Spectrum Smoothing : Why and How?

Sometimes data has spikes which are clearly artefacts of the processing or are due to some other external source. One is used to seeing these on time series but in some cases there are unrepresentative “spikes” in the frequency analysed data. Here we discuss how we can use spectrum smoothing to alleviate the problem. An example spectrum is shown below.

 

Figure 1: Spectrum with False “spikes”

There are clearly several spikes which at first sight could be mistaken for very narrow band resonant responses. Looking at the spectrum on a dB scale (or a log scale) shows the expected underlying spectral structure. This is basically a series of “bumps”.

Figure 2: Noisy data in dB

The objective is to obtain a better visualisation of this underlying structure. One could acquire more data to reduce the scatter and indeed this should be done in the present case as there was a very low number of statistical degrees of freedom. However as is sometimes the case the opportunity to take more data was not available and we have to make do with what we have! The underlying trend is obvious in the graph shown in Figure 2. However if we just trend the original signal the ‘spikes’ will cause a problem. We can however use a simple device. First take the loge (using Math Function) of the data. The result is shown below and this is clearly identical in form to that shown in Figure 2.

Figure 3: Log of Noisy Spectrum

Now we do the Mean Trend (Trend Functions) and in this case an integration length of 25 points was used. The resultant trend (Black) is then shown below overlaid on the loge data (Grey). It is clear that the trend line follows the principle structure.

Figure 4: Trend of Log Spectrum

To get back to the original plane we now just use the exponential function (Maths Functions). This gives the result shown below. There is still some degree of spikeness but it is much reduced and we have a much clearer indication of the underlying nature.

Figure 5: Smoothed Spectrum

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Dr Colin Mercer

Chief Signal Processing Analyst at Prosig
Dr Colin Mercer was formerly at the Institute of Sound and Vibration Research (ISVR), University of Southampton where he founded the Data Analysis Centre. He then went on to found Prosig in 1977. Colin retired as Chief Signal Processing Analyst at Prosig in December 2016. He is a Chartered Engineer and a Fellow of the British Computer Society.

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