Difference between Fourier transform of $\left(\frac{1}{3}\right)^{-k}$ with $k\le -1$ and $\left(\frac{1}{3}\right)^{k}$ with $k\ge 1$

When I want to calculate the discrete Fourier transform of $\frac{1}{3}^{-k}$ with $k\le-1$ I get that: $$\sum_{k=-{\infty}}^{-1}\left(\frac{1}{3}\right)^{-k}e^{-j\omega k} = \sum_{k'=1}^{\infty} \left(\frac{1}{3}e^{j\omega}\right)^{k'} = \frac{\frac{1}{3}e^{j\omega}}{1-\frac{1}{3}e^{j\omega}}$$ But if I change the variable before finding the Fourier transform I have the function $\left(\frac{1}{3}\right)^k$ with $k\ge1$ (both functions are the same). The discrete Fourier transform of the new function is as follows: $$\sum_{k=1}^{\infty}\left(\frac{1}{3}\right)^{k}e^{-j\omega k} = \sum_{k=1}^{\infty}\left(\frac{1}{3}e^{-j\omega}\right)^{k} = \frac{\frac{1}{3}e^{-j\omega}}{1-\frac{1}{3}e^{-j\omega}}$$ Are these two the same?

• The sequence $\{(1/3)^{-k}\}$ for $k\leq -1$ is not the same sequence as $\{(1/3)^{m}\}$ for $m \geq 1$. – Atul Ingle Nov 30 '16 at 23:02
• @AtulIngle you mean the order of the sequence? but the results of them in a definite point are equal, why is the order important? – user137927 Nov 30 '16 at 23:07
• See plots here. imgur.com/a/gVXpI The two sequences are mirror images in time i.e. flipped. This explains why their DFTs has a sign change on the frequency variable. – Atul Ingle Nov 30 '16 at 23:11
• @AtulIngle Oh you're right, I got it, I really appreciate your time. – user137927 Nov 30 '16 at 23:14

As Atul has pointed out, your both sequences are not the same, but they are time-reversed versions of each other:

k = np.arange(-15, 15)

s1 = (1/3.)**(-k)*(k <= -1);
s2 = (1/3.)**(+k)*(k >= +1)

plt.stem(k, s1, 'r')
plt.stem(k, s2, 'g')


Accordingly, what you see in your calculations is the time-reversal property of the Discrete-Time Fourier Transform:

$$x(t)=y(-t) \Leftrightarrow X(f)=Y(-f)$$