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3. (20%, Ch5: DFT/FFT) Let $X(e^{jw})$ denote the DTFT of the sequence $x(n) = (1/2)^n u(n)$. Let $y(n)$ denote a finite- duration sequence of length 9; i.e., $y(n) = 0, n < 0$, and $y(n) = 0, n \ge 9$. The 9-point DFT of $y(n)$, denoted by $Y(k)$, corresponds to 9 equally spaced samples of $X(e^{jw})$; i.e., $Y(k) = X(e^{j2\pi k/9})$. Please manually determine what is $y(n)$. Note: this is a manual derivation problem. Use the equation $y(n) = \sum_{r=-\infty}^{\infty} x(n - rN)$

          3. (20%, Ch5: DFT/FFT)
Let $X(e^{jw})$ denote the DTFT of the sequence $x(n) = (1/2)^n u(n)$. Let $y(n)$ denote a finite-
duration sequence of length 9; i.e., $y(n) = 0, n < 0$, and $y(n) = 0, n \ge 9$. The 9-point DFT
of $y(n)$, denoted by $Y(k)$, corresponds to 9 equally spaced samples of $X(e^{jw})$; i.e.,
$Y(k) = X(e^{j2\pi k/9})$. Please manually determine what is $y(n)$.
Note: this is a manual derivation problem. Use the equation $y(n) = \sum_{r=-\infty}^{\infty} x(n - rN)$

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3. (20%, Ch5: DFT/FFT)
Let X(e^jw) denote the DTFT of the sequence x(n) = (1/2)^n u(n). Let y(n) denote a finite-
duration sequence of length 9; i.e., y(n) = 0, n < 0, and y(n) = 0, n ≥ 9. The 9-point DFT
of y(n), denoted by Y(k), corresponds to 9 equally spaced samples of X(e^jw); i.e.,
Y(k) = X(e^j2π k/9). Please manually determine what is y(n).
Note: this is a manual derivation problem. Use the equation y(n) = ∑r=-∞^∞ x(n - rN)

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University Physics with Modern Physics

Hugh D. Young 14th Edition

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Solved by Expert Kajal Raghubanshi

Step 1

The DTFT of x(n) can be found by substituting the given sequence into the DTFT formula: X(e^jω) = Σ [x(n) * e^(-jωn)] Substituting x(n) = (1/2)^n u(n), we have: X(e^jω) = Σ [(1/2)^n * u(n) * e^(-jωn)] Since u(n) = 0 for n < 0, the summation can be simplified Show more…

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Let X(e^jÏ‰) denote the DTFT of the sequence x(n) = (1/2)^n u(n). Let y(n) denote a finite-duration sequence of length 9; i.e., y(n) = 0, n < 0, and y(n) = 0, n > 9. The 9-point DFT of y(n), denoted by Y(k), corresponds to 9 equally spaced samples of X(e^jÏ‰); i.e., Y(k) = X(e^j2Ï€k/9). Please manually determine what y(n) is. Note: this is a manual derivation problem. Use the equation y(n) = x(n - rN).

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