This is a section of the User's Manual, Chapter 3 - Background Information

 Windowing of Spectral Data

This has nothing to do with WindowsXX operating systems, but rather general background on the way the audio spectrum is estimated, based on a sequence of samples. Windowing functions are applied to the measured data to improve the detail of the spectrum display. But this is getting ahead of the story. Let's look in some detail at the whole process.

The Discrete Fourier Transform (DFT) is what everybody uses to make digital spectrum analyzers and the resulting waterfall displays, etc. This is a first cousin to the Fourier Transform, that the man by that name came up with. The full Fourier transform requires knowledge of the signal FOREVER, including ALL of the past. This is not particularly useful for measurements and so the DFT approximates it in two respects. First, only sampled data is used; in the case of the DSP-10 it is at a 9600 Hz rate. Secondly--and here is where the windowing comes in--we only look at the data for a short period of time. For the DSP-10, we have 1024 samples so it is a 0.1067 second sample.

The DFT behaves as though the signal went on forever, but with the assumption that the next 0.1067 second will look exactly like the one we measured.  And the next, as well...  This is all fine except that it is highly unlikely that the 1024 points will end on the same value, with the same slope, and the same curvature as it started. This almost always produces a major jump (OK-discontinuity) when passing through this point. The Fourier transform of this jump is spread over all frequencies, in general, and tends to overwhelm a low-level signal near the frequency of strong one. The jump causes a "sidelobe structure" that has a (sin x)/x shape in frequency, one that drops off very slowly; the first sidelobe is only down about 13 dB. The term "leakage" is often used, as the signal at one frequency appears to leak to other frequencies. This makes for a displayof limited utility.

The best solution to this jump problem is to taper the data towards zero in the region near the edges of the 0.1067 second sample period. If the data at the edges is zero, then the jump will also be zero.  There are endless ways to taper the data and they are called windowing functions. The DSP-10 allows you to operate with no windowing, or apply three functions differing in the amount of data taper. An classic curve is the Hamming window. It has a first sidelobe down 43 dB.  An extreme curve is the Blackman-harris 92 dB window (BH-92) which has a first sidelobe down 92 dB. In addition, there is a mild windowing function, the Tukey25.

At first, it would appear that one should always use the BH-92 window.  The reason for not doing this is  an observed increase in spectral width. The full sample of 1024 points produces a DFT 3 dB bandwidth of 0.89 bins or 0.89/0.1067=8.3 Hz. When a window is applied, the endpoints are virtually tossed out, and it is not surprising that the Fourier transform produces results that look like a  shorter length of data. For the Hamming window the 3 dB bandwidth is 12.2 Hz and for the BH-92 window it is 17.8 Hz. This makes signals appear wider, and also lets a little more noise in.  If the  signal is truly centered in a bin, this can cause some loss of S/N. If it is not centered, the use of a window does not cause a big penalty, since the loss for off-tune signals is less.

The noise bandwidth and sidlobe characteristics for the four options for windowing is summarized as follows, where SW=xxxx refers to the selected Spectral Width (ALT-J):

           SW=1200    SW=2400  SW=4800            Sidelobe/
Window    Bandwidth Bandwidth Bandwidth Rel S/N     falloff 
--------  --------- --------- --------- -------- -----------
None      2.3 Hz     4.7 Hz     9.4 Hz    0.0 dB -13 dB/slow 
Tukey25   2.6        5.2       10.3      -0.4 dB -14 dB/fast 
Hamming   3.2        6.4       12.8      -1.3 dB -43 dB/slow 
BH-92     4.7        9.4       18.8      -3.0 dB -92 dB
Bottom line: Most operation benefits from some windowing. I most often run with the Tukey25 or Hamming window. There are times when it is possible to see weak signals next to strong ones with the BH-92 window, and Alt W lets you find out if this is so. But, if there are no strong stations around, such as on a microwave band, the use of no windowing function at all maximizes the sensitivity.

1-D. J. DeFatta, J.G. Lucas, W.S. Hodgkiss, "Digital Signal Processing: A System Design Approach", John Wiley, 1988.  This is a GREAT book, if you comfortable with some college level math. It is not a "math book," though, like some!  Good data, examples; I use it all the time.

2-The DSP Chapter in 1999 and 2000 "Radio Amateurs Handbook", by Jon Bloom, (ARRL) has a good introduction, but I wouldn't recommend buying the book just for this chapter. But, everyone should have a Handbook around, anyway!

Notes for windowing:
A- The DSP-10 actually collects and processes every data point twice. The reason is windowing. When one set of data is tapering off, the alternate data set is reaching its peak. This has the effect of non-coherently integrating the data together, which is not as good as narrowing the bandwidth, but better than throwing the data away.  The "Spec Ave" number that is displayed is the "effective number" of DFTs being displayed, and the actual number being processed is twice that shown.

B- I keep referring to DFTs. When you hear the term FFT, it refers to the class of algorithms called Fast Fourier Transforms. This really is a computational detail. The important thing is that the FFT computes a DFT (efficiently).

C- I guess it is obvious that all of this has nothing to do with what you hear coming from the speaker of the DSP-10.

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