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Dip As4

The document is an individual assignment for Simad University's Computer Science Department, focusing on various mathematical proofs and calculations related to complex numbers and Fourier transforms. It includes tasks such as proving properties of the imaginary unit 'i', calculating powers of 'i', and performing discrete Fourier transforms (DFT) and their inverses (DIFT) for given data points. The assignment is due on March 11, 2024, and consists of multiple questions requiring mathematical proofs and transformations.

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OmarHaadnaan
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6 views1 page

Dip As4

The document is an individual assignment for Simad University's Computer Science Department, focusing on various mathematical proofs and calculations related to complex numbers and Fourier transforms. It includes tasks such as proving properties of the imaginary unit 'i', calculating powers of 'i', and performing discrete Fourier transforms (DFT) and their inverses (DIFT) for given data points. The assignment is due on March 11, 2024, and consists of multiple questions requiring mathematical proofs and transformations.

Uploaded by

OmarHaadnaan
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Simad University

Computer Science Dept.


Individual Assignment
DIP Assignment Four: [2 Marks]
Deadline:11 March 2024
Student Nam e:_____________________________________________ID#_____
Questions
1. Prove the following
i. 𝑖 4𝑛 = 1 ii. 𝑖 4𝑛+1 = 1 iii. 𝑖 4𝑛+2 = −1 iv. 𝑖 4𝑛+3 = −𝑖
2. Calculate powers of i:
𝒊𝟎 𝒊−𝟏 𝒊−𝟐 𝒊−𝟑 𝒊−𝟒 𝒊−𝟓 𝒊−𝟔

3. Using Fourier Transform of F(x) for the following points: x(0) =3, x(1) =4, x(2) =
6, and x(3) = 7.Calculate:-
i. DFT ii. DIFT
4. Using Fourier Transform of F(x) for the following points: x(0) =16, x(1) =12, x(2)
= 8, and x(3) = 4.Calculate:-
i. DFT ii. DIFT
5. Prove the followings:
−𝑗𝜋
i. 𝑒 2 = −𝑗
−𝑗3𝜋
ii. 𝑒 2 = 𝑗 iii. 𝑒 −𝑗27𝜋= − 1 iv. 𝑒 −𝑗2𝜋 = 1
3. Show that Discrete inverse Fourier Transform (DIFT) can be calculated directly
from the formula of Discrete Fourier Transform (DFT)
4. Consider the simple function f(x) given below and obtain the Fourier
Transform of F(u)

𝑀 𝑁
5. Prove that 𝐹((𝑥, 𝑦){−1}𝑥+𝑦 ) = 𝐹(𝑢 − ,𝑣 − )
2 2
END
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