Fourier's basic theorem is that any (non-pathological) waveform can be decomposed into the sum of sinusoids, even waveforms that look nothing like a sinusoid. In the finite-sized discrete case (DFT), there are a finite number of sinusoidal basis vectors, and any signal can be decomposed into them, including non-sinusoidal waveforms, as well as segments of sinusoidal waveforms of different frequencies from those of the basis vectors.

So your basic confusion might be in expecting the DFT to be a "search" for a single basis solution, instead of a decomposition into a bunch of sinusoids that individually might or might not resemble the input closely, or at all.

However, if there are some reasonably close matches among the basis sinusoids, searching x[i] of the DFT magnitude result might find the one with the closest match or fit, which, itself, again, need not be an exact match. So need not be your "expected" one.

Fourier's basic theorem is that any (non-pathological) waveform can be decomposed into the sum of sinusoids, even waveforms that look nothing like a sinusoid. In the finite-sized discrete case (DFT), there are a finite number of sinusoidal basis vectors, and any signal can be decomposed into them, including non-sinusoidal waveforms, as well as segments of sinusoidal waveforms of different frequencies from those of the basis vectors.

So your basic confusion might be in expecting the DFT to be a "search" for a single basis solution, instead of a decomposition into a bunch of sinusoids that individually might or might not resemble the input closely, or at all.

However, if there are some reasonably close matches among the basis sinusoids, searching x[i] of a DFT magnitude result might find the one with the closest match or fit, which, itself, again, need not be an exact match. So need not be your "expected" one.

Fourier's basic theorem is that any (non-pathological) waveform can be decomposed into the sum of sinusoids, even waveforms that look nothing like sinusoids. In the finite-sized discrete case (DFT), there are a finite number of sinusoidal basis vectors, and any signal can be decomposed into them, including non-sinusoidal waveforms, as well as sinusoidal waveforms of different frequencies from those of the basis vectors.

So your basic confusion might be in expecting the DFT to be a "search" for a single basis solution, instead of a decomposition into a bunch of sinusoids that individually might or might not resemble the input closely, or at all.

Fourier's basic theorem is that any (non-pathological) waveform can be decomposed into the sum of sinusoids, even waveforms that look nothing like a sinusoid. In the finite-sized discrete case (DFT), there are a finite number of sinusoidal basis vectors, and any signal can be decomposed into them, including non-sinusoidal waveforms, as well as segments of sinusoidal waveforms of different frequencies from those of the basis vectors.

So your basic confusion might be in expecting the DFT to be a "search" for a single basis solution, instead of a decomposition into a bunch of sinusoids that individually might or might not resemble the input closely, or at all.

However, if there are some reasonably close matches among the basis sinusoids, searching x[i] of the DFT magnitude result might find the one with the closest match or fit, which, itself, again, need not be an exact match. So need not be your "expected" one.