LabVIEW
Amplitude Spectrum and 1/3 Octave Analysis
Malcolm,
Fun with spectral analysis!
Several things are happening. Spectral leakage. Non-synchronous sampling. Finite data sets. Maybe some others.
The fundamental mathematics of the Fourier series assumes an infinitely repeating (in time) continuous periodic signal. You violate all of those assumptions and expect the result to be the same as if they are valid?
Look at the image below. This is from your VI with 1024 samples and the spectrum graph set to show the datapoint on a small segment of the spectrum.
Notice that there are about 6 non-zero points and that they are not symmetrically distributed around the peak. The total energy in those six points probably comes close to measuring the energy in your signal.
1. The Fourier transforms used in the LV spectral analysis VIs calculate the spectrum of a signal which consists of the finite dataset at the input as if it were concatenated onto itself infintely many times. If you look closely at the data you have generated, you will see that the signal does not start and end at the same voltage and that it does not contain a integer number of complete cycles. The image below shows 2 segments of the data as concatenated from a 128 sample signal. That signal clearly has some spectral components not at the fundamental frequency.
2. Windowing helps minimize the effect of the finite data set by reducing the values at the end of the dataset to near zero. Thus the effects of the discontinuities at the ends of the original datset are minimized. The choice of window functions is an art and depends on the nature of your signal, any noise or interference present and what you will be doing with the spectral results. If you look at the first element in the spectral output array (the ~zero-frequency or DC component) you will see that windowing reduces it substantially.
3. If the frequency of the signal is not exactly a subharmonic of the sampling rate, some of the energy "spills over" into adjacent frequency bins. Some of that effect is visible in the first image above. In general synchronized sampling does not occur. So to get amplitudes of spectral components you need to include the adjacent bins. But you cannot just sum them. Because of phase relations and possible amplitude scaling within the spectrum VIs, the correct way to get the amplitude has to be evaluated carefully. For example with samples set to 1024 the sum of 7 points centered around the peak is 1.27 compared to Max Vrms = 0.988 without a window and 1.01 compared to 0.498 with the window function. If you look at the windowed signal in the time domain compared to the generated sine you can see that the "average" energy must be much less. However the ratio of the peak spectral value to the sum of adjacent bins is higher with the windowed version because the windowed signal has less energy outside the "desired" spectral bins.
Are you more confused than before I started?
Lynn
Malcolm,
I will answer the question about the numbers in 3 now. The question about the 1/3 octave analysis will take a little time to put together some demos and make sure my thinking is clear.
The indicator named "sum" is the sum of 7 elements of the spectrum array centered on the highest peak. The Max Vrms is unmodified from your VI. The False case is just wired straight through to allow checking an unwindowed data sample. I did not need to duplicate the Array Max & Min - just not paying attention to the fact that it was already there. So I ran the VI twice, once with windowing and once without to get the numbers I posted.
You are right that synchronous sampling has its own issues. If you sample exactly two times per cycle (which fails the Nyquist criterion - a strict inequality), you could sample exactly on the zero crossings and every sample would be exactly zero. If you sample exaclty four times per cycle you could get samples of 0, 1, 0, -1 for a unit amplitude sine wave where the first sample was at zero phase. You could also get samples of 0.71, 0.71, -0.71, -0.71 when the first sample was at 45 degrees. Similarly, 30 degrees to the first sample generates 0.50, 0.87, -0.50, -0.87. So lots of samples per cycle make it easy to estimate the peak amplitude while fewer cycles require more complicated reconstruction. If you know that you are sampling at zero crossings and peaks, it is easy to get the amplitude. If you do not know the phase it is more difficult. When you are not synchronized the sample "phase" changes with each cycle.
The same thing is true of the spectrum. If you only have few spectral bins (coarse frequency resolution), it is more difficult to estimate the energy in each signal component. And if the bins do not align with the frequency of the signal, as they generally do not, the process requies some care to get valid results. The Fourier transform also calculates phase information as well as amplitude information and both need to be accounted for in determining the reults.
I will put something together later today or tomorrow to address the 1/3 octave and FFT size issues.
Lynn
