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If $X (n) = \{1 + 2j, 3 + 4j, 5+6j, 7 + 8j\}.$ Find DFT X(k) using DIF-FFT algorithm.

written 8.6 years ago by teamques10 ★ 69k

modified 3.2 years ago by sagarkolekar ★ 11k

(a) find X(k) using DIT-FFT/DIF-FFT algorithm

discrete time signal processing

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written 8.6 years ago by teamques10 ★ 69k

•   modified 8.6 years ago

$X[k]=∑\limits_{n=0}^{N-1}x[n]w_N^{nk} \text {where }$

$$1) N=4$$

$$\space\space \space 2) W_N^1=e^{-j 2π/N}$$

$$x(k) =\begin{bmatrix} W_N^0&W_N^0&W_N^0&W_N^0\\ W_N^0&W_N^1&W_N^2&W_N^3\\ W_N^0&W_N^1&W_N^2&W_N^3\\ W_N^0&W_N^3&W_N^6&W_N^9\end{bmatrix}\begin{bmatrix} X[0]\\ X[1]\\ X[2]\\ X[3]\end{bmatrix}$$

$$x(k)=\begin{bmatrix}1&1&1&1\\ 1&1&1&1\\ 1&1&1&1\\ 1&1&1&1\end{bmatrix}\begin{bmatrix}1+2j\\ 3+4j\\ 5+6j\\ 7+8j\end{bmatrix}\\$$ $$x(k)=\begin{bmatrix} 16+20j \ -8 \ -4-4j \ -8j \end{bmatrix}$$

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