A transient is used to refer to any signal or wave that is short lived. Transient detection has applications in power line analysis, speech and image processing, turbulent flow applications, to name just a few. For instance, in a power system signal, which is highly complicated nowadays because of its constantly changing loads and its dynamic nature, transients can be caused by lightnings, equipment faults, switching operations and so on. Transients are then observed as short lived high-frequency oscillations superimposed on the voltages or currents of fundamental frequency, which is 50/60 Hz, as well as exponential components.
A signal model incorporating these discontinuities is given by

\[ f(x) = \sum_{i=1}^t \alpha_i \exp(-\beta_i (x-a_i)) \cos(2\pi\sigma_i x + \tau_i) \mathbf{1}_{[a_i,z_i[}, \]

where \( \alpha_i \) is the amplitude, \( \sigma_i \) is the frequency (an integer multiple of the fundamental frequency), \( \tau_i \) is the phase angle, \( \beta_i \) is the damping factor and \( a_i \) and \( z_i \) are the starting and ending times of the component.

It is known to be difficult to take the discontinuities from \( \mathbf{1}_{[a_i,z_i[} \) into account, meaning the damped signals to start at different instants. Our sparsity based method can detect at each instant how many and which components are present in the signal.

As an illustration we model the synthetic signal with \(t=3\), \( [a_1, z_1[=[0, 0.0308[ \), \( [a_2, z_2[= [0.0308, 0.0625[ \), \( [a_3,z_3[= [0.0625, 0.1058[ \), or expressed in multiples of the sampling rate \( M = 1200 \) Hz, \( [a_1, z_1[ = [0/M,37/M[ \), \( [a_2, z_2[ = [37/M,75/M[ \), \( [a_3, z_3[ =[75/M,127/M[ \). So at every moment only one term is present in the signal, but the characteristics of that term may change. We furthermore have all \( \alpha_i=1 \), all \( \beta_i=0 \), all \( \sigma_i=60 \) and \( \tau_1=โ€“\pi/2 \), \( \tau_2=โ€“\pi/2 \), \( \tau_3=3\pi/4 \). In addition, uniformly distributed noise in \([โˆ’0.05,0.05]\) is added to the samples of the synthetic signal.

In the figure we merely show the result of the discontinuity detection (the actual values of the parameters \( \alpha_i \), \( \beta_i \), \( \sigma_i \), \( \tau_i \) can of course be determined simultaneously). Our test, based on a singular value decomposition, clearly reveals two nonzero and two (mostly) numerically zero singular values. The nonzero values represent the single constant 60 Hz background signal. This behaviour is disrupted at each phase change because then samples from two different signals are present in the monitored matrix. This happens at \(z_i = a_{i+1} \) or \( x = 37/M\) , \(75/M\), \(127/M\). The disruption is visible from \(x = (37-2t)/M \), \( (75-2t)/M \), \( (127-2t)/M \) on. The disruption disappears as soon as the samples are all in line again with the same underlying component signal.