4. Weighted Mixed Norm Minimization-based Joint Compressed Sensing
of the same channel are highly correlated. Consequently, the row vectors of the coefficient matrixA turn out to be correlated, and the algorithm performs better. But in the case of MECG signals, the correlation in each row of A is relatively low. This, in turn, might lead to moderate performance of the TMSBL for MECG signals. In the case of AWMNM, the assigned weights are adaptive in nature, unlike the BWMNM that uses static weights. This produces a more focal estimate of the signal as the algorithm progresses, but at the cost of extra iterations. Thus, it can be concluded that the BWMNM fares better in terms of the recovery speed. On the other hand, the AWMNM is preferable in terms of reconstruction accuracy.
4.5 Application Considerations
Table 4.4: Average runtime of the recovery algorithm of Figure 4.7 for jointly recovering one data packet (X ∈R512×8) at different number of measurements (M).
Techniques Average Runtime(s)
M=50 M=100 M=150
BWMNM 2.011 3.146 3.725
AWMNM 6.481 9.787 12.781
BP 5.011 5.346 6.012
MNM 1.991 2.918 3.810
TMSBL 18.972 39.235 65.057 ISL0 0.0936 0.1882 0.2295 SOMP 0.0028 0.0036 0.0046
performance can be observed for both AWMNM and BWMNM as compared to most of the techniques.
Thus, a higher data reduction (or a lower bit rate) can be achieved using the proposed technique, which results in transmission power savings in WBAN applications.
4.5.2 Computational requirements
The computational cost of the proposed method and other existing CS-based methods is studied in terms of average encoding and decoding time of the algorithms for one data packet (X ∈R512×8).
All simulations are done on the same platform using MATLAB R2010a environment running on a computer with 3.1GHz CPU and 4GB RAM. It is to be noted that the encoder is same in all the algorithms being compared. Therefore, all the algorithms take the same average time to encode a data packet. This average encoding time is found to be 0.163s, 0.257s and 0.334s whenM = 50, 100, and 150, respectively.
The average runtime of the proposed and other existing algorithms in recovering one data packet, is given in Table 4.4 for different number of measurements (M). It can be noted that out of the two proposed WMNM algorithms, AWMNM and BWMNM, the latter has almost the same runtime as the standard MNM since the weights are set a priori. Though, the AWMNM technique performs better than all the other existing algorithms (refer Figure 4.7), this performance gain is obtained with a moderate increase in the computational cost. The AWMNM is an iterative algorithm, which solves a WMNM problem in each iteration with updated weights each time. The runtime, therefore, required by the algorithm is approximately tmaxtimes the runtime of the standard MNM algorithm [34]. Here, tmax is the number of iterations used for the joint recovery (in all the simulations 3 iterations are
4. Weighted Mixed Norm Minimization-based Joint Compressed Sensing
taken). The TMSBL, which performs close to that of the proposed algorithm, is found to be the slowest one in terms of runtime. The SOMP and ISL0 are the fastest algorithms, with the SOMP having least runtime followed by the ISL0. However, both of these algorithms exhibit inferior recovery performances (refer to Figure 4.7).
From the above studies, it can be concluded that there is a trade-off between the recovery speed and the reconstruction accuracy. The proposed CS framework using the joint WMNM recovery transfers the computational burden from the encoder to the decoder. By using a binary sensing matrix it further reduces the computations involved at the resource-constrained encoder while moderately increasing the computations at the decoder. It is noteworthy that the decoders are usually resourceful since the decoding is performed on high performance computers. Therefore, the computational complexity of the decoder is not an important issue in the case of WBAN-enabled health monitoring applications [46].
4.5.3 Memory requirements
The memory requirements of the proposed method are same as that of the existing standard CS-based works [34], since we have not added any additional steps in the encoding process. We have calculated the memory requirements during the data compression process at the encoder. CS involves a matrix- vector multiplication for the data reduction task [1, 46]. Therefore, we need to calculate the memory required to store the sensing matrixΦ, which is multiplied with the arriving data packets X in order to compress them [61, 137]. Unlike the DWT-based compressors that work in the batch mode, here the complete data packets are not required during the compression process [61]. With y(0) = 0, the compressed measurement vector gets updated with each incoming data sample: y(i) =φixi+y(i−1). If Φ is a dense Gaussian matrix or a non-binary matrix [41, 44], the whole quantized matrix needs to be stored. This, in turn, increases the computational burden and the resource cost. However, in the proposed work, a binary sensing matrix is used with the entries being 0s and 1s only. This minimizes the computations (by replacing multiplication operations with additions) and also requires comparatively lesser memory to store. Each entry of Φ will require a maximum space of 1-bit.
Consequently, for a moderate sizeΦ∈R100×512, the memory required will be 100×512 bits = 50 kB.
The memory requirement can be further reduced if we store only the positions of 1s in each column of Φas suggested in [1,45]. For a signal of lengthN = 512, memory space of 42 kB is required by taking