Please do Exercise 1 and please show step by step and explain This text provides an explanation of the mathematical concepts related to the Discrete Fourier Transform (DFT) and its inverse (IDFT).Equations (1) and (2) define the relationship between sequences of complex numbers and their transformations.1. **Equation (1):** \( s_j = \sum_{k=0}^{N-1} a_k e^{2\pi i kj/N} \) This equation shows how to compute the sequence \(\{s_j\}\) from the sequence \(\{a_k\}\) using the DFT.2. **Equation (2):** \( a_n = \frac{1}{N} \sum_{j=0}^{N-1} s_j e^{-2\pi i jn/N} \) This equation shows how to recover the original sequence \(\{a_k\}\) using the IDFT.The text explains the symmetric relationship between the sequences \(\{a_n\}\) and \(\{s_j\}\). The notations introduced for convenience are:- **DFT Notation:** \( DFT(a_0, \ldots, a_{N-1}) := \sum_{k=0}^{N-1} a_k e^{2\pi i kj/N}, \quad j = 0, \ldots, N-1 \) - **IDFT Notation:** \( IDFT(s_0, \ldots, s_{N-1}) := \frac{1}{N} \sum_{j=0}^{N-1} s_j e^{-2\pi i jn/N}, \quad n = 0, \ldots, N-1 \) These notations simplify expressions for DFT and IDFT as:- \( \{s_j\} = DFT(\{a_j\}) \)- \( \{a_i\} = IDFT(\{s_j\}) \) where \(\{s_j\}\) and \(\{a_j\}\) denote the sequences \(\{s_0, ..., s_{N-1}\}\) and \(\{a_0, ..., a_{N-1}\}\), respectively.The importance of DFT and IDFT is emphasized in various fields such as modern science, engineering, and mathematics, where they are fundamental tools Now we can do something similar with the two 2-point DFT’s to get equations for \( a_j \) for \( j = 0, 1, 2, 3 \):\[\begin{align*}a_0 &= \frac{1}{2} \left( \text{DFT}(a_0, a_2)_0 + \text{DFT}(a_0, a_2)_1 \right) \\a_2 &= \frac{1}{2} \left( \text{DFT}(a_0, a_2)_0 - \text{DFT}(a_0, a_2)_1 \right) \\a_1 &= \frac{1}{2} \left( \text{DFT}(a_1, a_3)_0 + \text{DFT}(a_1, a_3)_1 \right) \\a_3 &= \frac{1}{2} \left( \text{DFT}(a_1, a_3)_0 - \text{DFT}(a_1, a_3)_1 \right)\end{align*}\]Exercise 1. Using the definition of DFT in Equation (3), prove Equations (11). ◊

Given: D.F.T.α0,.....αN-1j=∑k=0N-1αke2πikjN, j=0,1,....N-1To find: Expressions of two 2-point…

Now we can do something similar with the two 2-point DFT's to get equations for a; for j = 0,1,2,3: (DFT (αo, α2)o + DFT (αo, α2)1) (DFT (ao, a₂)o-DFT (αo, α₂)1) (DFT (α₁, α3)0 + DFT (α₁, α3)1) -(DFT (a1, a2)3 — DFT (α₁, α3)1). Exercise 1. Using the definition of DFT in Equation (3), prove Equations (11). αo 4₂= a₁ =

Now we can do something similar with the two 2-point DFT's to get equations for a; for j = 0,1,2,3: (DFT (αo, α2)o + DFT (αo, α2)1) (DFT (ao, a₂)o-DFT (αo, α₂)1) (DFT (α₁, α3)0 + DFT (α₁, α3)1) -(DFT (a1, a2)3 — DFT (α₁, α3)1). Exercise 1. Using the definition of DFT in Equation (3), prove Equations (11). αo 4₂= a₁ =

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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