Question
Download Solution PDFWhat is the discrete-time Fourier transform (DTFT) of the sequence given: x[n] = αnu[n], α < 1?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFThe Discrete-Time Fourier transform of a signal of infinite duration x[n] is given by
\(X\left( \omega \right) = DTFT\left\{ {x\left[ n \right]} \right\} = \mathop \sum \limits_{n = - \infty }^\infty x\left[ n \right]{e^{ - j\omega n}}\)
Given signal is, x[n] = αn u[n]
\(X\left( \omega \right) = \mathop \sum \limits_{n = - \infty }^\infty {\alpha ^n}u\left[ n \right]{e^{ - j\omega n}}\)
\( = \mathop \sum \limits_{n = 0}^\infty {\left( {\alpha {e^{ - j\omega }}} \right)^n}\)
\( = \frac{1}{{1 - \alpha {e^{ - j\omega }}}}\)
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