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Jan-Apr 2025: x n πn πn x n

The document outlines a homework assignment for the ECS204 course on Signals and Systems at IISER Bhopal, covering various topics in Fourier analysis. It includes tasks such as finding Fourier series coefficients, discrete Fourier transforms, inverse Fourier transforms, and proving relationships like Parseval’s relation. Additionally, it requires derivations related to periodic sequences and their Fourier series representations.

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Chirayu Sharma
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11 views1 page

Jan-Apr 2025: x n πn πn x n

The document outlines a homework assignment for the ECS204 course on Signals and Systems at IISER Bhopal, covering various topics in Fourier analysis. It includes tasks such as finding Fourier series coefficients, discrete Fourier transforms, inverse Fourier transforms, and proving relationships like Parseval’s relation. Additionally, it requires derivations related to periodic sequences and their Fourier series representations.

Uploaded by

Chirayu Sharma
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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ECS204 Course: Signals and Systems

Jan-Apr 2025 IISER Bhopal


HW6

1. Find the Fourier series coefficients for x[n] = cos 10πn + cos 20πn. How many complex
exponentials are present in x[n]?

2. Find the discrete Fourier transform and sketch the magnitude and phase spectrum of

(a) x[n] = δ[n]


(b) x[n] = u[n]
(c) x[n] = u[n] − u[n − 5]
(d) x[n] = an u[n], |a| < 1

3. Find the inverse Fourier transform of X(Ω), which is a rectangular pulse:


(
1, −W0 < Ω < W0
X(Ω) =
0 elsewhere

4. If x[n] ↔ X(Ω) are Fourier transform pairs, find the Fourier transform of nx[n].

5. If x1 [n] ↔ X1 (Ω) and x2 [n] ↔ X2 (Ω) are two Fourier transform pairs, then prove that

x1 [n] ? x2 [n] = X1 (Ω)X2 (Ω)

6. Prove the Parseval’s relation for discrete-time aperiodic signals:


∞ Z π
X
2 1
|x[n]| = |X(Ω)|2 dΩ
n=−∞
2π −π

7. From the textbook (Alan. V. Oppenheim)


Let x[n] be a periodic sequence with period N and Fourier series representation:
2π
X
x[n] = ak ejk N n
k=hN i

The Fourier series coefficients for each of the following signals can be expressed in terms of
the ak in the equation above. Derive these expressions for:

(i) x[n − n0 ]
(ii) x[n] − x[n − 1]
(iii) x? [−n]

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