Please do exercise 3 and please show step by step and explain The image presents a series of equations related to the Discrete Fourier Transform (DFT), specifically the 4-point DFT and its decomposition into two 2-point DFTs.### Equations Explanation1. **Equation (9): Decomposition of 4-point DFT** We summarize equations using the following notation: \[ DFT(a_0, a_1, a_2, a_3)_j = DFT(a_0, a_2)_j + e^{2\pi ij/4}DFT(a_1, a_3)_j \] \[ DFT(a_0, a_1, a_2, a_3)_{j+2} = DFT(a_0, a_2)_j - e^{2\pi ij/4}DFT(a_1, a_3)_j \] These equations apply if \( j = 0, 1 \).2. **Equation (10): Recovery of 2-point DFTs** By adding and subtracting the two equations from (9), we obtain: \[ DFT(a_0, a_2)_j = \frac{1}{2} \left(DFT(a_0, a_1, a_2, a_3)_j + DFT(a_0, a_1, a_2, a_3)_{j+2}\right) \] \[ DFT(a_1, a_3)_j = \frac{e^{-2\pi i/4}}{2} \left(DFT(a_0, a_1, a_2, a_3)_j - DFT(a_0, a_1, a_2, a_3)_{j+2}\right) \] These equations hold for \( j = 0, 1 \).3. **Equation (11): Calculation of \( a_i \)** Using the two 2-point DFTs, the following are derived for \( a_i \) where \( i = 0, 1, 2, 3 \): \[ a_0 = \frac{1}{2} \left( DFT(a_0, a_2)_0 + DFT(a_ **Exercise 3.** We derived Equation (10) which expresses the two 2-point DFT's \( DFT(a_0, a_2) \) and \( DFT(a_1, a_3) \) in terms of the 4-point DFT, \( DFT(a_0, a_1, a_2, a_3) \). Use a similar argument to express the two 4-point DFT's \( DFT(a_0, a_2, a_4, a_6) \) and \( DFT(a_1, a_3, a_5, a_7) \) in terms of the 8-point DFT, \( DFT(a_0, a_1, \ldots, a_7) \).

Exercise 3. We derived Equation (10) which expresses the two 2-point DFT's DFT (ao, a2) and DFT (α₁, 3) in terms of the 4-point DFT, DFT (ao, a1, a2, a3). Use a similar argument to express the two 4-point DFT's DFT (0, 2, 4, α6) and DFT (a1, a3, a5, a7) in terms of the 8-point DFT, DFT (ao, 1,..., α7). 0

Exercise 3. We derived Equation (10) which expresses the two 2-point DFT's DFT (ao, a2) and DFT (α₁, 3) in terms of the 4-point DFT, DFT (ao, a1, a2, a3). Use a similar argument to express the two 4-point DFT's DFT (0, 2, 4, α6) and DFT (a1, a3, a5, a7) in terms of the 8-point DFT, DFT (ao, 1,..., α7). 0

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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