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Suppose <math>X(w)\,\!</math> is the DTFT of a discrete-time signal <math>x[n]\,\!</math>. | Suppose <math>X(w)\,\!</math> is the DTFT of a discrete-time signal <math>x[n]\,\!</math>. | ||
| − | What is the DTFT of the time-reversal <math>x[-n]</math>? | + | What is the DTFT of the time-reversal <math>x[-n]\,\!</math>? |
<math>\begin{align} & \sum_{n=-\infty}^{\infty} x[-n]e^{-jwn} \\ & \quad (\text{change of variable} \;\; m=-n) \\ = & \sum_{m=-\infty}^{\infty} x[m]e^{jwm} = X(-w) \\ \end{align}</math> | <math>\begin{align} & \sum_{n=-\infty}^{\infty} x[-n]e^{-jwn} \\ & \quad (\text{change of variable} \;\; m=-n) \\ = & \sum_{m=-\infty}^{\infty} x[m]e^{jwm} = X(-w) \\ \end{align}</math> | ||
Revision as of 11:16, 17 November 2010
Solution to Q1 of Week 13 Quiz Pool
Suppose $ X(w)\,\! $ is the DTFT of a discrete-time signal $ x[n]\,\! $.
What is the DTFT of the time-reversal $ x[-n]\,\! $?
$ \begin{align} & \sum_{n=-\infty}^{\infty} x[-n]e^{-jwn} \\ & \quad (\text{change of variable} \;\; m=-n) \\ = & \sum_{m=-\infty}^{\infty} x[m]e^{jwm} = X(-w) \\ \end{align} $
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