# Which Fourier Transform properties should be applied first when they are combined together?

A sequence $x[n]$ has Fourier Transform $X(e^{jw})$, and the Time shfiting, Frequency shifting and Time reversal properties are shown below, respectively : $$x[n-n_d] \Rightarrow e^{-jwn_d}X(e^{jw})$$ $$e^{jw_On}x[n] \Rightarrow X(e^{j(w-w_O)})$$ $$x[-n] \Rightarrow X(e^{-jw})$$

Suppose we were given that the Fourier transform of a signal $Y(e^{jw})$ is $3e^{j4w}X(e^{-{j(w-w_O)}})$, and we want to work backwards to find the original sequence. If I use Time reversal first then: $$x[-n] \Rightarrow X(e^{-jw})$$ followed by Frequency shifting: $$e^{jw_On}x[-n] \Rightarrow X(e^{-j(w-w_0)})$$ and then finally Time shfiting and Linearity: $$3e^{jw_O(n+4)}x[-n+4] \Rightarrow 3e^{j4w}X(e^{-j(w-w_0)})$$

So, I have obtained $y[n] = 3e^{jw_O(n+4)}x[-n+4]$,but if I had changed the order of properties which I applied,I would have gotten a slightly different answer. Should certain properties always be applied first when dealing with such problems?

If you do things right the order doesn't matter. Note that replacing $n$ by $n+4$ in $x[-n]$ results in $x[-(n+4)]=x[-n-4]$ and not in $x[-n+4]$.