I understand that when we introduce a linear time shift using DFT on a finite sequence, the algorithm assumes that the signal repeats itself outside of the given range. Here is an example explaining this circular shift property of DFT (Oppenheim, 1998):
Now say if I sampled a finite length (X samples) of a pure sinusoid, but the sinusoid itself is not perfectly repeating when joining the beginning and end together. Hence, when I want to introduce a time shift in the frequency domain, the DFT algorithm will assume that my signal repeats itself every X number of samples. So I get this:
The output signal (red) is not a smooth sinusoid because my original signal does not repeat itself perfectly.
My question is, is there a way for me to apply the time shift to the pure sinusoid without having this discontinuity in the output? That is, can I have another algorithm that continues my signal smoothly for me after the time shift has been applied?
Thank you all for your time!