Difference Fourier Shift Theorem and Cross Correlation Theorem

Question

can someone explain me the difference between the Fourier Shift Theorem and the Cross Correlation Theorem?

The shift theorem says that a delay in the time domain corresponds to a linear phase term in the frequency domain

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So in case of image processing I can find out the shift of two images, which are shifted against each other. I only need to calculate the normalized cross power spectrum of this two images and do an invers Fourier Transform to get the peak, which represents the shift of these to images.

The Cross correlation Theorem says

the cross-correlation between two signals is equal to the product of fourier transform of one signal multiplied by complex conjugate of fourier transform of another signal.

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So what is the difference between these two theorems?

With both I can for example calculate the shift between two images. I can do it, in both ways, with the Phase Correlation (cross-power-spectrum), right?

Answer 1

First of all, what you quote as the Cross correlation Theorem

the cross-correlation between two signals is equal to the product of fourier transform of one signal multiplied by complex conjugate of fourier transform of another signal

doesn't make much sense: the product mentioned above gives the _cross_power spectral density function, and not the cross-correlation function at all. The cross-correlation function is the inverse Fourier transform of the cross-power spectral density function, namely, the product mentioned above.

That being said, the key difference between what you are doing and what the Cross-Correlation Theorem is that you are using a special case that is perhaps better termed the Autocorrelation function theorem. Notice the use of the word another (next to last word in the highlighted wording above). When another is replaced by the same, we get that the autocorrelation function is the inverse Fourier transform of the power spectral density.