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Chapter 5: Problem 10

Write out the Fourier matrix \(F_{8} .\) Show that \(F_{8} P_{8}\) can be partitioned into block form: $$\left(\begin{array}{cc} F_{4} & D_{4} F_{4} \\ F_{4} & -D_{4} F_{4} \end{array}\right)$$

Short Answer

Expert verified
The Fourier matrix \(F_8\) can be found using the formula \(F_{N}[k, n] = \frac{1}{\sqrt{N}}e^{-\frac{2\pi i}{N}kn}\). After finding matrices \(F_4\) and \(D_4\) using similar calculations, we create the permutation matrix \(P_8\). Then, we calculate \(F_8P_8\) and verify the relationship: \[ F_8P_8 = \left(\begin{array}{cc} F_{4} & D_{4} F_{4} \\ F_{4} & -D_{4} F_{4} \end{array}\right) \] Thus, \(F_8P_8\) can be partitioned into the given block form.

Step by step solution

01

Determine the Fourier matrix \(F_8\)

To find the Fourier matrix \(F_8\), we can use the following formula for the elements of the Fourier matrix: \[ F_{N}[k, n] = \frac{1}{\sqrt{N}}e^{-\frac{2\pi i}{N}kn} \] where \(k, n = 0, 1, 2, ..., N-1\) and \(i\) is the imaginary unit. Using this formula, we can find the elements of the Fourier matrix \(F_8\).
02

Find the \(F_4\) matrix

Before we can partition \(F_8P_8\) into the given block form, we must find the \(F_4\) matrix. To do this, we can apply the same formula as in Step 1, but with \(N=4\).
03

Find the diagonal matrix \(D_4\)

The diagonal matrix \(D_4\) is a diagonal matrix with elements \(\frac{1}{\sqrt{8}}e^{-\frac{2\pi i}{8}(0)}, \frac{1}{\sqrt{8}}e^{-\frac{2\pi i}{8}(1)}, \frac{1}{\sqrt{8}}e^{-\frac{2\pi i}{8}(2)}, \frac{1}{\sqrt{8}}e^{-\frac{2\pi i}{8}(3)}\) on the main diagonal.
04

Create permutation matrix \(P_8\)

To create the permutation matrix \(P_8\), we can organize it as follows: \[ P_8 = \left[\begin{array}{cccccccc} 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & -1 \\ \end{array}\right] \]
05

Calculate \(F_8P_8\)

Now, we should multiply the Fourier matrix we found in Step 1 by the permutation matrix we found in Step 4.
06

Demonstrate equivalency

Finally, to show that \(F_8P_8\) can be partitioned into the given block form, we simply need to demonstrate the following equivalent relationship: \[ F_8P_8 = \left(\begin{array}{cc} F_{4} & D_{4} F_{4} \\ F_{4} & -D_{4} F_{4} \end{array}\right) \] If we have successfully shown this equivalency, we will have proved that \(F_8P_8\) can be partitioned into the desired block form.

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Most popular questions from this chapter

Show that the product of two orthogonal matrices is also an orthogonal matrix. Is the product of two permutation matrices a permutation matrix? Explain.

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