I am having a bit of trouble understanding why, after we defined the sinusoid as (2.1), we changed the
sin to a
cos in (2.2).
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The typical complex Fourier transform or decomposition has the cosine function representing the real components and the sine function representing the imaginary components of the decomposition.
The above cosine normalization allows the DC term, or cos(0), to be real and non-zero, given strictly real input (because ωt == 0 for all t if ω == 0). It would be a bit strange to input a strictly real signal and get an imaginary DC term out of a Fourier analysis.
Another feature of the cosine normalization is that the cosine terms decompose a signal into a strictly even symmetric function. The remaining sine terms are all odd functions. Only an even function can represent a constant (DC) non-zero signal.