FFT on non-uniformally sampled signals
"Does the interpolation of a signal alter the FFT?"
Yes, it does.
The key point is that in a system that uses sampling, you simply do not know what does the signal "do" between two sampling instances. For a high enough sampling frequency Fs, it is just assumed that the corresponding sampling period is high enough to be approximated by a series of sinc pulses. For more information please see: http://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem

By using interpolation, you only assume that the signal behaves in a certain way during the time that it is not observed. Therefore, the result of the FFT will be different, even for different interpolation methods.

Furthermore, interpolation has a "natural" smoothing action because all it can do is "degrade" the existing data, it cannot add anything more to it. For more information about this notion, please see the Data Processing Theorem (https://books.google.co.uk/books?id=F4j8EHBKS1IC&amp;lpg=PA313&amp;ots=tHO2C1ZJs1&amp;dq=the%20data%20processing%20theorem&amp;pg=PA313#v=onepage&amp;q&amp;f=false)

This is why, interpolation will tend to represent lower frequencies (for which you have more data anyway, due to the sampling frequency) better.

By the way, 'interp1' by default uses linear interpolation, that is, intermediate values between sampling instances are assumed to lie on a straight line that is defined by the two known points. An alternative to this would be spline interpolation which takes into account the slope by which the interpolated curve should enter and depart from the interpolating area. For more information about this, please see http://en.wikipedia.org/wiki/Cubic_Hermite_spline

(Please also note Trigonometric Interpolation, a technique very closely associated with the Fourier Transform. For more information please see: http://en.wikipedia.org/wiki/Trigonometric_interpolation)

From the Time Signal, the FFT on the complet signal is closer to reality. but it still doesn't show all frequencies. Why?

Because it is not exactly the same signal, as far as the FT is concerned.

A key property of the Discrete Fourier Transform is "Periodicity". According to this, the signal at the input of the Fourier Transform is assumed...periodic. That is, you may be passing a voice recording of "One Two Three" to the FT to recover its harmonic component but the FT assumes that the signal, in reality sounds like "...One Two Three One Two Three One Two Three One Two Three..." extending to infinity both forwards and backwards. For more information about this please see: http://en.wikipedia.org/wiki/Discrete_Fourier_transform (Periodicity property).

The signal that you call "Time Signal", contains a brief period of silence in the beginning and ends abruptly, that is, it would be discontinuous if it were to be a periodic signal. By the way, you may also be interested in the technique of windowing, for more information please see: http://en.wikipedia.org/wiki/Window_function#Symmetry_and_asymmetry

I almost can't extract any information fro the phase plot. What am I missing?

The phase spectrum is simply the angle between the imaginary and real components of the signal spectrum. However, having seen how the imaginary and real components of the FT look like, their phase content is not expected to be a smooth and well behaved function, except perhaps for very simplistic cases.

By the way, you may want to look up the concept of phase unwrapping, to improve your phase plots. For more information, please see https://ccrma.stanford.edu/~jos/fp/Phase_Unwrapping.html

Hope this helps.