So there is an example in my book where g(k) is converted to G(f) and its written

$$g(k)\Longleftrightarrow G(f)$$


$$a^ku(k)\Longleftrightarrow \frac{1}{1-ae^{-j2\pi f}}$$

My question is, how do we find that $a^ku(k)\Longleftrightarrow 1/(1-ae^{-j2 \pi f})$ ?


From the definition of the Fourier transform for series you have:

$$G(f)=\sum_{k=-\infty}^{\infty}g(k)e^{-j2\pi kf}=\sum_{k=0}^{\infty}a^ke^{-j2\pi kf}=\frac{1}{1-ae^{-j2\pi f}},\quad |a|<1$$

where the last equality follows from the formula of the geometric sum, which is valied if $|a|<1$ holds.


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