If signal sample length is N, it can stand for at most N/2 + 1 kinds of frequencies

Question

1. How to prove the statement in the title?

Here is how I understand it.

Let's say we have 10 samples.

Signal Peak: 1, 3, 5, 7, 9

Signal Valley:2, 4, 6, 8, 10.

Frequencies we can extract from it is: (What are in the bracket are the waveform)

freq (1, 2, 3) freq(1, 4, 5), freq(1, 6, 9), Freq(1, 8, ?) Freq(1, 10, ?) and DC freq(1, 3, 5, 7, 9)

Here it is N/2+1 = 10/2+1 = 6

2, Can we extract a frequency from just 2 signal sample? like Freq(1, 10, ?) in the above.

Answer 1

The statement it wrong.
On the contrary, unless the signal is zero, if it has finite number of samples, generally it has infinite number of frequencies (Finite in Time Domain -> Infinite in Frequency Domain, Finite in Frequency Domain -> Infinite in Time Domain).

If the signal is periodic, and we have N real samples of a full period of the signal, then it is true.

It can be easily seen using the Discrete Fourier Series / Discrete Fourier Transform.

What is right in general is that any N samples can be represented by N/2 + 1 frequencies. Namely, Applying DFT on a signal of length N and then IDFT will result in the exact same N samples.

This question should really be moved into Signal Processing Forum.