Can two different signals have the same Fourier transform?

Can two different signals have the same Fourier transform?

Of course they can. Simply imagine that a signal is composed by two sine waves with different frequencies. The magnitude of the Fourier-transform of both signals will be identical.

What is the relationship between the energy spectrum and the autocorrelation function?

Energy spectral density measures signal energy distribution across frequency. Autocorrelation function of an energy signal measures signal self-similarity versus delay: can be used for synchronization.

What is the spectra of a signal?

The signal spectrum describes a signal’s magnitude and phase characteristics as a function of frequency. The system spectrum describes how the system changes signal magnitude and phase as a function of frequency. For example, at around 100 Hz the transfer function has a magnitude value of around 0.707.

Is the Fourier transform one to one?

That is, the Fourier transform determines the function. The inverse Fourier transform gives a continuous map from L1(R ) to C0(R). This is also a one-to-one transformation. One useful fact is that if f is in L1(R) and g is in L2(R), then the convolution f ∗g is in L2(R).

How can I get autocorrelation of the spectrum?

That means autocorrelation should be able to be obtained by inverse Fourier transform the spectrum. The following code (run within Octave, with “pkg load signal”) shows the Fourier transform of the autocorrelation DOES look like the spectrum, but the inverse Fourier transform of the spectrum does not look like the autocrrelation.

How to calculate autocorrelation of a Fourier transform?

Take the inverse Fourier transform using the first transform pair above and you get the autocorrelation function Rx(τ) Which is the result from the post you linked. You can take this the other way and calculate the autocorrelation first and then take the Fourier transform to yield the power spectral density.

What does the autocorrelation look like in octave?

The following code (run within Octave, with “pkg load signal”) shows the Fourier transform of the autocorrelation DOES look like the spectrum, but the inverse Fourier transform of the spectrum does not look like the autocrrelation. What did I do wrong?

How to calculate autocorrelation of periodic power signals?

And, in the end, can we conclude that for a periodic power signal (FRx)(f) = Sx(f) = | X(f) | 2 = ∞ ∑ n = − ∞ | cn | 2δ(f − nf0) where Sx(f) is PSD, holds? Where cn are the complex Fourier coefficients. This is also a common Fourier transform pair.