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I am asked to find Fourier transform of y[n] in terms of Fourier transform of x[n].. attached is my attempt! I have a specific question at the end of the attempt.
The definition of y[n] is not valid! if n = 1 y[n] = x[1/3]??? No! You probably have to specify that y[n] = x[n/3] for n = 3k, where k is and integer. This is called upsampling. In that case the second sum you wrote is correct. However, your parenthesis: e^-jw(3k) is misleading! You should have written e^-j(3w) k. You have to realize that X(e^jw) while standard, it a bit of an abuse of notation. Really the DTFT is a function of w. The sum you wrote is just the DTFT with 3w replacing w. Therefore we can do the same in X(e^jw) -> X(e^3jw), number 2.
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