(New page: Using the general formula, where k = 0, ... (N/2 - 1), <math> \begin{align} X(k) &= X_0(k) + W^k_NX_1(k) \\ X(k+N/2) &= X_0(k) - W^k_NX_1(k) \end{align} </math> So using appropriate not...) |
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<math> | <math> | ||
\begin{align} | \begin{align} | ||
| − | W_8^1(1)F_1(1) &= 1/\sqrt{2}(1 - j) \sqrt{2}( | + | W_8^1(1)F_1(1) &= (1/\sqrt{2})(1 - j) \sqrt{2}(2+2j) = 4\\ |
W_8^2(2)F_1(2) &= -j(4j) = 4\\ | W_8^2(2)F_1(2) &= -j(4j) = 4\\ | ||
\\ | \\ | ||
k &= 0: X_8(0) = 0, X_8(4) = 0 \\ | k &= 0: X_8(0) = 0, X_8(4) = 0 \\ | ||
k &= 1: X_8(1) = F_0(1) + W_8^1(1)F_1(1) = 4 + 4 = 8\\ | k &= 1: X_8(1) = F_0(1) + W_8^1(1)F_1(1) = 4 + 4 = 8\\ | ||
| − | k &= 1: X_8( | + | k &= 1: X_8(5) = F_0(1) - W_8^1(1)F_1(1) = 4 - 4 = 0\\ |
k &= 2: X_8(2) = F_0(2) + W_8^2(2)F_1(2) = 4 + 4 = 8\\ | k &= 2: X_8(2) = F_0(2) + W_8^2(2)F_1(2) = 4 + 4 = 8\\ | ||
k &= 2: X_8(6) = F_0(2) - W_8^2(2)F_1(2) = 4 - 4 = 0\\ | k &= 2: X_8(6) = F_0(2) - W_8^2(2)F_1(2) = 4 - 4 = 0\\ | ||
Latest revision as of 05:02, 5 October 2010
Using the general formula, where k = 0, ... (N/2 - 1),
$ \begin{align} X(k) &= X_0(k) + W^k_NX_1(k) \\ X(k+N/2) &= X_0(k) - W^k_NX_1(k) \end{align} $
So using appropriate notation and substituting N = 8, we obtain
$ \begin{align} X_8(k) &= F_0(k) + W^k_8F_1(k) \\ X_8(k+4) &= F_0(k) - W^k_8F_1(k) \\ \end{align} $
Substituting k and using values from the table,
$ \begin{align} W_8^1(1)F_1(1) &= (1/\sqrt{2})(1 - j) \sqrt{2}(2+2j) = 4\\ W_8^2(2)F_1(2) &= -j(4j) = 4\\ \\ k &= 0: X_8(0) = 0, X_8(4) = 0 \\ k &= 1: X_8(1) = F_0(1) + W_8^1(1)F_1(1) = 4 + 4 = 8\\ k &= 1: X_8(5) = F_0(1) - W_8^1(1)F_1(1) = 4 - 4 = 0\\ k &= 2: X_8(2) = F_0(2) + W_8^2(2)F_1(2) = 4 + 4 = 8\\ k &= 2: X_8(6) = F_0(2) - W_8^2(2)F_1(2) = 4 - 4 = 0\\ k &= 3: X_8(3) = 0, X_8(7) = 0 \\ \\ X_8(k) &= [0,8,8,0,0,0,0,0]\text{ or}\\ X_8(k) &= 8 \delta[k-1] + 8 \delta[k-2] \end{align} $
