(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}(1+j) = 4\\
+
W_8^1(1)F_1(1) &= (1/\sqrt{2})(1 - j) \sqrt{2}(1+j) = 4\\
 
W_8^2(2)F_1(2) &= -j(4j) = 4\\
 
W_8^2(2)F_1(2) &= -j(4j) = 4\\
 
\\  
 
\\  

Revision as of 15:57, 3 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}(1+j) = 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(7) = 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} $


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