discrete FT

Question 1

Relate fourier matrix to f and C

Answer 1

$$\vec{f} = F \vec{C}$$

where vectors f and C 0 -> N-1

Question 2

How does one define the fourier matrix

Answer 2

NxN fourier matrix

F =
[1 1 1 1  --- 1
 1 w $w^2$ --- $w^{N-1}$
 1  $w^2$ $w^4$ --- $w^{2(N-1)}$
---
 1 $w^{(N-1)}$ $w^{2(N-1)}$ --- $w^{(N-1)^2}$]

where $w = e^{2\pi i/N}$

Question 3

rel foureir matrix to its inverse

Answer 3

since
$F\bar{F} = NI$

$F^{-1} = \frac{1}{N} \bar{F}$

Question 4

DFT in matrix notation

Answer 4

$\vec{C} = \frac{1}{N} \bar{F} \vec{f}$

Question 5

IDFT in matrix notation

Answer 5

$$\vec{f} = F \vec{C}$$

Question 6

Def of DFT

Answer 6

The action of converting a discrete set of function values f0, f1, …, fN−1 into the discrete set
of coefficients C0, C1, …, CN−1

Question 7

When are the discrete coefficients Cn meaningful

Answer 7

they are only meaningful trapezium rule approx of the continuous coefficients cn fpr $$|n| \leq \frac{N}{2}$$

Question 8

How do we arrive at the IDFT

Answer 8

he truncated complex Fourier series with trapezium rule approximations for its coefficients,
returns the exact function values at the node points xk = 2pk/N

Question 9

IDFT

Answer 9

he action of converting a discrete set of coefficients C0, C1, …, CN−1 into the discrete set
of function values f0, f1, …, fN−1