# I'm having problems simplifying this discrete-time fourier tranform

I have this problem, and I can't get to the solution. $$X(e^{j\omega}) = \sum_{n=-\infty}^{\infty} {(0.6)^{|n|}[u(n + 10) − u(n − 11)]}e^{-j\omega n}$$

The solution is $$X(e^{j\omega}) = \frac{0.64 − 2(0.6)^{11} \cos(11ω) + 2(0.6)^{12} \cos(10ω)}{1.36 − 1.2 \cos(ω)}$$

But I don't know how to get there, i think it is related to a series(geometric, maybe) and things like Euler formula but still not able to find a way to get there. This is an exercise(3.3.2) from Proakis and Inge book, Digital Signal Processing using Matlab.

• Homework? I can see three separate properties of the Fourier Transform that you can use to simplify this if you apply them in turn -- can you name one? Nov 29, 2021 at 0:31
• Linearity, I think. Nov 29, 2021 at 7:42
• The very outermost operation going on there is summing $x_2(n) = {(0.6)^{|n|}[u(n + 10) − u(n − 11)]}e^{-j\omega n}$ from $-\infty$ to $+\infty$. If you knew the Fourier transform of $x_2$, what would be your overall Fourier transform? (Yes, the Socratic Method is named after a guy who was forced by his colleagues to drink poison. None the less, I'm hoping you'll have an "aha" moment here). Nov 29, 2021 at 21:41

$$X(e^{j\omega}) = \sum_{n=-10}^{\infty} {(0.6)^{|n|}}e^{-j\omega n} - \sum_{n=11}^{\infty} {(0.6)^{|n|}}e^{-j\omega n}$$