (see Figure 4). The device is initially in a sleep state, then it becomes active to perform a successful message transmission, and finally, it returns to the sleep state.
Remarkably, while the device is active, it spends an average of 6.0 s involved in tasks that do not correspond to payload transmission such as device processing, protocol overhead, etc. This time is significant, even when compared with the nominal data transmission time of up to 2.72 s for the maximum payload size allowed in SBD (i.e., 340 bytes). Recall that, as mentioned in Section 3, despite the theoretical user capacity of 1.78 kbit/s, Iridium SBD offers an actual user capacity of 1 kbit/s. Payload transmission time is determined as the payload size in bits divided by the user capacity of 1 kbit/s. On the other hand, the device consumes a greater amount of current after transmission has finished, during which the device returns to the sleep state and its current consumption decreases slowly. Such a slow decrease is due to the supercapacitor included in the device. The average current consumption of the considered successful message transmission (computed since the device receives an AT command to start message transmission, until 3 min later), denoted as I OK , is 7.66 mA. The current consumption of the device when it returns to sleep mode tends asymptotically to be 77.0 µA, after more than 24 h in the absence of communication activity. Remarkably, while the device is active, it spends an average of 6.0 s involved in tasks that do not correspond to payload transmission such as device processing, protocol overhead, etc. This time is significant, even when compared with the nominal data transmission time of up to 2.72 s for the maximum payload size allowed in SBD (i.e., 340 bytes). Recall that, as mentioned in Section 3, despite the theoretical user capacity of 1.78 kbit/s, Iridium SBD offers an actual user capacity of 1 kbit/s. Payload transmission time is determined as the payload size in bits divided by the user capacity of 1 kbit/s. On the other hand, the device consumes a greater amount of current after transmission has finished, during which the device returns to the sleep state and its current consumption decreases slowly. Such a slow decrease is due to the supercapacitor included in the device. The average current consumption of the considered successful message transmission (computed since the device receives an AT command to start message transmission, until 3 min later), denoted as IOK, is 7.66 mA. The current consumption of the device when it returns to sleep mode tends asymptotically to be 77.0 μA, after more than 24 h in the absence of communication activity.
We next focused on the current consumption behavior of the Iridium IoT device when it performs a message transmission in apparently good visibility conditions, but the message cannot be successfully delivered, for L = 1 byte ( Figure 5). In line-of-sight conditions, such an event may mainly occur due to an intersatellite handover or due to contention for communication resources. We observe that, in this case, the average current consumption of the device increased; we speculate that the device performs lower layer retransmission attempts, and after unsuccessful message delivery, the message transmission is considered failed, and the device returns to sleep mode. However, additional upper-layer retransmissions of the same message can be triggered subsequently by the application. We next focused on the current consumption behavior of the Iridium IoT device when it performs a message transmission in apparently good visibility conditions, but the message cannot be successfully delivered, for L = 1 byte ( Figure 5). In line-of-sight conditions, such an event may mainly occur due to an intersatellite handover or due to contention for communication resources. We observe that, in this case, the average current consumption of the device increased; we speculate that the device performs lower layer retransmission attempts, and after unsuccessful message delivery, the message transmission is considered failed, and the device returns to sleep mode. However, additional upper-layer retransmissions of the same message can be triggered subsequently by the application. Based on our measurements, we found that the average time since the start of the operation until the instant in which the device returns to sleep mode, when message delivery is unsuccessful, is 20.0 s. This duration, which is greater than the one in a successful message transmission, is consistent with our speculation that lower layer retries may have been performed.
The average current consumption corresponding to the considered unsuccessful message transmission, since the device receives the command to start message transmission until 3 min later, denoted IKO, is 10.1 mA. Based on our measurements, we found that the average time since the start of the operation until the instant in which the device returns to sleep mode, when message delivery is unsuccessful, is 20.0 s. This duration, which is greater than the one in a successful message transmission, is consistent with our speculation that lower layer retries may have been performed.
The average current consumption corresponding to the considered unsuccessful message transmission, since the device receives the command to start message transmission until 3 min later, denoted I KO , is 10.1 mA.
We next calculated the average current consumption of the considered Iridium device, denoted I Avg , taking into account that unsuccessful message transmission may occur, with probability p, and assuming that an application-layer mechanism triggers a message retransmission after T Retry if the last message transmission attempt has failed. We also assume that successive retries of a failed transmission attempt are mutually independent. Figure 6 illustrates the device current consumption in an example where three transmission attempts are needed for successful delivery of a message. Based on our measurements, we found that the average time since the start of the operation until the instant in which the device returns to sleep mode, when message delivery is unsuccessful, is 20.0 s. This duration, which is greater than the one in a successful message transmission, is consistent with our speculation that lower layer retries may have been performed.
The average current consumption corresponding to the considered unsuccessful message transmission, since the device receives the command to start message transmission until 3 min later, denoted IKO, is 10.1 mA.
We next calculated the average current consumption of the considered Iridium device, denoted IAvg, taking into account that unsuccessful message transmission may occur, with probability p, and assuming that an application-layer mechanism triggers a message retransmission after TRetry if the last message transmission attempt has failed. We also assume that successive retries of a failed transmission attempt are mutually independent. Figure 6 illustrates the device current consumption in an example where three transmission attempts are needed for successful delivery of a message. Let TAct denote the duration of the interval since the Iridium device starts the process to perform a message transmission, until the message is successfully delivered. Note that, after an unsuccessful message transmission, there will be an application-layer-triggered Let T Act denote the duration of the interval since the Iridium device starts the process to perform a message transmission, until the message is successfully delivered. Note that, after an unsuccessful message transmission, there will be an application-layer-triggered message retransmission. The time between the start of two consecutive message transmission attempts is equal to T Retry (see Figure 6).
Let T Sleep denote the time since successful message transmission until the start of the first transmission attempt of the next message. We assume T Act < T, which will hold in a practical scenario. Therefore, T Sleep + T Act = T.
Let I Act and I Sleep indicate the average current consumption of the device during T Act and T Sleep , respectively.
Based on the defined variables, the average current consumption of the device, I avg , can be calculated as follows: where E[·] indicates the expected value of the corresponding variables. We started to calculate the terms in (1) by first determining E[I Act ]. For the sake of analytical simplicity, we assumed that T Act comprises an integer number of intervals of duration T Retry including the interval where the successful message transmission occurs (i.e., the last interval, see Figure 6). Let I OK and I KO denote the average current consumption of an interval of duration T Retry , where successful and unsuccessful message transmission have occurred, respectively. Let n indicate the number of intervals of duration T Retry that may occur within T.
The above equation can be expressed as follows: We next determined E[T Act ], under the considered assumptions. E[T Act ] can be determined as follows: The above equation can be expressed as shown next: The next term from (1) that we determined is E[T Sleep ], which can be easily found as follows: Finally, E[T Sleep ] can be determined empirically. This value decreases with T, and it tends asymptotically to 80.7 µA.
Therefore, I avg can be calculated based on the presented Equations ((1)- (6)) and values. Assuming that the Iridium IoT device runs on a battery of capacity C batt , the theoretical lifetime of the device can be calculated as follows: Note that the theoretical device lifetime calculated as per (7) assumes a battery of ideal characteristics. A real battery will typically exhibit non-ideal features (e.g., a certain degree of self-discharge) that will lead to a lower actual device lifetime. Therefore, the theoretical device lifetime must be considered as an upper bound of the actual device lifetime.
The last performance metric th we considered was the energy efficiency of data transmission when using the Iridium device. This metric can be calculated in terms of the energy cost per delivered user data bit, EC db , as shown next, where V indicates the battery voltage:

Evaluation
On the basis of the models provided in the previous section, this section evaluates the average current consumption and the lifetime of the considered Iridium IoT device as well as the corresponding energy cost per delivered bit. The section is organized into three subsections, which focus on each of the aforementioned performance parameters, respectively.
We performed the evaluation for a wide range of T values. We also considered different values for p, which allowed us to capture ideal and non-ideal conditions (note that, in absence of application-triggered retries, we experimentally observed a value of p ≈ 0.1). We assumed T Retry = 3 min. Figure 7 illustrates the average current consumption of the device as a function of T, for different p values, based on the model in Section 4. As expected, the average current consumption decreased with T as the influence of sleep intervals increased, and the average current consumption tends asymptotically to sleep state current consumption. sumption), and retries have a relatively greater impact than for smaller values of T. As T increases further, the active intervals related with message transmission become less relevant compared with very long sleep intervals, rendering the active intervals asymptotically negligible.

Device Current Consumption
We also analyzed the impact of the payload size on the average current consumption. Note that payload transmission time contributes to both IOK and IKO, therefore, it also contributes to E [IAct]. As payload size increases, E[IAct] also increases, which in turn increases the average current consumption. While such an impact decreases with T, we found that payload size actually had a low influence on the average current consumption for the whole range of considered T values (of 6.5% for T = 10 min, and below 1% for the rest of the considered T values). For the sake of illustration clarity, Figure 6 only shows the results for a 1-byte message payload size, since curves for the rest of the considered payload sizes (L ≤ 300 bytes) show negligible differences.  Figure 8 also depicts the average current consumption of the device as a function of T. In this case, the figure compares the results provided by the model with results obtained by measuring the average current consumption over intervals of a duration one order of magnitude greater than each corresponding value of T (except for T = 10,000 min, where the duration was equal to T). In the measurements, all transmissions were successful, therefore we show the model results for p = 0. As shown in Figure 8, the model generally predicts current consumption with high accuracy, especially for T ≥ 100 min. For T = 10 min, the difference between the model and the measured results is greater (equal to 10%), since the transmission part, which is prone to variability depending on communication conditions, becomes more relevant. The average current consumption increased with p, since the number of applicationlayer retries also increases. However, impact of p on the average current consumption varies depending on the value of T. For p = 0.1, and relatively low values of T (e.g., T = 10 min), the average current consumption increases by a factor of~20%. As T increases (e.g., for T = 100 min), such a factor reaches~31%, since intervals where the device slowly return to sleep become more significant (with greater duration and lower current consumption), and retries have a relatively greater impact than for smaller values of T. As T increases further, the active intervals related with message transmission become less relevant compared with very long sleep intervals, rendering the active intervals asymptotically negligible.
We also analyzed the impact of the payload size on the average current consumption. Note that payload transmission time contributes to both I OK and I KO , therefore, it also contributes to E[T Act ]. As payload size increases, E[T Act ] also increases, which in turn increases the average current consumption. While such an impact decreases with T, we found that payload size actually had a low influence on the average current consumption for the whole range of considered T values (of 6.5% for T = 10 min, and below 1% for the rest of the considered T values). For the sake of illustration clarity, Figure 6 only shows the results for a 1-byte message payload size, since curves for the rest of the considered payload sizes (L ≤ 300 bytes) show negligible differences. Figure 8 also depicts the average current consumption of the device as a function of T. In this case, the figure compares the results provided by the model with results obtained by measuring the average current consumption over intervals of a duration one order of magnitude greater than each corresponding value of T (except for T = 10,000 min, where the duration was equal to T). In the measurements, all transmissions were successful, therefore we show the model results for p = 0. As shown in Figure 8, the model generally predicts current consumption with high accuracy, especially for T ≥ 100 min. For T = 10 min, the difference between the model and the measured results is greater (equal to 10%), since the transmission part, which is prone to variability depending on communication conditions, becomes more relevant.

Device Lifetime
We next evaluate the theoretical lifetime of an Iridium device, based on (7), and considering a battery capacity of 2400 mAh, as a function of T and p.
As shown in Figure 9, lifetime increases asymptotically with T. The maximum lifetime was 3.39 years. Remarkably, a lifetime of one year could only be achieved for values of T greater than 100 min. As T decreased below 100 min, device lifetime decreased abruptly. For example, the device lifetime for T = 10 min was only 43.8 days.
As expected, device lifetime also decreases with p due to the greater current consumption caused by retries. The impact of a non-zero p on asymptotic device lifetime, for practical link qualities, is negligible.  Figure 10 illustrates the energy cost per delivered bit of user data, as a function of T, L, and p, based on (8). This performance parameter increases with T due to the energy consumed during sleep intervals. On the other hand, increasing the user data payload size amortizes the energy required to deliver a user data bit. The latter may be reduced by up to between two and three orders of magnitude.

Device Lifetime
We next evaluate the theoretical lifetime of an Iridium device, based on (7), and considering a battery capacity of 2400 mAh, as a function of T and p.
As shown in Figure 9, lifetime increases asymptotically with T. The maximum lifetime was 3.39 years. Remarkably, a lifetime of one year could only be achieved for values of T greater than 100 min. As T decreased below 100 min, device lifetime decreased abruptly. For example, the device lifetime for T = 10 min was only 43.8 days.

Device Lifetime
We next evaluate the theoretical lifetime of an Iridium device, based on (7), and considering a battery capacity of 2400 mAh, as a function of T and p.
As shown in Figure 9, lifetime increases asymptotically with T. The maximum lifetime was 3.39 years. Remarkably, a lifetime of one year could only be achieved for values of T greater than 100 min. As T decreased below 100 min, device lifetime decreased abruptly. For example, the device lifetime for T = 10 min was only 43.8 days.
As expected, device lifetime also decreases with p due to the greater current consumption caused by retries. The impact of a non-zero p on asymptotic device lifetime, for practical link qualities, is negligible.  Figure 10 illustrates the energy cost per delivered bit of user data, as a function of T, L, and p, based on (8). This performance parameter increases with T due to the energy consumed during sleep intervals. On the other hand, increasing the user data payload size amortizes the energy required to deliver a user data bit. The latter may be reduced by up to between two and three orders of magnitude. As expected, device lifetime also decreases with p due to the greater current consumption caused by retries. The impact of a non-zero p on asymptotic device lifetime, for practical link qualities, is negligible. Figure 10 illustrates the energy cost per delivered bit of user data, as a function of T, L, and p, based on (8). This performance parameter increases with T due to the energy consumed during sleep intervals. On the other hand, increasing the user data payload size amortizes the energy required to deliver a user data bit. The latter may be reduced by up to between two and three orders of magnitude. Message delivery errors increase the energy cost per delivered bit due to the additional cost of retransmissions as well as the greater energy consumption of a failed message transmission attempt.

Comparison with Other IoT Technologies
In order to assess the energy performance results obtained in the previous section from a broader perspective, this section compares the energy performance of Iridium satellite IoT technology with that of prominent long-range terrestrial IoT technologies such as LoRaWAN or Sigfox. The two latter technologies offer a communication link range typically 1-2 orders of magnitude lower than the Iridium one. While energy performance of a given technology depends on how the technology is configured, and on how it is implemented on real hardware, this section aims to explore the energy performance of the different communication technologies on representative examples using reasonably common settings. Figure 11 depicts the lifetime of a 2400 mAh-battery-operated device sending a message (carrying a 12-byte payload) for different values of T and for p = 0, for Iridium, Sigfox, and LoRaWAN. In all cases, an ideal line-of-sight channel has been assumed. In addition, a single user was assumed (i.e., there was no interference from other users). For Sigfox and LoRaWAN, the calculations were performed by using published energy models and device settings for operation in the 868 MHz band [20,21]. More specifically, for the Sigfox evaluation, radio configuration (RC) number 1 (i.e., RC1) and unidirectional data message transmission were assumed [21]. In RC1, the uplink and downlink bit rate for RC1 was 100 bit/s. For the LoRaWAN evaluation, the sender device belonging to Class A, and data rate (DR) 0, which corresponds to spreading factor (SF) 12 and a bit rate of 250 bit/s, has been used [20]. Under the considered conditions, and due to spectrum access regulations, both LoRaWAN and Sigfox need to enforce a duty cycle below 1%. Note that Iridium satellite IoT technology is also assumed to be duty cycled in this study, since for feasibility and for a practical battery lifetime, the IoT device must return to sleep mode after message transmission.
For a fair comparison of the asymptotic device lifetime (which is independent of communication activity), the same battery capacity and asymptotic sleep current have been assumed for all three considered technologies. Note that while the transmission power used by the Iridium device was 32 dBm, the transmit power considered in the Message delivery errors increase the energy cost per delivered bit due to the additional cost of retransmissions as well as the greater energy consumption of a failed message transmission attempt.

Comparison with Other IoT Technologies
In order to assess the energy performance results obtained in the previous section from a broader perspective, this section compares the energy performance of Iridium satellite IoT technology with that of prominent long-range terrestrial IoT technologies such as LoRaWAN or Sigfox. The two latter technologies offer a communication link range typically 1-2 orders of magnitude lower than the Iridium one. While energy performance of a given technology depends on how the technology is configured, and on how it is implemented on real hardware, this section aims to explore the energy performance of the different communication technologies on representative examples using reasonably common settings. Figure 11 depicts the lifetime of a 2400 mAh-battery-operated device sending a message (carrying a 12-byte payload) for different values of T and for p = 0, for Iridium, Sigfox, and LoRaWAN. In all cases, an ideal line-of-sight channel has been assumed. In addition, a single user was assumed (i.e., there was no interference from other users). For Sigfox and LoRaWAN, the calculations were performed by using published energy models and device settings for operation in the 868 MHz band [20,21]. More specifically, for the Sigfox evaluation, radio configuration (RC) number 1 (i.e., RC1) and unidirectional data message transmission were assumed [21]. In RC1, the uplink and downlink bit rate for RC1 was 100 bit/s. For the LoRaWAN evaluation, the sender device belonging to Class A, and data rate (DR) 0, which corresponds to spreading factor (SF) 12 and a bit rate of 250 bit/s, has been used [20]. Under the considered conditions, and due to spectrum access regulations, both LoRaWAN and Sigfox need to enforce a duty cycle below 1%. Note that Iridium satellite IoT technology is also assumed to be duty cycled in this study, since for feasibility and for a practical battery lifetime, the IoT device must return to sleep mode after message transmission. Sigfox and LoRaWAN energy models used were 14 dBm and 11 dBm, respectively. These transmit power settings stem from the need to comply with European regulations on the frequency bands used by Sigfox and LoRaWAN (which limit transmit power to 14 dBm). Figure 11. Lifetime of a device running on a 2400 mAh battery for Iridium, Sigfox. and LoRaWAN, as a function of T and for p = 0.
As shown in Figure 11, as T increases, device battery lifetime becomes less dependent on the communication technology, since sleep intervals become dominant. However, as T decreases, the device lifetime for Iridium becomes significantly lower than that achieved for LoRaWAN and Sigfox (e.g., by factors of 7.8 and 5.8 for T = 10 min, respectively). Such a difference cannot only be attributed to the transmission power of the Iridium device, which leads to a current consumption of up to 45.8 mA (Figure 4), since the current consumption in transmit state for the LoRaWAN and Sigfox devices is 82.8 mA and 27.5 mA, in their respective energy models. There are two main contributors to Iridium device energy consumption underperformance: (i) the current consumption penalty introduced by its supercapacitor, especially when the device returns to sleep mode after communication activity, and (ii) the large processing and communication overhead, which corresponds to 6.0 s (for a nominal transmission time of 2.72 s for a 340-byte user payload) during which the device consumes a significant amount of current.

Conclusions
To our best knowledge, in this work, we provided the first energy performance study of a satellite IoT technology. We first performed a current consumption characterization of an Iridium device, based on real hardware. On that basis, we derived a current consumption model and evaluated the energy performance of an Iridium device in terms of average current consumption, device lifetime, and energy cost of data delivery. We also compared the energy performance of Iridium technology for IoT use cases with that of well-established long range IoT technologies such as LoRaWAN and Sigfox.
Energy-wise, Iridium is a suitable technology for relatively infrequent message transmission. The asymptotic lifetime of the battery-operated Iridium IoT device considered, for a battery capacity of 2400 mAh, is 3.55 years. For very low message rates (e.g., once per day), it performed similarly to LoRaWAN or Sigfox. However, Iridium energy underperformance increases with message rate, compared to the latter technologies. However, Iridium still offers nearly a one-year lifetime for a rate of one message every 100 min. For a fair comparison of the asymptotic device lifetime (which is independent of communication activity), the same battery capacity and asymptotic sleep current have been assumed for all three considered technologies. Note that while the transmission power used by the Iridium device was 32 dBm, the transmit power considered in the Sigfox and LoRaWAN energy models used were 14 dBm and 11 dBm, respectively. These transmit power settings stem from the need to comply with European regulations on the frequency bands used by Sigfox and LoRaWAN (which limit transmit power to 14 dBm).
As shown in Figure 11, as T increases, device battery lifetime becomes less dependent on the communication technology, since sleep intervals become dominant. However, as T decreases, the device lifetime for Iridium becomes significantly lower than that achieved for LoRaWAN and Sigfox (e.g., by factors of 7.8 and 5.8 for T = 10 min, respectively). Such a difference cannot only be attributed to the transmission power of the Iridium device, which leads to a current consumption of up to 45.8 mA (Figure 4), since the current consumption in transmit state for the LoRaWAN and Sigfox devices is 82.8 mA and 27.5 mA, in their respective energy models. There are two main contributors to Iridium device energy consumption underperformance: (i) the current consumption penalty introduced by its supercapacitor, especially when the device returns to sleep mode after communication activity, and (ii) the large processing and communication overhead, which corresponds to 6.0 s (for a nominal transmission time of 2.72 s for a 340-byte user payload) during which the device consumes a significant amount of current.

Conclusions
To our best knowledge, in this work, we provided the first energy performance study of a satellite IoT technology. We first performed a current consumption characterization of an Iridium device, based on real hardware. On that basis, we derived a current consumption model and evaluated the energy performance of an Iridium device in terms of average current consumption, device lifetime, and energy cost of data delivery. We also compared the energy performance of Iridium technology for IoT use cases with that of well-established long range IoT technologies such as LoRaWAN and Sigfox.
Energy-wise, Iridium is a suitable technology for relatively infrequent message transmission. The asymptotic lifetime of the battery-operated Iridium IoT device considered, for a battery capacity of 2400 mAh, is 3.55 years. For very low message rates (e.g., once per day), it performed similarly to LoRaWAN or Sigfox. However, Iridium energy underperformance increases with message rate, compared to the latter technologies. However, Iridium still offers nearly a one-year lifetime for a rate of one message every 100 min.
This work focused on the use case of a battery-operated IoT device using Iridium to send messages periodically such as a sensor. However, the work also allows us to conclude that a battery-operated actuator using Iridium might only be feasible for latency-tolerant applications. Since delivering a message (e.g., a command) to such an Iridium device requires the device itself to previously send a data message, and the time between uplink message transmissions needs to be relatively long for a practical battery lifetime, sending a command to an actuator may involve a significant delay (e.g., in the order of hours, or even greater).