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I am trying to learn this part but I dont seem to understand. if I have a finite length sequence, and a n-point DFT of it (in an interval)... is it possible to evaluate expressions of the n-point DFT without computing them explicitly?

the exercise is:

Let $x[n]$ be finite-length sequence such that: $x[n] = \{2,1,1,0,3,2,0,3,4,6\}$ for $0<=n<=9$ and $X[k]$ be the 10-point DFT of $x[n]$ for $0\le k\le9$. evaluate the following without explicitly computing the DFT:

$$I) \space X[0]$$ $$II) \space \sum_{k=0}^9 X[k]$$ $$III) \space \sum_{k=0}^9 |X[k]|^2 $$ $$IV) \space \sum_{k=0}^9 X[k]e^{-j \frac {(4\pi)}{5}k}$$

and then it continues:

let $X[k]$ denote N-point DFT of an N-point sequence $x[n]$ ($n$ is even). Two N/2-point sequence are defined as:

$$g[n] = a_1 x[2n] + a_2 x[2n+1] \space where \space 0\le n\le(N/2)-1$$ $$h[n] = a_3 x[2n] + a_4 x[2n+1] \space where \space 0\le n\le(N/2)-1$$

constant $a_i$ satisfies: $a_1 a_4 \neq a_2 a_3$

let $G[k]$ and $H[k]$ denote the N/2-point DFT's of $g[n]$ and $h[n]$ respectively. Determine N-point DFT $X[k]$ in terms of $G[k]$ and $H[k]$ .

any clues on the solution?

any input is highly appreciated. thank you

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Parts of the answer.

I) is the average value (DC coefficent) of the signal, 2.2

III) is, by Parzeval's theorem, equal to the square sum of the signal, 80

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    $\begingroup$ III) not exactly, only valid if the unitary form of the DFT is used. II) and IV) can be solved by the inverse DFT, the additional task relates to the linearity of the DFT and the invertibility of the sequence construction. $\endgroup$ Jan 27 '14 at 14:39

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