A Comparison with Linear Filtering

Although the performance o f the noise reduction scheme in the last two sections appears to be excellent this is a little misleading. This is because the signal we are cleaning is predominantly low frequency, whereas the additive noise is chosen to be broad band. To be fair, we should therefore compare the performance o f our noise reduction method with that o f a traditional linear filter. Here we compare it to a 15th order low-pass Butterworth filter with the cut-off frequency set at the point where the noise floor becomes visible in the power spectrum o f the noisy data (Other comparisons between nonlinear noise reduction and linear filters can be found in Kostelich and Yorke [1990] and Grassberger et al [1992]). A butterworth low-pass filter is commonly used in linear signal processing and has the following frequency response function, H(u))\

|/ï( “)l = ,

^

=

(6.1.2)

y 1

f"

where is the cut-off frequency and N is the order o f the filter. For the periodic orbit in the last section the cut-off frequency was chosen to be 0.08, where a frequency o f 1.0

20 1 5 V2 1 GO 5 0 O S O 1 OO 1 5 0 VI V2 •■ÿ; O 5 0 VI

Figure 6.1.6: The top pictire shows the plot of a limit cycle from the Lorenz equations using the first two singular directions.. The bottom picture shows the same data with 10% additive noise.

20 1 5 1 O V2 — 5 — 1 O — 1 5 — 1 OO — 5 0 O 5 0 1 OO 1 5 0 VI V2 %.. O 5 0 VI

Figure 6.1.7: The top picture shows a singular systems plot of the noisy limit cycle filtered with the Levenberg-Marquardt algorithm. The bottom picture shows the same signal filtered with a 15th order Butterworth low pass filter.

corresponds to half the sampling rate. The state space plot for this filtered orbit is given in the bottom picture o f figure 6.1.7 and a comparison o f the power spectra for the noisy and clean orbits is given in figure 6.1.8. From this it is clear that the low-pass filter has successfully removed the high frequency noise floor. However the state space plot (figure 6.1.7) allows us to see the difference between the two filtered signals more clearly. The low-pass filter has merely smoothed the orbit. This has not produced a clear periodic orbit. In contrast our noise reduction algorithm concentrates on making the signal deterministic. Hence, although our filtered orbit is more deterministic than the linearly filtered one the power spectrum contains more energy at the higher harmonics (see figure 6.1.8).

Although our noise reduction method has produced a more periodic orbit this does not mean that it is necessarily closer to the original clean orbit. These comparisons are made in the table below. We also compare the measurement error for the two chaotic signals considered in section 6.1.1. For the chaotic signal with 10% noise the cut-off frequency was set at 0.08 and for the chaotic signal with 50% noise the cut-off was 0.06

We should mention that a Butterworth filter is an HR linear filter an therefore we do not expect it to preserve the dynamics. This also has the effect o f introducing transients at the ends o f the filtered time series that degrade the signal. To overcome this we compare the all the signals both with and without the end effects (the end effects were removed by simply excluding the first and last 50 points from the error calculations). The order o f the Butterworth filter was chosen as a compromise between producing a filter with a sharp frequency cut-off and minimising the end effects.

Below we list the measurement error (with and without end effects) for the three cases examined above as well as the performance o f the relevant linear filter in each case. It is clear from this table that the end effects play an important part in the performance o f the Butterworth filter. However, even taking the end effects into account, we can see that our nonlinear noise reduction procedure performs better than the linear filtering in each case.

1 o = 1 o° o 0.5 Frequency (U

1

I

Figure 6.1.8: This plot shows the frequency spectra for, clockwise from the top left: the clean limit cycle, the noisy limit cycle, the noisy limit cycle filtered using a nonlinear noise reduction method and the noisy limit cycle filtered with a linear 15th order low-pass Butterworth filter.

Signal

(initial Noise level)

Measurement Error

With End Effects End Effects Removed

Raw Chaotic Signal (10%) 5.48x10^ 5.36x10"

Chaotic Signal (10%) after application o f noise reduction

3.25 x l( F 3.06x10^

Chaotic Signal (10%) after filtering with a 15th order Butterworth filter

1.14X10" 4.44X10^

Raw Chaotic Signal (50%) 1.37x10" 1.34X10" Chaotic Signal (50%) after

application o f noise reduction

7.23x10" 6.67x10^

Chaotic Signal (50%) after filtering with a 15th order Butterworth filter

3.17x10" 2.71x10"

Raw Limit Cycle (10%) 1.42x10" 1.40x10"

Limit Cycle (10%) after application o f noise reduction

6.44x10^ 6.13x102

Limit Cycle (10%) after application o f 15th order Butterworth filter

7.14X10" 1.11X10"