calculating “inner product”

[Aidan P]

On page 765, Ellis states that “It turns out that finding the Fourier series coefficients – the optimal scale constants and phase shifts for each harmonic – is very straightforward: All you have to do is multiply the waveform, point-for-point, with a candidate harmonic, and sum up (i.e., integrate) over a complete cycle; this is known as taking the inner product between the waveform and the harmonic”

But, how do we determine the amplitude of the basis function/candidate harmonic that we multiply the waveform by before integration? Is it just 1?

[Catherine Lai]

Yes, the peak amplitude of the candidate harmonic (basis function) is 1.

You can see this from the DFT equation: each term in the sum is the nth input value (x[n]) times the corresponding point on the basis function (the phasor in the signals notebooks), the latter is represented by a complex number (e^{-jnk 2pi/N}, for DFT[k]) which has magnitude 1. This means the corresponding sinusoid has a peak amplitude of 1.

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