Zooming in with Chirp Z

The previous page illustrated how Bluestein's identity, with all those chirps in it, enables a more flexible transform type than Discrete Fourier Transform. I was impressed by the accuracy of the method when inputsize N = outputsize K. Despite the dizzying rotations it seems to be as precise as regular DFT. This encouraged me to go ahead and explore the zooming options of chirp Z transform.

For zoom chirp Z, the transform formula is slightly modified with two extra phase parameters. The first one is an angle phi in the definition of W:

With this parameter, the acceleration of the chirp can be set. Below unity, the spectrum will be covered partly instead of completely. Above unity, the spectrum will fold over itself.

The other thing is an extra rotation of the input signal with a factor:

This factor with theta is an addition to the chirp that already rotates the signal. It is a fixed frequency offset. This complex exponential is sometimes abbreviated A^-n to keep formulas simpler. Even with all complex exponentials hidden behind capitals, it looks complicated enough:

The phi and theta parameters together outline the region of the spectrum that will be covered. With theta being a frequency offset, it sets the lower boundary. Phi acts as the reciprocal of a zoom factor: when phi gets smaller, you zoom in on the spectrum. Adding these arguments to Bluestein's algorithm, if you have that already implemented, is relatively simple.

I can hardly wait to test this. So here we go. On the left below is the dull non-zoomed spectrum of cosines with periodicity 5 and 6. In the plot on the right I zoom in with a factor 20. Wahh.. the two points are heavily smeared out.

Maybe it is better to do amplitudes instead of real and imaginary parts for one time. Then we do not have to see those dreadful phase rotations. To compute amplitudes I do Pythagoras on real and imaginary parts, and a factor 1/2^(1/2) for complex to real normalisation, plus a factor 2 compensation for the input/FFTframesize ratio of 1/2. Below I did harmonic cosines 5 and 6 again but now with zoom factor 100:

From this spreaded data, the correct values can not be retrieved anymore. Actually, the non-zooming transform was more precise, although it's plot was not very helpful for inspection. But in real life, one does not always start out with such perfect harmonic cosines. In case the original data was smeared out anyway, a zoom-in may offer a relatively favourable analysis.

For a next testcase I entered cosines with periodicity 5.32 and 100.32. A non-zooming transform shows the major peaks at index 5 and 100. Smaller peaks at index 6 and 101 tell that the actual frequencies must be somewhat higher. One may do an educated guess about it, that is all.

0    0.021106    1    0.022590    2    0.027635    3    0.039069    4    0.069606    5    0.294313    6    0.142466    7    0.059311    8    0.038178

Let us now zoom in on the lowest ten harmonics. The middle of the peak is at index 532. The amplitude at index 532 is around 0.35. Recall that the other half of the amplitudes are in the conjugates, which are now far beyond our scope. The amplitude of sinusoids is 1/2^(1/2) or around 0.707.

527    0.348276    528    0.348803    529    0.349217    530    0.349519    531    0.349709    532    0.349786    533    0.349750    534    0.349601    535    0.349340    536    0.348967    537    0.348481

To inspect the coefficients for the cosine at periodicity 100.32, I choose an offset of periodicity 95. So that is where the view starts. With zoom factor 100, harmonic 100 will be at index 500. The peak is at index 532 and the amplitude there is slightly over 0.35.

527    0.351905    528    0.352463    529    0.352906    530    0.353233    531    0.353443    532    0.353538    533    0.353516    534    0.353378    535    0.353124    536    0.352753    537    0.352267

These results are amazingly accurate about periodicity. When zooming in, the coefficients are spreaded out over more bins. So, zoom chirp Z is a weird interpolation technique. It is quite effective, though. In the testcase, a small segment of the spectrum was in fact upsampled with a factor 100. That is not even the limit.

It is sometimes stated that chirp Z transform can expand a spectrum's resolution. That is why I got interested in the first place. Yet some people firmly contradict this statement. Now that I have played around with chirp Z, I think that both views miss the point partly. Chirp Z has two kinds of resolution: input resolution and output resolution. About the input, chirp Z can not change anything. When you feed it a mix of two frequencies which are not orthogonal within the framesize, there is no way to resolve these frequencies. It is not a defect of chirp Z transform that it can not see such frequencies seperately, because they simply are not separate frequencies over that period. Extending the framesize is the only thing that would help.

Let me do a case with cosines of periodicity 5.2 and 5.8, to check what happens. The plot shows one single blob, with it's highest point at index 551. The blob is not higher but certainly fatter than the ones in the previous case.

546    0.357322    547    0.357450    548    0.357548    549    0.357619    550    0.357661    551    0.357675    552    0.357661    553    0.357619    554    0.357551    555    0.357455    556    0.357332

For comparison I entered two cosines of periodicity 5.5. The result is completely different, peaking at amplitude 0.7.

547    0.704130    548    0.705355    549    0.706347    550    0.707107    551    0.707632    552    0.707924    553    0.707980    554    0.707802    555    0.707388    556    0.706739    557    0.705855    558    0.704736

Apparently, for non-orthogonal frequencies chirp Z transform can at least compute an energy distribution within a single bin of the input resolution. The increased output resolution brings information that you would not have otherwise. 

Let me push it to the extreme by zooming in on only five of the original 1024 bins. Cosines of periodicity 2.1 and 2.9 were entered. They are peaking at 'harmonics' 1.95 and 3.2 however. That is not a consequence of the zoom factor, but of these frequencies being near to each other. Their ripples mutually distort their correlations. This means you can not trust the peak positions in cases like these.

The ripples in the zooming transform do not represent frequencies in the signal input. They are not so much consequence of regular leakage, but an artefact introduced by spectrum upsampling. The ripples represent discontinuities in the downsampled signal which is the time domain equivalent of the upsampled spectrum. Details of that are on the page Chirp Z Mechanism. Smoothing the input with a window will eliminate boundary discontinuities, but at the same time introduce it's own distortion of the spectrum.

All together, I have no idea if or when chirp Z is useful for zooming in on the spectrum. Anyway, there is an option to zoom out on the spectrum, thereby upsampling the signal. That may be a method to get a better view of details in time domain. Resampling with chirp Z transform is discussed on the next page.