Discrete temporal signal
Similar to spatial signals, the definition of the sensed data as vector can apply to the time-domain signals as well. A SN that regularly samples r times every t seconds has a temporal sampling rate of r/t samples per seconds. In a typical SN, the sensed data is then quantized and converted to binary data. Depending on the specific application requirements, more data processing steps may be applied before or after calculating the measurements. The discrete temporal signal recorded by the exemplified SN records a discrete signal of size r every t seconds that can be represented by a real vector f ∈ Rr.
Spatiotemporal signal
Spatiotemporal sampling is a direct extension of spatial and temporal sam- pling. For example, a WSN consisting of n SNs each of which recording r samples in every t time units, produces a discrete spatio-temporal signal that can be represented by a vector f ∈ RN where N = nr.
In case of sole spatial sampling, N and n are equal, because during each time period unit only one sample is recorded by a SN. Note that in parts of this text, our discussion may only focus on spatial signals without considering the temporal data. In such cases, we may use either N or n to refer to the size of the signal. According to our definition, r = 1 in pure spatial sampling mode, and hence, N = n when no temporal sampling is taking place.
3.2
Basics of Compressive Sensing
Throughout this text we may use the terms signal and vector interchange- ably. Both of them refer to discrete values of a spatiotemporal phenomenon.
3.2.1
Sparse and compressible signals
To begin formal description of the CS theory, we need some mathematical definitions. The formal definitions apply to the original signal in time or space domain as well as its projection on some frequency domain. Therefore, we use the notation v as a general vector which may refer to a signal or its compressive transformation. The first three definitions are general mathe- matical definitions which may apply to any vector. Definition 4 refers to the CS theory in a more specific manner. This clarification was required to avoid ambiguity in using the mathematical notation for the signal vectors.
Definition 1. Vector v ∈ RN is said to be S-sparse if kvk0= S, i.e., v has
Definition 2. S-sparse vector vS ∈ RN is made from non-sparse vector
v ∈ RN by keeping S largest entries of v and zeroing its all other N − S
entries.
Definition 3. Vector v is said to be compressible if most of its entries are near zero. More formally, v ∈ RN is compressible when kv−vSk2 is negligible
for some S N .
Note that k·k2 denotes the norm-2 operator1. The above definitions may
apply to any vector which can be a signal vector or its projection on some orthonormal basis. In the coming sections, we may use the definitions 1 to 3 to both time-domain or space-domain signals or their projections on bases like Fourier, DCT, wavelet, etc. In any case, the vector v in the definitions above can be substituted with the signal vector f or its compressive projection x whichever is discussed in the context.
Definition 4. Signal f is compressible under orthonormal basis Ψ when f = Ψx and x is compressible. The matrix Ψ is a real complex orthonor- mal matrix with the basis vectors of Ψ as its rows. We also say that Ψ is a compressive basis for signal f .
Compressibility of WSN signals
Most signals recorded from natural phenomena are compressible under Fourier, DCT and the family of wavelet transforms. This is the fundamen- tal fact behind every traditional compression technique. Audio signals are compressible under Fourier transform. Images are compressible under DCT or wavelet. WSN also records a distributed spatiotemporal signal from a natural phenomenon which can be compactly represented under the family of Fourier or wavelet orthonormal bases.
3.2.2
Compressive measurement
CS is distinguished from traditional compression techniques in signal acqui- sition method as it combines compression into the data acquisition layer and tries to recover the original signal from fewest possible measurements. This is very advantageous in applications where acquiring individual samples is infeasible or too expensive. WSN is an excellent application for CS since acquiring every single sample from the whole network leads to a large traffic that can be beyond the capacity of the resource-limited SNs.
1
For a real vector v ∈ RN, norm-1 of v is defined as kvk1=P N
i=1|vi| and norm-2 of v
is defined as kvk2=
q PN
3.2. BASICS OF COMPRESSIVE SENSING 33
Definition 5. Measurement matrix Φm is an m × N real or complex matrix
consisting of m < N basis vectors randomly selected from orthonormal mea- surement basis Φ that produces an incomplete measurement vector y ∈ Cm
such that y = Φmf .
Definition 6. Coherence between the measurement basis Φ and the com- pressive basis Ψ is denoted by µ(Φ, Ψ) and is equal to max
1≤i,j≤N|hφi, ψji| where
for each 1 ≤ i, j ≤ N , φi’s and ψj’s are basis vectors of Φ- and Ψ-domain
respectively and h·i is the inner product operation.
3.2.3
Signal reconstruction
The novelty of the CS theory is that, it provides conditions for accurate recovery of the original data by applying a polynomial-time reconstruction algorithm on the collected compressive measurements.
Theorem 1. Suppose signal f ∈ RN is S-sparse in Ψ-domain, i.e., f = Ψx
and x is S-sparse. We acquire m linear random measurements by randomly selecting m basis vectors of the measurement basis Φ. Assume that y ∈ Cm
represents these incomplete measurements such that y = Φmf where Φm is
the measurement matrix. Then it is possible to recover f exactly from y by solving the following convex optimization problem:
ˆ
x = argmin
x∈CN
kxk1 subject to yk = hΨx, φki for all k = 1, . . . , m.
(3.1) where φk’s are the rows of the Φm matrix. Recovered signal will be ˆf = Ψˆx
Candes and Tao [2006].
In case of real measurement and compressive bases, problem (3.1) can be simplified to a linear program Vanderbei [2001]. Noiselet measurement matrices involve complex numbers and thus a linear program can not solve problem (3.1) Coifman et al. [2001]. Convex optimization problem (3.1) with complex values can be cast to a Second Order Cone Program (SOCP) Winter et al. [2005].
Sparsity and incoherence
Accurate signal recovery is possible when the number of measurements fol- lows
where C > 1 is a small real constant Candes and Romberg [2007].
From Equation (3.2) it is clear why sparsity and incoherence is important in CS. In order to efficiently incorporate CS theory in a specific sampling scenario, we need the measurement and compressive bases to be incoherent as maximum as possible to decrease parameter µ in (3.2). Moreover compressive basis must be able to effectively compress the signal f to decrease S in (3.2). When these two preconditions hold for a certain sampling configuration, it is possible to recover the signal f from m measurements where m can be much smaller than the dimension of the original signal Candes and Tao [2006].
Random measurements
Interestingly, random matrices such as a Gaussian matrix with independent and identically distributed (i.i.d.) entries from a N (0, 1) have low coherence with any fixed orthonormal basis Candes and Romberg [2007]. The elements of such a random matrix can be calculated on the fly using a pseudo-random number generator which is common between SNs and the sink. When the normal random generator at every SN is initialized by the id-number of that SN, the sink can also reproduce the measurement matrix. Note that in this case, there is no need for a centralized control to update the measurement matrix and the values of the measurement matrix are not needed to be stored inside the SN. Therefore, using random measurement matrices gives us more flexibility and requires less memory on the SNs. Instead, because of slightly more coherence between random measurement matrix and the fixed compres- sive basis, the number of required measurements m will increase according to (3.2).
Noisy measurements
CS is also very stable against noisy measurement and can also handle signals that are not strictly sparse but compressible. It is a very idealistic condition being able to transform signal f to a strictly sparse vector in Ψ-domain. Instead, f is usually transformed into a compressible form with many near zero entries. Candes et al. have shown that if kx − xSk2< for some
integer S < N and a small real constant , then the recovery error by solving problem (3.1) will be in order of O(). Similarly, in a noisy environment if the measurement vector is added by an Additive White Gaussian Noise (AWGN) ∼ N (0, σ2), the recovery error is bounded by O(σ2) Candes et al. [2006a,b].