In a WSN application scenario, the P Radonlike projections correspond to randomly weighted sums of the values sensed by different subsets of WSN sensors. The sums can be evaluated using different techniques.
Following the approach in [24], the sensors within a subset can synchronously transmit their weighted sensed values, and the sum can be realized at the FC by onair analogical superimposition of received signals. This protocol requires strict control of the power received by the FC from each sensor. Precisely, each sensor node needs to estimate the channel seen towards the FC in order to precompensate the transmitted value according to the channel attenuation. Thereby, although feasible in principle, this approach requires a sophisticated processing and tight power control by the sensor nodes.
According to a data gathering paradigm, the projections are computed within the network by a subset of sensors while they are forwarding their sensed values to the FC. The sums in (7) can then be realized by routing and accumulating values of z[n
_{1},n
_{2}] over suitably tilted paths in the network grid discrete support.
Here we refer to such a data gathering approach, and we infer some consequences from the peculiar sparse structure of the matrix Φ
_{
R
} on the computation procedure. Firstly, we observe that in every row, the nonzero coefficients of the matrix Φ
_{
R
} are arranged so as to obtain {y}^{({\vartheta}_{p})}[m] as the sum of the values of a column of the rotated image {\mathbf{z}}^{({\vartheta}_{p})}. When collecting the measurement in a WSN, each projection {y}^{({\vartheta}_{p})}[m] can then be computed by accumulating measurements throughout a specific, suitably tilted, grid path. Secondly, we observe that in every column of the matrix Φ
_{
R
}, there are only P nonzero values. Hence, each value z[i,j] shall contribute only to P out of M projections {y}^{({\vartheta}_{p})}[m]. When realizing the Radonlike CS in the sensor network grid, each sensor shall transmit its value P times.
Based on these premises, we recognize that the sparse structure of the matrix Φ
_{
R
} results into two main features of Radonlike projection computation in a WSN:

The computation of each projection {y}^{({\vartheta}_{p})}[m] is performed in a distributed way within the WSN, and it requires signaling among grid sensors which are adjacent along a WSN path.

The accumulation along different paths can then be realized in parallel, provided that the distance between contemporaneously signaling nodes is kept large enough.
The data gathering scheme shown in Figure 3 is a case in point. A square network grid is employed to flood the data towards the FC. While forwarding the data along the linear paths (in red in Figure 3), the node partially accumulates them and therefore cooperates to incremental computation of the Radonlike CS measurements. To alleviate the load of the nodes in the FC nearby, the network can be divided into four quadrants that sequentially contribute to the data collection. As we will show in the following, this quadrantbased approach may be used to derive a specific data gathering procedure.
We now turn to quantifying the energy consumption and bandwidth occupancy required for realizing the Radonlike CS by means of a data gathering procedure in a WSN. In Section 6.1, we evaluate the energy consumption and bandwidth occupancy of the Radonlike CS scheme in a WSN. Next, we compare these results with selected stateoftheart schemes, namely those of the random sensing approaches described in Section 1, detailed in Section 6.2.
6.1 Radonlike CS efficiency
We now evaluate the allocated bandwidth and consumed energy for entirely collecting the measurements in a time T
_{
c
} in the WSN scenario in Section 2. The actual bandwidth occupation and energy consumption depend not only on the WSN structure but also on the adopted data gathering procedure. Without loss of generality, we refer to the suboptimal data gathering algorithm sketched out in Appendix 2, here briefly summarized for the reader’s convenience.
According to the algorithm in Appendix 2, the sensors transmit their readings to the FC by data gathering through suitable multihop paths. Figures 4 and 5 illustrate, as an example, the multihop paths selected for the computation of the projections y
^{(π/2)}[m],y
^{(π/4)}[m] within a network quadrant.
A deterministic (collisionfree) time division multiple access (TDMA) is adopted. Signaling takes place between adjacent nodes only. Parallel transmission of nodes sufficiently apart is considered. In Appendix 2, we evaluate two parameters that directly affect the energy and bandwidth consumption of the data gathering algorithm, namely the total number of singlehop transmissions (N
_{TX}) and the total number of time slots (N
_{TS}) needed to collect all the sensors’ readings to the FC.
The total number of node transmission, N
_{TX}, for the algorithm under concern comes out to vary linearly with the network size:
{N}_{\text{TX}}\approx {\gamma}_{P}\xb7N,
(13)
γ
_{
P
} being a scalar factor depending on the adopted P projections. The guidelines for calculating γ
_{
P
} are given in Appendix 2 where we also evaluate the value of γ
_{
P
} for different sets of directions ϑ
_{
p
}.
Based on the same data gathering algorithm, we have evaluated the number of time slots needed to collect all the sensors’ readings to the FC. The number of time slots turns out to vary just with the square root of the network size, that is,
{N}_{\text{TS}}\approx {\delta}_{P}\xb7\sqrt{N},
(14)
δ
_{
P
} being a factor depending on the considered projection directions ϑ
_{
p
}. Appendix 2 reports the guidelines for the evaluation of the parameter δ
_{
P
} for different sets of directions ϑ
_{
p
}.
We observe that, as a consequence of the sparsity of the Radonlike sensing matrix, (1) the number of transmission varies only linearly with the network size N, and (2) the algorithm being parallelized, the number of time slots varies linearly with the square root of the network size N. With these results, we are able to evaluate the allocated bandwidth and consumed energy for entirely collecting the measurements in a time T
_{
c
} in the WSN scenario described in Section 2.
Projection evaluation according to the data gathering scheme detailed in Appendix 2 accounts for a series of transmissions among neighboring nodes. The energy spent for a singlehop transmission is given by {E}_{\text{SH}}=G{d}_{\text{SH}}^{2}{T}_{p}^{(\text{RL})}, with d
_{SH} = d or \sqrt{2}d being the distance for horizontally, vertically, and diagonally adjacent nodes (a scale factor depending on the actual transmission parameters, namely the sensitivity at the FC receiver and the transmitter and receiver antenna gains) and {T}_{p}^{(\mathit{\text{RL}})} the time needed for packet transmission.
Overall, we can express the total consumed energy^{c} for the Radonlike CSbased approach as
{E}_{\text{RL}}={N}_{\text{TX}}G\phantom{\rule{2.77626pt}{0ex}}{d}^{2}{T}_{p}^{(\text{RL})}.
(15)
The packet duration {T}_{p}^{(\text{RL})} depends on the design of the selected sensing system. If the overall sensing framework is designed under the system constraint of having a fixed occupied bandwidth B, the packet duration time will be evaluated as {T}_{p}^{(\text{RL})}=L/{B}_{\text{RL}} with B
_{RL} = B.
This approach is suited to an application scenario where the bandwidth devoted to intersensor communications is fixed in advance. Other possible system constraints concern the time interval during which a whole set of measurement is acquired. In this case, the time for refreshing of the measurement is fixed to T
_{
c
}. Stemming on such a design constraint, the packet duration time needed to assure that the measurements are collected within a maximum time of T
_{
c
} is written as follows:
{T}_{p}^{(\text{RL})}\le \frac{{T}_{c}}{{N}_{\text{TS}}}.
(16)
Under this setting, the minimum occupied bandwidth, defined as the packet length in bits L over the packet transmission time, for the Radonlike approach is evaluated as
{B}_{\text{RL}}=\frac{L}{{T}_{p}^{(\text{RL})}}\ge \frac{L}{{T}_{c}}{\delta}_{P}\xb7\sqrt{N}.
(17)
The relation providing the energy consumption in (15) can then be exploited either by considering an assigned packet time arising from a system bandwidth constraint or by assuming a given sensing rate 1/T
_{
c
}. In the latter case, the packet duration is evaluated as in (16) so that the consumed energy reads as follows:
{E}_{\text{RL}}=\frac{{\gamma}_{P}}{{\delta}_{P}}{T}_{c}G\phantom{\rule{2.77626pt}{0ex}}{d}^{2}\xb7\sqrt{N}.
(18)
Such system design choices should be carefully taken into consideration when comparing energy consumption of different schemes possibly comprising different numbers of singlehop transmissions N
_{TX}.
When performing numerical simulations, we have considered both the two aforementioned cases, namely:

Fixed packet duration for the different compared schemes: this corresponds to a fixed system bandwidth constraint (cfr. results in Figure 6).

Fixed sensing procedure duration T
_{
c
}: this corresponds to different packet durations (cfr. results in Figure 7).
6.2 RS efficiency
We now present a few results on the energy and bandwidth consumption of the approaches proposed in [10]. Therein, a RS procedure, allowing only a randomly chosen subset of sensors to acquire the measurement, is coupled with both a TDMA scheme and a random access scheme. Herein we elaborate on these results and add a few details. With respect to the computation in [10], where the energy consumption for each sensor to transmit to the network sink is approximated by a constant, here we explicitly take into account the dependence of the energy with respect to (w.r.t.) the spatial sensor location. Secondly, therein an approximate relation is established between two key system parameters, namely (1) the minimal fraction of sensing data that must be correctly received at the sink to allow CS reconstruction and (2) the minimal bandwidth. Herein we extend the relation to different ranges of sensing probability, better suited to RS of a spatially sparse field.
In the RS/deterministic access (RD) scheme, the FC randomly chooses a set of M sensors, M being a sufficient number of measurements for satisfactory field reconstructions and broadcasting the addresses of eliged nodes through the network. The selected nodes acquire the measurements and transmit their readings to the FC via a TDMA deterministic access scheme. As in this scheme only M nodes need to share the TDMA frame, the packet transmission time is {T}_{p}^{(\text{RD})}={T}_{c}/M. Consequently, the occupied bandwidth for the RD scheme is {B}_{\text{RD}}=M\frac{L}{{T}_{c}}.
In order to evaluate the consumed energy, let us consider the set {\mathcal{C}}_{M;N} collecting all the possible configurations of M out of N nodes. The energy consumption of a given configuration c\in {\mathcal{C}}_{M;N} of M is expressed as {E}^{(c)}=\sum _{{k}_{c}}G\phantom{\rule{2.77626pt}{0ex}}{d}_{{k}_{c}}^{2}{T}_{p}^{(\text{RD})} where the sum over the index k
_{
c
} spans the M sensors within the combination c. Thereby, the energy consumption of each combination depends on the distances of the M nodes from the FC. In this respect, we evaluate the energy consumption of the RD scheme as the average over all the possible combinations: {E}_{\text{RD}}=\frac{1}{{K}_{M;N}}\sum _{c\in {\mathcal{C}}_{M;N}}\sum _{{k}_{c}}G\phantom{\rule{2.77626pt}{0ex}}{d}_{{k}_{c}}^{2}{T}_{p}^{(\text{RD})} where {K}_{M;N}=\left(\genfrac{}{}{0ex}{}{N}{M}\right) is the cardinality of {\mathcal{C}}_{M;N}. We recognize that in the overall sum over the K
_{
M;N
} combinations, the energy spent by each and every network sensor appears in \left(\genfrac{}{}{0ex}{}{N1}{M1}\right) terms, corresponding to the combinations it belongs to. Therefore, denoting {N}_{1}\stackrel{\text{def}}{=}{\alpha}_{1}\sqrt{N}, {N}_{2}\stackrel{\text{def}}{=}{\alpha}_{2}\sqrt{N}, the above sum can be rewritten as
{E}_{\text{RD}}=M\frac{{\alpha}_{1}{\alpha}_{2}({\alpha}_{1}^{2}+{\alpha}_{2}^{2})}{48}{\mathit{\text{NT}}}_{p}^{(\text{RD})}.
The energy consumption and occupied bandwidth performance of the scheme in [10] need to be addressed in a slightly different way w.r.t. the previous cases. CS theoretic results determine the number M of measurements needed at the FC to correctly restore the sensed field; within an observation time, this constraint corresponds to a required percentage q
_{
s
} of correctly received samples at the FC. Because of possible collisions, the required percentage q
_{
s
} needed at the FC does not translate straightforwardly into a sensing probability p
_{
s
}. In [10], the authors establish a relationship among the sensing probability p
_{
s
} and the probability q
_{
s
} of correct packet reception and demonstrate that, given q
_{
s
}, a minimum bandwidth B is required in order to assure that a feasible value of p
_{
s
} exists. Thereby, if the available bandwidth is not accurately dimensioned, small values of p
_{
s
} do not provide enough measurements at the FC, whereas large values of p
_{
s
} cause too many collisions.
Besides, in [10], the authors provide an expression, standing for small values of q
_{
s
}, of the minimum bandwidth as a function of the desired q
_{
s
}. Following the guidelines in [10], we have extended such results to accommodate for large values of q
_{
s
}, too. Specifically, we came up with the relation {B}_{\text{RR}}^{\text{min}}=\frac{L}{{T}_{c}}\left(2N{[ln{q}_{s}]}^{1}+1\right)\phantom{\rule{2.77626pt}{0ex}}\text{if}\phantom{\rule{2.77626pt}{0ex}}{q}_{s}\ge {e}^{1}. The RS settings are therefore assigned as follows. Firstly, the desired q
_{
s
} is fixed according to the reconstruction quality constraints. Secondly, the minimum needed bandwidth is evaluated. Finally, the selected bandwidth value is employed to derive the needed sensing probability p
_{
s
}[10]. With these positions, the packet transmission time {T}_{p}^{(\text{RR})} is determined from the employed bandwidth as {T}_{p}^{(\text{RR})}=L/{B}_{\text{RR}}^{\text{min}}. Besides, the energy consumption is determined by the value of p
_{
s
}; the average network consumed energy is evaluated as
{E}_{\text{RR}}=\frac{{p}_{s}}{24}{N}^{2}{T}_{p}^{(\text{RR})}.
(19)
6.3 Further remarks
Before turning to the numerical performance evaluation, a few remarks are in order. The above analysis has pointed out that the consumed energy and allocated bandwidth adopting the Radonlike CS scheme grow only with the square root \sqrt{N} of the network size, whereas those of selected stateoftheart approaches vary with the power N. The impact of these trends on energy consumption depends on all system parameters and mostly on p
_{
s
}. For a spatially sparse field, where p
_{
s
} and consequently q
_{
s
} tend to be high, since a large fraction of sensors shall transmit their values using RS with random access to allow proper reconstruction, a random sensing scheme is prone to exhibit a large energy consumption and bandwidth occupancy. The Radonlike CS scheme then yields a reduced energy consumption for each node, as well as a parsimonious bandwidth use for collecting data over the entire grid. The gain is more and more evident as the network size (i.e., the covered area) increases.
Let us point out that, on nonspatially sparse fields, the conditions for accurate reconstruction, e.g., the values of p
_{
s
} and P, may differ, leading to different relative performances. The investigation of this issue is left for further studies.
The actual gain in terms of energy and bandwidth depends on the constants γ
_{
P
},δ
_{
P
} which grow with the number of considered projections. The advantages of the Radonlike CS scheme are expected to be evident on spatially sparse signals, where low values of P (e.g., P = 3) enable reconstruction, whereas the RS data gathering algorithm requires a high percentage of samples to reach the FC in order to achieve satisfying reconstruction results [18].
As far as the medium access scheme is concerned, the hereindevised Radonlike CS scheme is realized via a TDMA technique, just as the RD scheme. Therefore, it implies an effort of synchronization and scheduling. Nonetheless, different data gathering procedures can be envisaged realizing the Radonlike CS using a random access criterion. This issue is left for further studies. Finally, these results depend on the peculiar structure of the Radonlike sensing matrix Φ
_{
R
} and, although derived for a particular data gathering algorithm, can be generalized to different Radonlike CS measurement computation schemes.