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I am studying for my Digital Signal Processing course and I am stucking on the following exercise:

Given an $\text{LTI}$-system with input signal $$x(n)=\frac{1}{4^n}u(n)$$ and impulse response $$h(n)=\frac{1}{2^n}u(n),$$ calculate the output $y(n)$ of the system using a DTFT.

So, even though this isn't cleared up in the exercise, I imagine $u(n)$ is the discrete-time step function, so it is defined as $$u(n) = \left\{ \begin{array}{ll} 1 & n\geq 0 \\ 0 & n < 0 \\ \end{array} \right. $$ Well, shouldn't the answer be the convolution $(x*h)(n)$? I have tried to use the convolution theorem but I am unable to calcualte the inverse DTFT for the product of those two DTFT's. What should I do now? I am very much confused and do not know how to proceed. Thank you in advance.

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NoHomotopy
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1 Answers1

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As seen here at Wikipedia, the DTFT of signals of the form $a^n u[n]$ is

$$ a^n u[n] \leftrightarrow \frac{1}{1 - a e^{-i \omega}} $$

To perform convolution of two sequences, it is enough to use to the convolution theorem (also mentioned in the above Wikipedia article)

$$ x * y = \rm{DTFT}^{-1} \left[\rm {DTFT} \{x\}\cdot \rm {DTFT}\{y\}\right] $$

AJN
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