It might also be worth pointing-out that the "imaginary" part of a complex signal is not a signal that only exists in the "4th-dimension of hyperspace". If you take a "real" cosine wave signal ("real" in both senses: the real part of a complex signal, and also one that you can create with the signal generator on your bench), and phase-shift it by 90° so that it becomes "imaginary", that only means that it is changed from a cosine wave to a sine wave. It is still a signal that you can create with a signal generator. What's the difference between a cosine wave and a sine wave? The cosine wave has value "1" when time t=0. The sine wave has value "0" when time t=0.Please note that at only 0 and pi phase shift is this not a complex signal. The i is the 'imaginary number' representing the quadraphase signal.
This is why your head hurts so very much trying to do a 90 degree phase shift of a real signal. This is also why 'DC' makes your head ache, because the DC and Pi components of the FFT are STRICTLY REAL.
What does it mean to phase shift DC by 90°? The equation for a discrete-time cosine wave is x[n]=cos(2*pi*n/N), where N is the period, in samples. At DC, N approaches infinity and thus (2*pi*n/N) approaches zero, so x[n]=cos(0)=1 for all n. Phase-shift that by 90° and x[n]=sin(0)=0 for all n.
Similarly, at half the sampling frequency, N=2 (two samples per cycle). Thus, x[n]=cos(2*pi*n/2)=cos(pi*n), and since n is an integer the value of x[n] alternates between +1 and -1. (You can think of this as sampling the cosine wave at its positive and negative peaks.) Phase-shift that by 90° and x[n]=sin(pi*n)=0 for all n. (You can think of this as sampling the sine wave at its zero-crossings.)
This is why the concept of using an FFT/DFT to create imaginary values at DC and pi makes no sense. There is no way to interpret the results.
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