[latexpage]The following article was inspired by a question asked by a reader regarding my previous article – Data Windows : What, why and when?. Specifically, the reader asked *“Could you please put an example about overlapping technique? Maybe with the same 10 or 9.5 Hz sinusoidal wave example so that we can see how overlapped window could make a difference in processed frequency spectrum.” *

Overlap analysis, just as the name suggests, means consecutive analysis blocks of the time data are overlapped by the designated percentage of the time record. In the case of the examples presented here, the time block is selected as 1 second corresponding to a ${\Delta}f$ of 1 Hz.

As stated in the previous article, analysis of a stationary signal, Figure 1, with a selected overlap of 0% would consist of analysis of time blocks with an applied Hanning windowing function (Figure 2).

Once we apply our window then the signal would look like that shown in Figure 3. The spectrum would be calculated from this signal.

Now, what happens if the signal is not stationary, such as displayed in Figure 4. You can see an event that occurs at periodic intervals (1 second intervals) that might well be important to the analysis. (NOTE: This signal was created by adding an impulsive component containing a 100 Hz sinusoid)

Overlaid in red in Figure 4 is the Hanning window function. Applying this window to the signal with 0% overlap would result in the analysis signal being almost exactly the same as in Figure 3 because the Hanning window function zeros out the beginning and end of each time record. And, in our example case, this coincides with the periodic impulse.

Overlap processing (50% used for this example) would capture the events of interest (100 Hz sinusoid pulsed at 1 second intervals) because with the overlapping of the time records used to calculate the spectrum, the Hanning window function would align with the events thus including them in the calculated spectrum. This is shown in figure 5.

The difference in the resulting spectrum is shown in Figure 6. The blue curve is with no overlap and the red curve is with the 50% overlap as shown in the above Figures 4 & 5 respectively.

It is plain that with no overlap all of the 100Hz information is missing. This shows the potential for missing events by not using adequate overlap when applying a windowing function.

#### John Mathey

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Thank you for the example.

With 50% overlapped window, it could pick up 100Hz information which is missed with 0% overlapped window.

Therefore, is the true information an addition of two results shown in Figure6, or I guess overlapping?

If there is 50Hz content occurring at 0.5 second with 1 second interval,

the 50% overlap window will miss this 50Hz content, but 0% window will catch.

So, how the true information will be estimated? – 0% overlap will catch10 & 50 Hz, and 50% will do 10 & 100Hz. Neither window will get true contents of 10, 50 & 100 Hz.

Thank you again,

SeanL,

Good observation. The example presented in the response is a very special case and not typically encountered in real life. It was presented to demonstrate the extreme case.

Windowing functions are designed to measure stationary signals meaning the signals are repetitive and predictable. The spectra used in the averaging are calculated on time blocks modified by the windowing function. It is good to keep in mind that, just as the spectra calculated for each of the time blocks are averaged together, the individual spectra calculated for each of the time blocks are based on the average energy of the signal throughout that individual time block.

You may have noticed the amplitude of the Hanning Window function is actually 2 in the center of the window meaning the portion of the signal near the center of the time block is amplified. This compensates for both ends of the windowing function being less than 1 and consequently the signal in the time block being attenuated. This compensation means the total energy of a windowed repetitive predictable signal is estimated to be equivalent to that of the true energy in the time block before windowing.

With the Hanning Windowing function, as with many other winding functions, if transient events occur throughout the signal, they may be attenuated or amplified depending on when they occur relative to the beginning and end of the time blocks used to calculate the spectra. I typically use 75% overlap as default to accommodate this. Although this is not perfect, it is a good approximation. The price paid for using a higher overlap processing is time.

With all this said, typically transient events are better served by other types of windowing functions or averaging techniques.

I hope this additional discussion clarifies more than confuses your understanding.

John Mathey

PROSIG-USA

Hi John,

Thank you very much for your clear explanation. I have some questions regarding the postprocessing:

1. When we consider overlapping windows, we obtain a matrix from which we can plot the spectrogram. Can you explain how you have obtained the frequency spectrum shown in figure 6?

2. Should the length of the window be equal to the length of the FFT? For example if my FFT length is 512, can my window length be 400 or so?

3. If yes, how to maintain a particular frequency resolution of the windowed signal? For example if I calculate the window length with 2 Hz frequency resolution (in my case it is important to obtain window length from frequency resolution and sampling frequency) and then my FFT length is the next power of 2, then the frequency resolution will be lesser than 2 Hz.

Thanks a lot again.