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Let's say we have some 5 different sine waves, each having a frequency of 10 Hz but differing in phase i.e. 0, Pi/6, Pi/4, Pi/3, and Pi/2. Each of these signals is sampled at twice the signal frequency, therefore at 20 Hz.

Now upon the reconstruction of the samples, we notice that the amplitude of the reconstructed signal increases as the phase increases. When the phase is 0, the signal is essentially zero and when the amplitude is Pi/2, the signal amplitude is that of the original signal.

Now I can see why this happens by putting test values into the sine samples. For example, for the sine wave with 0 phase, all the samples at 20 hertz are essentially 0 and therefore the reconstruction is poor. I am trying to understand the general idea or a mathematical notion behind this. Is there something that I am missing from the sampling theorem?

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Sampling at twice the highest frequency is the (unreachable) lower limit for when sampling for an infinite length of time (longer than the lifetime of the known universe).

Since you sampled for a clearly shorter amount of time, you have to sample at a higher rate. The shorter the length of sampling, or the shorter the reconstruction filter, the higher the sample rate needed for a given aliasing stop-band ripple floor. This will only be an approximation to sampling a bandlimited signal, since any signal with only finite support in the time domain has infinite support in the frequency domain.

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Is there something that I am missing from the sampling theorem?

Yes. The sample rate must be LARGER than the highest frequency in the signal. Equal doesn't work.

You will ALWAYS get aliasing at the Nyquist frequency itself. The effect of the aliasing depends on the phase, so that's why you see the different amplitudes.

In practice you want some margin between the highest frequency and the Nyquist frequency so you can use non-ideal real world filters to manage the aliasing.

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