Docsity
Docsity

Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity


Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan


Guidelines and tips
Guidelines and tips

Assignment 3 signal and systems, Assignments of Signals and Systems

Questions get repeated sometimes from previous year papers or similar sort of questions are given.This could be helpful.

Typology: Assignments

2019/2020

Available from 08/08/2021

meetendra-singh
meetendra-singh 🇮🇳

17 documents

1 / 1

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
EED-201 (Signals and Systems): Assignment 3
1. A continuous-time signal x(t) is obtained at the output of an ideal low-pass filter with cutoff
frequency ωc= 1000π. If impulse-train sampling is performed on x(t), which of the following
sampling periods would guarantee that x(t) can be recovered from its sampled version using
an appropriate low-pass filter?
i) T= 0.5×103ii) T= 2 ×103iii) T= 104.
2. Let x(t) be a signal with Nyquist rate ωo. Determine the Nyquist rate for each of the
following signals:
i) x(t) + x(t1) ii) dx(t)
dt iii) x2(t) iv) x(t) cos(ω0t).
3. Determine the Fourier transform for πω < π in the case of each of the following periodic
signals:
i) sin π
3n+π
4ii) 2 + cos π
6n+π
8.
4. An LTI system with impulse response h1[n] = 1
3nu[n] is connected in parallel with another
causal LTI system with impulse response h2[n]. The resulting parallel interconnection has
the frequency response:
He =12+5e
127e+ej2ω.
Determine h2[n].
5. Compute the Fourier transform of each of the following signals: (use properties where needed)
i) x[n] = 1
3|n|u[n2] ii) x[n] = 1
2|n|cos π
8(n1)iii) x[n] = sin(πn/5)
πn cos 7π
2n
iv) x(t) = 1 + cos πt, |t| 1
0,|t|>1v) x(t) = P+
n=−∞ e−|t2n|vi) d
dt {u(2t)+u(t
2)}.
6. Find the signal x(t) or x[n] corresponding to the following Fourier transforms:
i) Xe = cos2ω+ sin23ω(discrete)
ii) Xe =P
k=−∞(1)kδωπ
2k(discrete)
iii) X() = 2 sin[3(ω2π)]
(ω2π)(continuous)
iv) X() = 2[δ(ω1) δ(ω+ 1)] + 3[δ(ω2π) + δ(ω+ 2π)] (continuous)
v) X() = (sin2(3ω))cos ω
ω2(continuous).
7. The input and the output of a stable and causal LTI system are related by the differential
equation
d2y(t)
dt2+ 6dy(t)
dt + 8y(t)=2x(t)
i) Find the impulse response of the system.
ii) What is the response of this system if x(t) = te2tu(t)?
8. Impulse-train sampling of x[n] is used to obtain:
g[n] = P
k=−∞ x[n]δ[nkN ].
If X(e ) = 0 for 3π/7 |ω| π, determine the largest value for the sampling interval N
which ensures that no aliasing takes place while sampling x[n].
1

Partial preview of the text

Download Assignment 3 signal and systems and more Assignments Signals and Systems in PDF only on Docsity!

EED-201 (Signals and Systems): Assignment 3

  1. A continuous-time signal x(t) is obtained at the output of an ideal low-pass filter with cutoff frequency ωc = 1000π. If impulse-train sampling is performed on x(t), which of the following sampling periods would guarantee that x(t) can be recovered from its sampled version using an appropriate low-pass filter? i) T = 0. 5 × 10 −^3 ii) T = 2 × 10 −^3 iii) T = 10−^4.
  2. Let x(t) be a signal with Nyquist rate ωo. Determine the Nyquist rate for each of the following signals: i) x(t) + x(t − 1) ii) dx dt(t ) iii) x^2 (t) iv) x(t) cos(ω 0 t).
  3. Determine the Fourier transform for −π ≤ ω < π in the case of each of the following periodic signals: i) sin

( (^) π 3 n^ +^

π 4

ii) 2 + cos

( (^) π 6 n^ +^

π 8

  1. An LTI system with impulse response h 1 [n] =

3

)n u[n] is connected in parallel with another causal LTI system with impulse response h 2 [n]. The resulting parallel interconnection has the frequency response:

H

ejω^

= −12+5e

−jω 12 − 7 e−jω^ +e−j^2 ω^.

Determine h 2 [n].

  1. Compute the Fourier transform of each of the following signals: (use properties where needed)

i) x[n] =

3

)|n| u[−n−2] ii) x[n] =

2

)|n| cos

( (^) π 8 (n^ −^ 1)

iii) x[n] = sin( πnπn/ 5)cos

( (^7) π 2 n

iv) x(t) = 1 + cos πt, |t| ≤ 1 0 , |t| > 1 v) x(t) =

n=−∞ e

−|t− 2 n| (^) vi) d dt {u(−^2 −^ t) +^ u(t^ − 2)}.

  1. Find the signal x(t) or x[n] corresponding to the following Fourier transforms: i) X

ejω^

= cos^2 ω + sin^2 3 ω (discrete) ii) X

ejω^

k=−∞(−1) kδ (ω − π 2 k

(discrete) iii) X(jω) = 2 sin3((ω−ω 2 −π)^2 π) iv) X(jω) = 2[δ(ω − 1) − δ(ω + 1)] + 3[δ(ω − 2 π) + δ(ω + 2π)] (continuous) v) X(jω) = (sin^2 (3ω)) cos^ ω ω^2 (continuous).

  1. The input and the output of a stable and causal LTI system are related by the differential equation d^2 y(t) dt^2 + 6^

dy(t) dt + 8y(t) = 2x(t) i) Find the impulse response of the system. ii) What is the response of this system if x(t) = te−^2 tu(t)?

  1. Impulse-train sampling of x[n] is used to obtain: g[n] =

k=−∞ x[n]δ[n^ −^ kN^ ]. If X(ejω^ ) = 0 for 3π/ 7 ≤ |ω| ≤ π, determine the largest value for the sampling interval N which ensures that no aliasing takes place while sampling x[n].