{"id":5676,"date":"2019-09-17T13:20:03","date_gmt":"2019-09-17T11:20:03","guid":{"rendered":"http:\/\/www.tonmeister.ca\/wordpress\/?p=5676"},"modified":"2019-09-17T13:43:52","modified_gmt":"2019-09-17T11:43:52","slug":"dfts-part-6-windowing-artefacts","status":"publish","type":"post","link":"https:\/\/www.tonmeister.ca\/wordpress\/2019\/09\/17\/dfts-part-6-windowing-artefacts\/","title":{"rendered":"DFT’s Part 6: Windowing artefacts"},"content":{"rendered":"\n

Links to:
DFT\u2019s Part 1: Some introductory basics<\/a>
DFT\u2019s Part 2: It\u2019s a little complex\u2026<\/a>
DFT\u2019s Part 3: The Math<\/a>
DFT\u2019s Part 4: The Artefacts<\/a>
DFT’s Part 5: Windowing<\/a><\/p>\n\n\n\n

In Part 5, we talked about the idea of using a windowing function to “clean up” a DFT of a signal, and the cost of doing so. We talked about how the magnitude response that is given by the DFT is rarely “the Truth” – and that the amount that it’s not True is dependent on the interaction between the frequency content of the signal, the signal envelope, the windowing function, the size of the FFT, and the sampling rate. The only real solution to this problem is to know what-not-to-believe when you look at a DFT output.<\/p>\n\n\n\n

However, we “only” looked at the artefacts on the magnitude response in the previous posting. In this last posting, we’ll dig a little deeper and NOT throw away the phase information. The problem is that, when you’re windowing, you’re not just looking at a screwed up version of the magnitude response, you’re also looking at a screwed up phase response as well.<\/p>\n\n\n\n

We saw in Part 1<\/a> and Part 2<\/a> how the phase of a sinusoidal waveform can be converted to the sum of a real and an imaginary component. (In other words, if you add a cosine and a sine of the same frequency with very specific separate gains applied to them, the result will be a sinusoidal waveform with any amplitude and phase that you want.) For this posting, we’ll be looking at the artefacts of the same windowing functions that we’ve been working on – but keeping the real and imaginary components separate.<\/p>\n\n\n\n

Rectangular windowing<\/p>\n\n\n\n

We’ll start by looking at a plot from the previous post, which I’ve duplicated below.<\/p>\n\n\n\n

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Figure 1: The magnitude responses calculated by a DFT for 6 different frequencies. Note that the bin centre frequency is 1000.0 Hz. <\/figcaption><\/figure>\n\n\n\n

The way I did the plot in Figure 1 was to create a sine wave with a given frequency, do a DFT of that, and plot the magnitude of the result. I did that for 6 different frequencies, ranging from 1000 Hz (exactly on a bin centre frequency) to 999.5 Hz (halfway to the adjacent bin centre frequency).<\/p>\n\n\n\n

There’s a different way to plot this, which is to show the result of the DFT output, bin by bin, for a sinusoidal waveform with a frequency relative to the bin centre frequency. This is shown below in Figure 2.<\/p>\n\n\n\n

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Figure 2: Rectangular window: <\/strong>
The relationship between the frequency of the signal, the frequency centres of the DFT bin, and the resulting magnitude in dB. Note that the X-axis is frequency, measured in distance between bin frequencies or “bin widths”.<\/figcaption><\/figure>\n\n\n\n

Now we have to talk about how to read that plot… This tells me the following (as examples):<\/p>\n\n\n\n