DEALING WITH MEASUREMENT NOISE

(A gentle introduction to noise filtering)

CONTENTS

Frequency Characteristics

Low-pass filters allow the low frequency components of an input signal to pass through while attenuating (reducing) high frequency components, hence the term low-pass. There are other filters, such as band-pass and high-pass filters, where the classification is based on the frequency ranges that the filter allows to pass through.

Measurement noise fall into the high frequency range of the signal spectrum, while the underlying process signal usually lies towards the low frequency end. Thus filters that are used to remove noise from measurements are of the low-pass types. The exponentially weighted moving average filter (or equivalently the 1st-order low-pass filter) are but one of many possible types of low-pass filters.

A signal can be decomposed into components of different frequencies. Thus, one method of examining the capabilities of filters is to look at what happens to inputs of different frequencies when they are passed through the filter. There are two features of the output signals that we can investigate:

1. what are the differences between the magnitudes of the input and the output signals (given by the amplitude ratio) and
2. whether there is a time lag between input and output signals (given by the phase-shift)

Plots of amplitude-ratio and phase-shift against frequency gives the frequency response of the system. For simplicity, in this set of introductory notes, we will consider only the amplitude ratio characteristics of low-pass filters.

The amplitude ratio can be regarded as a frequency dependent gain of the filter. The job of the low-pass filter is to filter out high frequency components, therefore its amplitude ratio should be low at high frequencies. At low frequencies, the low-pass filter should allow the input signal to pass through undistorted, and so its amplitude ratio at the low frequency end of the signal spectrum should be unity. The red lines in the figure below show the frequency dependent amplitude ratio plots of 3 first-order low-pass filters, with time-constants 10, 20 and 30.

Figure 5. Amplitude ratio plots of 1st-order low-pass filters with differen time-constants

Note that the amplitude ratios in the above figure are expressed in terms of decibels (dB).

To convert a number, x, to dB, simply apply a log and multiply the result by 20, that is, . Therefore 0 dB is equivalent to an amplitude ratio of 1, while a large negative dB indicates that the amplitude ratio is very small.

From Figure 5, we can see that first-order filters are capable of attenuating high frequency components. In particular, the larger the time-constant of the filter, the higher the degree of filtering. This is easily discerned by examining the amplitude ratios of the three filters at any particular frequency (see the three vertical, dashed blue lines in Figure 5).

A measure of the efficiency of a filter is its bandwidth. This is defined as the frequency range of a signal that a filter allows to pass through with minimal attenuation; signal attenuation is considered to be significant when the amplitude ratio is less then -3dB, (approximately 0.7 as a ratio). The -3dB line is the horizontal magenta coloured line in Figure 5, from which it can be seen that the bandwidth of a filter decreases with increasing values of filter time-constants.

Low-pass filters need not be limited to first-order types. Figure 6 shows the frequency responses of 3 low-pass filters with the following Laplace transfer functions:

The second and third-order filters are respectively, implemented by placing 2 and 3 first-order filters in series.

Figure 6. Amplitude ratio plots of 1st, 2nd and 3rd-order low-pass filters

We can see from Figure 6 that as the order of the filter increases, the slopes of the respective amplitude ratio plots becomes steeper. What this indicates is that higher order low-pass filters provide higher rates of signal attenuation. From the -3dB line (magenta line), it is also clear that the bandwidth of a filter decreases with increasing order. Thus a higher degree of filtering can be achieved by employing higher order filters.

Theoretically, we can design a filter such that its amplitude ratio beyond some frequency, , is a vertical line, while the amplitude ratio below this threshold frequency is horizontal at 0 dB. A filter with this kind of frequency response characteristics is called an ideal low-pass filter, and is called the cut-off frequency; that is any signal with frequency components higher than this will be completely removed by the filter. Thus, the bandwidth of the ideal filter is equal to its cut-off frequency.

As mentioned previously, detrimental lags are introduced into the processed signal by low-pass filters. This aspect can also be studied by plotting the filter’s frequency dependent phase shift characteristics. However, the effects are not as graphic as time-response plots, and so phase-shift plots are not covered in this set of introductory notes. Those who are interested can find further details in standard signal processing texts, or read the appropriate section of the notes on Applications of Frequency Response (requires the free Acrobat Reader).

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