An example: simulated signal with noise

In Figure 1a a simple signal is constructed from continuous sine wave ($A$), one-point discontinuity (Dirac's delta, $B$) and three Gabor functions ($C$, $D$ and $E$) of the same time positions ($C$ and $D$) or the same frequencies ($D$ and $E$). Figure 1b gives time-frequency energy distribution obtained for this signal from MP decomposition by means of eq. (12). In the left, 3-dimensional plot, energy is proportional to the height. Right 2-dimensional plot presents energy in shades of gray. Perfect representation of all the signal structures is due to the fact that the signal was constructed as a sum of dictionary's elements only.

Figure 1: (a): left--components of the simulated signal: sine A, Dirac's delta B and Gabor functions C, D and E. Right--signals, labelled b, c and d, constructed as sum of structures A-E and white noise, and decomposed in corresponding panels (b), (c) and (d). (b): time-frequency energy distribution (eq. 12) obtained for sum of structures A-E; in 3-D representation on the left energy is proportional to the height, in right panel--to the shades of gray. Panels (c) and (d): decompositions of signals with linear addition of noise, S/N = 1/2 ($-3$ dB) in (c) and $-6$ dB in (d), the same realization of white noise was used in both cases. Exact parameters of presented time-frequency structures are given in Table 1.

In Figure 1c and 1d a white noise of energy twice and four times the signal's energy is added. In both cases the same realization of white noise was used (with different weights). Table 1 presents parameters of simulated time-frequency structures $A$-$E$ compared to parameters of corresponding time-frequency atoms fitted by MP to the mentioned signals, with S/N ratio from $\infty$ (simulated signal $b$ without noise) through 1/2 ($-3$ dB, signal c) to 1/4 ($-6$ dB, signal d).

Table 1: Parameters of structures A-E (Figure 1)--original values in simulated signal and parameters recovered by MP decomposition for signals with different S/N ratios. Time position of sine wave ($^1$) and frequency of Dirac's delta ($^2$) are set by convention as half of the time and frequency ranges, respectively

parameters	amplitude	scale	position	frequency
structure A (sine)
original	1.00	512	--	0.5000
S/N=$\infty$	0.62	512	256$^1$	0.5031
	0.98	256	506	0.5031
	0.9	256	14	0.5031
S/N=1/2	1.40	256	506	0.5031
	1.23	256	43	0.5031
S/N=1/4	1.41	256	506	0.5031
	1.16	256	43	0.5031
structure B (Dirac)
original	10.00	0	64	--
S/N=$\infty$	10.37	0	64	1.57$^2$
S/N=1/2	8.03	0	64	1.57$^2$
S/N=1/4	-	-	-	-
structure C (Gabor)
original	3.00	64	128	2.40
S/N=$\infty$	2.96	64	132	2.39
S/N=1/2	2.98	64	118	2.39
S/N=1/4	3.10	64	111	2.39
structure D (Gabor)
original	3.00	64	128	1.20
S/N=$\infty$	2.96	64	124	1.19
S/N=1/2	2.75	64	132	1.19
S/N=1/4	1.72	256	234	1.18
structure E (Gabor)
original	3.00	128	320	1.20
S/N=$\infty$	2.97	128	326	1.20
S/N=1/2	2.35	128	294	1.19
S/N=1/4	1.12	256	361	1.13
parameters	amplitude	scale	position	frequency

We can observe some characteristic properties of decomposition:

• Sine wave $A$ retains its frequency up to the noisiest signal, however, even in the absence of noise additional shorter structures are fitted near the beginning and the end of the signal to account for border effects. In noisy signal the 'infinite' sine tends to be explained by two shorter structures, due to border effects again.
• Dirac's delta disappears in the strongest noise, but till then preserves its exact time position.
• Gabor function $C$, being the only structure in this frequency, retains relatively well all of the parameters with a slight flow of time position at higher noise levels.
• Finally, the two Gabor functions $D$ and $E$ lying in the same frequency with different time positions and widths exhibit slight flow of parameters for S/N=1/2, and in higher noise melt in two longer structures: one in between (in Table 1 assigned to structure $D$) and one more to the right.

The above example demonstrates robustnes of the method in presence of linearly added white noise--in this case most of the basic time-frequency characteristics were represented even in S/N=1/4 ($-6$ dB).