Examples of event distribution by weighting

Several cases are shown here that illustrate the effects of the weighting in the previous discussion. The first case shows the effects of varying $\lambda$ in equation (). Figure  shows the events from which the signal filter $\st S$ and the noise filter $\st N$are calculated. The data in Figure  are therefore taken as definitions of signal and noise, the signal being the flat event and the noise being the dipping event. Figure  shows the data to be separated into signal and noise. In addition to the signal and noise seen in Figure , an event with a dip of intermediate slope has been added in Figure . This event is only slightly attenuated by the filters $\st S$ and $\st N$.

By solving equation () with $\lambda=1$,the results seen in Figure  are obtained. The event with the intermediate slope has been about evenly distributed between the signal and the noise.

Next, equation () is solved with $\lambda=10$.Increasing $\lambda$ increases the weight given to the top part of equation (), $\sv 0 \approx \st S\sv s$, so events that do not fit $\st S$ extremely well get eliminated from $\sv s$.As expected, Figure  shows the event with intermediate slope has been almost completely moved to the noise.

When $\lambda$ is decreased to 0.1, the weight given to the top part of equation () is decreased so any event that does not fit the lower part of equation () extremely well is pushed into $\sv s$.This can be seen in Figure , where the event of intermediate slope is almost entirely contained in the signal.

 

signalnoise.a Figure 7 The event on the left is defined as signal, the event on the right is defined as noise.

 

data.a Figure 8 The data, made up of both signal and noise, and an added event.

 

separ7sal.a.1 Figure 9 The calculated signal and noise using $\lambda=1$.

 

separ7sal.a.10 Figure 10 The calculated signal and noise using $\lambda=10$.

 

separ7sal.a.0.1 Figure 11 The calculated signal and noise using $\lambda=0.10$.

In the previous examples, equation () has been solved with a zero estimated value of $\sv s$.This was possible since the signal was not significantly attenuated by the filter $\st N$.In the next examples, equation () has been solved with a preliminary estimate of $\sv s$ being the data $\sv d$,since both filters $\st S$ and $\st N$ can completely eliminate one part of the data. For Figures  to , the signal filter $\st S$ is a two-dimensional prediction-error filter with the form  

$$\begin{array} {ccccc} 0 & a_{-2,1} & a_{-2,2} & a_{-2,3} & ... ... a_{2,0} & a_{2,1} & a_{2,2} & a_{2,3} & a_{2,4} \end{array}.$$ (97)

The noise filter $\st N$ is a one-dimensional prediction-error filter with the form  

$$\begin{array} {c} 1 \\ a_{1} \\ a_{2} \end{array}.$$ (98)

Figure  shows the events defined as the signal and noise. The signal is a series of horizontal events with random amplitudes. The noise is mono-frequency sine waves with random shifts. Both filters () and () will eliminate the sine waves, since a prediction is done along the time axis, but only filter () can predict the signal, since the amplitudes in time are random and unpredictable by filter (). To allow any of the noise in the output, equation () must be solved with a preliminary estimate of $\sv s$ being the data, or all the sine waves will be removed from the system.

When equation () is solved with $\lambda=1$,Figure  shows that the noise is evenly distributed between the calculated signal and the calculated noise. Increasing $\lambda$ to 10 moves the sine waves into the noise section, producing the excellent separation of signal and noise seen in Figure . Decreasing $\lambda$ to 0.1 moves the sine waves almost completely into the signal. This weighting gives a useful tool in distributing events between calculated signal and noise.

 

input
Figure 12 The events on the left are defined as signal, the events on the right are defined as noise.

 

signois.1
Figure 13 The calculated signal and noise using $\lambda=1$.

 

signois.10
Figure 14 The calculated signal and noise using $\lambda=10$.

 

signois.0.1
Figure 15 The calculated signal and noise using $\lambda=0.10$.