Data truncation and estimation

Further to the above “ Filtering and distortion ” after , We also need to examine the problem is , Whether the truncated signal can be used to estimate the original signal . Of course , The reverse inference here is not a complete reduction , Because for random signals , We tend to pay more attention to its statistical characteristics , Rather than specific values . This requires an article on the spectrum .

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• Data truncation and estimation

  • The spectrum of the truncated signal
  • Dripping water hides the sea

The spectrum of the truncated signal

Real signals often have messy power spectrum . And when the signal is truncated , The power spectrum will be shortened accordingly .

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Simply speaking , For a signal ,

its FFT Transformation （ Because the real signal is often not a periodic signal , So ignore Fourier series ） It can always be expressed as

in other words , A digital signal passes FFT The length of the transformed power spectrum is the same as that of itself . So when the signal is truncated , The power spectrum of a small segment of signal will shorten the corresponding length .

here , If we adopt this assumption , Let's assume that between small pieces of signal “ be similar ” Of , in other words , We believe that random signals are stationary , When it is sampled in different time periods , These sampled signals have the same digital characteristics . Then we hope to restore the original signal from a small signal .

Dripping water hides the sea

The specific method is interpolation . We assume that the original signal has a power spectrum

So truncate （ Cut into M paragraph ） It is equivalent to downsampling it

therefore , The most intuitive idea is that we can put back the downsampled power spectrum in the order of truncation

Let's see the effect , Although there will inevitably be edge effects caused by truncation in the data , However, the truncated short data can still restore the distribution characteristics of the original signal to a large extent . From the comparison of different frequencies , The truncation effect decreases as the frequency decreases .

High freq.

High freq.

Middle freq.

Middle freq.

Low freq.

Low freq.

The code of this article can be seen in my front-end notebook

Reconstruction by segments^[1]

The front-end program can choose a variety of noise forms and filter window functions .

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Reference material

[1]

Reconstruction by segments: https://observablehq.com/@listenzcc/reconstruction-by-segments