Let us assume that a sampled signal belongs to a class of signals of bounded variation. A signal x(t) defined over a time interval [a,b] is said to be of bounded variation if the following sum is bounded [46,47]: (1) ∑ j = 1 k | x ( t j ) − x ( t j − 1 ) | < ∞for every partition of the interval [a,b] into subintervals (t_j−1, t_j), where j = 1,2…,k; and t₀ = a, t_k = b. The measure denoted by V a b ( x ) and defined as: (2) V a b ( x ) = def sup ∑ j = 1 k | x ( t j ) − x ( t j − 1 ) |is said to be a total variation of a signal x(t) on the interval [a,b].

A signal of bounded variation is not necessarily continuous but it is differentiable almost everywhere. Moreover, x(t) can have discontinuities of the finite-jump type at most [46,47]. If x(t) is continuous, the total variation V a b ( x ) might be interpreted as the vertical component (ordinate) of the arc-length of the x(t) graph, or alternatively, as the sum of all consecutive peak-to-valley differences. The definition of the signal of bounded varation is exemplified in Figure 1 where the partition of the interval [a,b] into a number of k = 9 subintervals is arranged, and the graphical interpretation of V a b ( x ) is illustrated.

All the signals which appear in the physical reality or are generated in the laboratory can be approximated by a class of signals of bounded variations.

By the definition, a signal x(t) is sampled at the instant t_i according to the energy criterion if the energy of a difference between the signal value x(t_i) and the value included in the most recent sample x(t_i−1) accumulated during the ith sampling interval (t_i−1,t_i) reaches a certain constant threshold ζ > 0: (3) ∫ t i − 1 t i [ x ( t ) − x ( t i − 1 ) ] 2   dt = ζwhere i = 1,2,…,n is a number of samples taken during the analyzed time interval [a,b].

We assume also that the zero-order hold is used to keep the value of the most recent sample between sampling instants.

A detection of events that trigger sampling operations according to the error energy criterion may be implemented in the analog circuit, or using digital processing where the continuous-time signal is represented by periodic samples taken with high frequency.

The former implementation needs a circuit squaring the difference between the instantaneous signal value x(t) and the value corresponding to the most recent sample x(t_i−1) followed by the integration circuit and the comparator. The squaring circuits usually take advantage of the squaring feature of MOSFETs [48]. The latter needs the upward event-driven architecture where the event-triggered communication is implemented on the top of the background time-triggered signal acquisition [49].

The motivation to introduce the energy criterion for event-based sampling is similar to the argumentation for using the integral criteria [8,23]. As stated in [8,23], in the linear send-on-delta algorithm, the signal oscillations or the steady-state error are not detected by the sampling operations if these oscillations remain within the threshold [8,23]. In the sampling according to the error energy criterion, similarly like in the integral sampling schemes, the signal is “tracked” continuously since signal variations are accumulated during the sampling interval.

As follows from (3), in the error energy sampling criterion similarly as in the integral sampling, the sampling time is a function of the intersampling signal behavior since the signal variations that occur within the intersampling interval are successively accumulated [8]. The difference is that the successive sampling error values are squared before integration in the error energy based sampling criterion. Compared to the integral sampling criterion [8] and to the area-triggered sampling [23]), the error energy criterion gives more weight to extreme sampling error values.

Furthermore, the energy is one of fundamental signal measures with an important physical interpretation. The performance of monitoring or control system is usually defined as integral or energy of the error. The sampling according to energy criterion is useful in case of indirect measurements, in particular, when the measurements in the energy domain are performed. For example, it occurs if the electric energy is measured by sampling of current intensity signal or if the mechanical energy is estimated on a basis of tracking velocity.

In the event-based schemes, the current sampling rate depends on the input signal variations and the sampling resolution defined as 1/ζ. Intuitively, the high sampling resolution (small ζ) causes the number of samples in a time unit to increase, and conversely. Now we will derive the relationship between the sampling resolution ζ (threshold) and the mean rate of samples taken during the specified time interval [a,b].

By letting the signal x(t) be approximated by a truncated Taylor series, if the threshold ζ is small enough we have: (4) x ( t ) − x ( t i − 1 ) ≅ x ′ ( t i − 1 ) ( t − t i − 1 )   for   t i − 1 ≤ t ≤ t iwhere x′(t_i−1) is the signal time-derivative at the instant t_i−1.

Hence: (5) ξ = ∫ t i − 1 t i − 1 + Δ t i [ x ( t ) − x ( t i − 1 ) ] 2   dt ≅ ∫ 0 Δ t i [ x ′ ( t i − 1 ) ] 2   t 2   dt = [ x ′ ( t i − 1 ) ] 2 3   ( Δ t i ) 3where Δt_i = t_i − t_i−1 is the length of the ith sampling interval.

The Δt_i can be derived from (5) as a function of the signal derivative x′(t_i−1) at the point t_i−1 and the sampling resolution ζ: (6) Δ t i = 3 ζ [ x ′ ( t i − 1 ) ] 2 3

By the definition, the mean sampling rate s_L, defined as the average number of samples in a time unit, in the uniform sampling in the energy domain is given by: (7) s L = def n ∑ i = 1 n   Δ t i

For the sake of simplicity but without loss of generality, assume that the analyzed time interval is a sum of the sampling intervals, i.e., t₀ = a, t_n = b, and: (8) ∑ i = 1 n Δ t i = b − a

Furthermore, let us assume that the signal derivative x′(t) is constant at any time instant t within the ith sampling interval (t_i−1,t_i) and equal to the derivative at the point t_i−1 in the beginning of this interval: (9) ∀ t ∈   ( t i − 1 ,   t i ) : x ′ ( t ) ≅ x ′ ( t i − 1 )

Hence: (10) ∑ i = 1 n   ∫ t i − 1 t i   [ x ′ ( t i − 1 ) ] 2 3   dt ≅ ∑ i = 1 n   [ x ′ ( t i − 1 ) ] 2 3   Δ t i

This approximation (10) is more accurate if the threshold ζ is small.

On the basis of the equation (6), we obtain: (11) ∑ i = 1 n   ∫ t i − 1 t i   [ x ′ ( t i − 1 ) ] 2 3   dt = 3 ζ 3 n

On the basis of (8), (10), and (11), we obtain the expression for the mean sampling rate: (12) s L = ∑ i = 1 n ∫ t i − 1 t i [ x ′ ( t i − 1 ) ] 2 3   dt 3 ζ 3 ( b − a )

If the lengths of the sampling intervals Δt_i = t_i − t_i−1 are small enough, the expression [ x ′ ( t i − 1 ) ] 2 3 is constant in time so the sum of integrals on the right side of the formula (12) might be approximated by the single integral of the function [ x ′ ( t ) ] 2 3 taken over the whole time interval [a,b] ≡ [t₀,t_n]: (13) s L ≅ 1 3 ζ 3 [ x ′ ( t ) ] 2 3 ¯where [ x ′ ( t ) ] 2 3 ¯ denotes the mean of the cubic root of the signal derivative square in the time interval [a,b] as follows: (14) ∫ a b [ x ′ ( t ) ] 2 3 dt b − a = [ x ′ ( t ) ] 2 3 ¯

Summing up, the mean sampling rate according to the uniform event-driven sampling in the energy domain is expressed by two factors:

- the mean of the cubic root of the signal derivative square [ x ′ ( t ) ] 2 3 ¯, which is a measure of the sampled signal x(t),

To estimate how effective the sampling according to energy criterion is in terms of the sampling rate, we compare it with the conventional periodic scheme. As a criterion of comparison we choose the same maximum energy of sampling error in both sampling schemes.

As a sampling period T we select such value that the energy of a difference between the current signal value x(t) and the value included in the most recent sample x(t_i−1) = x((i−1)T) accumulated during each sampling period cannot exceed a given threshold ζ > 0 (compare with (3)): (15) ∫ ( i − 1 ) T iT [ x ( t ) − x ( ( i − 1 T ) ) ] 2   dt ≤ ζ

The sampling period T in the uniform scheme is adjusted to the fastest change of a signal during a time interval [a,b] so the following relationship is valid: (16) ξ = ∫ 0 T [ x ′ ( t ) max ] 2   t 2   dt = [ x ′ ( t ) max ] 2 3   T 3where |x′(t)|_max = max{|x′(t)|; t ∈ [a,b]} is the maximum of the first signal derivative in relation to the time during an interval [a,b].

Taking into account (16), the sampling rate s_R in the periodic scheme is selected according to the following formula: (17) s R = def 1 T = [ x ′ ( t ) max ] 2 3 ζ 3

The energy of the sampling error in the sampling period T reaches its maximum ζ only if the signal x(t) changes its value with the maximum rate defined by |x′(t)|_max. During slow signal variations, the energy of the sampling error accumulated in the sampling period T is lower. Instead, the energy of the sampling error in the sampling scheme according to the energy criterion equals ζ in each sampling interval.

The effectiveness of event-based sampling according to energy criterion is defined similarly to the classical send-on-delta [1] and the integral sampling schemes [8] as the ratio of the periodic s_R and the mean rate s_L of the event-based sampling according to energy criterion measured during a certain time interval [a,b]: (18) χ = s R s Lwhere the maximum energy of the sampling error ζ is the same in both schemes.

By setting (12) and (17) to (18), we have: (19) χ = [ x ′ ( t ) max ] 2 3   ∑ i = 1 n   ∫ t i − 1 t i [ x ′ ( t ) ] 2 3   dt b − a

Since the sampled signal x(t) is approximated by the first two terms of the Taylor series (see (4)), the formula (19) gives only an approximation of the real sampling effectiveness. This approximation is more accurate if the number of samples, n, taken during a specified time interval [a,b] is high.

In particular, if the sampling resolution is infinite (ζ→0), then n→∞, and the sum in the numerator of the formula (20) approaches the integral as follows: (20) lim n → ∞ ∑ i = 1 n   ∫ t i − 1 t i   [ x ′ ( t ) ] 2 3   dt = ∫ a b   [ x ′ ( t ) ] 2 3   dt

Let us define the asymptotic effectiveness χ_∞ of the sampling in energy domain in relation to the periodic sampling as: (21) χ ∞ = def lim 1 / ζ → ∞   χ

By setting (19) and (20) to (21), we have: (22) χ ∞ = lim 1 / ζ → ∞ χ = lim n → ∞ s R s L = ( b − a ) [ x ′ ( t ) max ] 2 3 ∫ a b [ x ′ ( t ) ] 2 3   dt

Finally, taking into (14) in (22), the final formula for the asymptotic effectiveness is obtained: (23) χ ∞ = [ x ′ ( t ) max ] 2 3 [ x ′ ( t ) ] 2 3

As follows from (23), the asymptotic effectiveness χ_∞ is independent of the sampling resolution defined by ζ, and constitutes the embedded feature of the sampled signal similarly as for the send-on-delta scheme [1], and the integral sampling [8].

Comparison of the asymptotic effectiveness χ_∞ of sampling according to energy criterion defined by (23) to the asymptotic effectiveness q_∞ of integral sampling [8], and the minimum effectiveness p_min of the send-on-delta scheme [1] shows that the sampling in energy domain is less effective than the send-on-delta scheme but more effective than integral sampling (for the send-on-delta scheme, the asymptotic effectiveness is the minimum effectiveness at the same time [1]): (24) q ∞ ≤ χ ∞ ≤ p minwhere p min = | x ′ ( t ) | max | x ′ ( t ) | and q ∞ = | x ′ ( t ) | max | x ′ ( t ) |.

Moreover, the inequality (24) is reduced to the equality (q_∞ = χ_∞ = p_min) only for a pure linear signal. For the other continuous-time signals, the inequality (24) is strong, i.e., q_∞ < χ_∞ < p_min.

On the other hand, it is straightforward to show that higher effectiveness in sampling according to the selected event-based criterion is obtained at the cost of increasing the total sampling error defined as the sum of errors for all the samples taken during a specified time interval.

Thus, the ratio of the sum of energy of all the sampling errors in sampling according to energy criterion to the energy of all the sampling errors in periodic sampling is higher than the ratio of the total integral error in the integral sampling to the total integral error in the periodic sampling. On the other hand, the opposite relationship exists for the sampling according to energy criterion compared to the linear send-on-delta scheme (the ratio of the sum of error energy in sampling in energy domain and periodic sampling is lower than the ratio of the total linear error in send-on-delta scheme and in periodic sampling).

The signal-dependent event-based sampling criteria might be approximated by a unified integral criterion as follows. The continuous-time signal x(t) is sampled at the instant t_i according to the event-based sampling criterion if the characteristic function z[x(t) − x(t_i−1)] of a difference between the signal value x(t_i) and the value included in the most recent sample x(t_i−1) accumulated during the ith sampling interval (t_i−1,t_i) reaches a certain constant threshold θ > 0: (25) z [ x ( t ) − x ( t i − 1 ) ] = 1 ( t i − t i − 1 ) k ∫ t i − 1 t i [ x ( t ) − x ( t i − 1 ) ] l   dt = θwhere k, l > 0 are integers.

In particular, the integral sampling scheme is modeled for k = 0, l = 1, and the sampling according to energy criterion for k = 1, l = 2, respectively. For k = 1, l = 1, the generalized sampling criterion approximates the model of the send-on-delta principle for high sampling resolution (i.e., if θ is small). More precisely, modeling the send-on-delta scheme by the generalized event-based sampling criterion (25) is accurate provided that the approximation stated by (4) is satisfactory.

The generalized criterion defines also a set of the other event-based sampling criteria, for example, the time-weighted integral criterion for k = −1, l = 1. However, some event-based sampling schemes can be unstable, especially for slowly varying signals and high θ, meaning that the events triggering sampling operations might not occur for long time.

The generalized criterion (25) for event-based sampling is similar to the generalized criteria for adaptive sampling formulated in [15,18].

The concept of event-based temporal sampling can be easily extended to sampling signals in the space domain. The difference is that continuous sensing of signals in the space, similarly as may be carried out in the time domain by the use of an analog sampling circuit, is impracticable in general. Instead, the continuous space signals are sampled by sensors deployed at discrete locations. Thus, the upward event-driven architecture is recommended to implement the distributed event-based system to sample continuous-space signals (see Section 2.2) [49].

Event-based reporting in the sensor networked system with the upward event-driven architecture is a two-step process corresponding to fine and coarse quantization of the signal value. First, the continuous-space signal is sampled by sensors deployed along the space being monitored (e.g., uniformly every unit distance) which results in conversion of the continuous-space to the discrete-space signal. Next, the sensors process the samples and communicate together in order to detect a significant change of a characteristic parameter defined by the event-based reporting criterion. The specified change of the selected parameter is searched along the one-dimensional space in relation to the selected reference position. The locations of the sensing devices corresponding to the positions where the specified changes have been detected create the (one-dimensional) discrete-space picture of the characteristic parameter distribution. The density of sensors deployment should be high enough referred to the intended resolution of the event-based reporting. The concept of sampling continuously-spaced signals may be extended to two-dimensional and three-dimensional space domains.

Let as assume that the signal x(s) is a function of a one-dimensional space s. In order to keep the analysis as general as possible, we assume that continuous sensing of signals in the space domain is practicable. Furthermore, let us assume that the continuous-space signal x(s) is sampled at the position s > s_i−1 if a characteristic parameter y[x(s) − x(s_i−1)] accumulated along the distance (s_i−1,s), where s_i−1 is a position where the previous sample has been taken, reaches a certain constant threshold ζ > 0.

The definition of the characteristic parameter y[x(s) − x(s_i−1)] depends on the sampling criterion as follows:

- spatial (linear) sampling error [x(s) − x(s_i−1)] in the spatial send-on-delta sampling scheme,

The mean spatial sampling densities are expressed by the formulae equivalent to that derived for the corresponding temporal sampling criteria. In particular, the mean spatial sampling density s_L according to the error energy criterion depends on the signal space-derivative [ x ′ ( s ) ] 2 3 ¯ along the sampling distance on the one hand, and the sampling resolution ζ on the other : (26) s L ≅ 1 3 ζ 3   [ x ′ ( s ) ] 2 3 ¯where [ x ′ ( s ) ] 2 3 ¯ denotes the mean of the cubic root of the signal space-derivative square along the sampling distance [a,b] as follows: (27) ∫ a b [ x ′ ( s ) ] 2 3   dt b − a = [ x ′ ( s ) ] 2 3 ¯

The mean spatial sampling densities according to the send-on-delta and to integral criterion may be defined similarly as for the corresponding event-based temporal schemes, see [1,8].