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HW10

The document outlines a course on Signals and Systems, detailing various tasks related to Fourier and Laplace transforms, including their similarities and differences, properties of regions of convergence (ROC), and specific signal transformations. It includes problems requiring the determination of Laplace transforms for given signals, analysis of ROC, and true/false statements regarding properties of ROC and transforms. Additionally, it presents relationships between time-domain signals and their Laplace transform pairs.

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Chirayu Sharma
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15 views2 pages

HW10

The document outlines a course on Signals and Systems, detailing various tasks related to Fourier and Laplace transforms, including their similarities and differences, properties of regions of convergence (ROC), and specific signal transformations. It includes problems requiring the determination of Laplace transforms for given signals, analysis of ROC, and true/false statements regarding properties of ROC and transforms. Additionally, it presents relationships between time-domain signals and their Laplace transform pairs.

Uploaded by

Chirayu Sharma
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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ECS204 Course: Signals and Systems

Jan-Apr IISER Bhopal


2025HW10

1. State the similarities and differences between Fourier transform and Laplace

transform 2. Mention the properties that ROC should satisfy for

1. Right-sided signals
2. Left-sided signals
3. Finite-duration signals

3. Determine the Laplace transform of x(t) = e−tu(t)+e−2tu(t). Also, draw the ROC. Does
the Fourier transform of x(t) exist?

4. Determine the Laplace transform of x(t) = e−atu(t) + ebtu(−t), assuming that a and b
are real such that b > −a. Also, draw the ROC. Does the Fourier transform of x(t) exist?

5. Determine the Laplace transform of the following signals. Also, draw the ROC.

(a) x(t) = δ(t)


(b) x(t) = δ(t + 1) + δ(t − 1)
(c) x(t) = e−atu(t) − u(t − 10)

6. Consider the Laplace transform X(s) of a signal x(t) having three poles at -1,1, and
3. De termine all possible legal ROCs.

7. State True or False

(a) ROC can contain poles of the function X(s) of a signal x(t).
(b) ROC comes in strips in the s-plane.
(c) ROC is a connected region in the s-plane.
(d) Fourier transform is Laplace transform evaluated on the jω axis.

8. If x(t) ↔ X(s) are Laplace transform pairs, then show that

(a) x(t − t0) ↔ e−st0X(s)


(b) es0tx(t) ↔ X(s − s0)
(c) ddtx(t) ↔ sX(s)
t
(d) R −∞ x(τ )dτ ↔ X(s)
s
(e) x(−t) ↔ X(−s)

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