Week 12 - Fourier Series - II and KL Transform - Nptel

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Courses » Mathematical Methods and Techniques in Signal Processing

Week 12 - Fourier

Series - II and KL Transform

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Week 0 - Background and Prerequisites

Week 1 - Introduction to Signal Processing, State Space Representation and Vector Spaces - I

Week 2 - Vector Spaces - II

Week 3 - Vector Spaces - III and Signal Geometry

Week 4 - Probability and Random Processes

Week 5 - Sampling Theorem and Multirate Systems - I Week 6 -

Due on 2019-04-24, 23:59 IST.

1)

1.5 points

2) 2 points

Assignment 12

The due date for submitting this assignment has passed.

As per our records you have not submitted this assignment.

Instructions:

Attempt all questions.

1.

Submission deadline: 24th April 2019 23:59 IST 2.

Solutions to be posted: 25th April 2019 3.

Older browsers might show unnecessary vertical bars at the end of math equations.

4.

Consider the vector . Let be the KL transform obtained from a set of vectors containing . What is the norm of the vector obtained after performing KL Transform on the vector ? Provide your answer upto two decimal places.

No, the answer is incorrect.

Score: 0

Accepted Answers:

(Type: Numeric) 9.00

Which of the following functions belong to ?

v = [1 4 8]

^T

A

v u = Av

v

[0, 1]

L

²

(x) = { where 100 − |k| > 0

f

1

k

0 0 ≤ x ≤ 1 else

(x) = where m > 0 and m ∈ Z f

2 1

x^m

(x) = sin mx where m ≥ 0 and m ∈ Z f

3

A project of In association with

Funded by

Mathematical Methods and Techniques in Signa... https://onlinecourses-archive.nptel.ac.in/noc19_...

1 of 4 Wednesday 22 May 2019 10:28 AM

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Powered by Week 8 -

Multirate Systems - IV

Week 9 - Wavelets - I

Week 10 - Wavelets - II and Continuity of Functions

Week 11 - Fourier Series - I

Week 12 - Fourier Series - II and KL Transform

Interaction Session

Convergence in norm of Fourier series Convergence of Fourier series for all square integrable periodic functions Problem on limits of integration of periodic functions Matrix Calculus KL transform Applications of KL transform Demo on KL Transform Quiz : Assignment 12 Assignment 12 - Solutions

3) 2 points

4) 2 points

5) 2 points

6) 2 points

(True/False): Consider the function . Let be the

orthogonal projection of onto the space , as defined in Lecture 71. As is a scaled version of a basis function in the Fourier basis, there exists such

that for and for .

True False

No, the answer is incorrect.

Score: 0

False

Consider the following optimization problem:

.

Consider 's to be the Lagrange multipliers. Then, the Lagrangian is obtained

as .

Using this Lagrangian, we reduce the problem to the following problem .

As this is an unconstrained problem, we solve it by equating the derivative of the Lagrangian with respect to to 0. Find the value of which gives the optimal solution.

No, the answer is incorrect.

Score: 0

(True/False): The following points are uncorrelated

: .

True False

No, the answer is incorrect.

Score: 0

False

Which of the following vectors are the dominant eigenvectors of the covariance

(x) = { where 100 − |k| > 0 f

1

k

0 0 ≤ x ≤ 1

(x) = sin mx where m else ≥ 0 and m ∈ Z f

3

(x) = where m > 0, m is finite and m ∈ Z f

4

x

^m

f(x) = sin 45x cos 45x f

N

(x)

f(x) V

N

f(x)

L ∈ Z (x) = f(x)

f

N

N = L f

N

(x) = 0 N ≠ L

minimize x

^T

x

subject to { a

^T_i

x = } for i b

i

∈ {1, 2, … , n} where , a

i

x ∈ R

^m

, ∈ b

i

R, { } ν

i

L(x, , … , ) = ν

1

ν

n

x

^T

x + ∑

ⁿ_i=1

ν

i

( a

^T_i

x − ) b

i

minimize x x

^T

x + ∑

ⁿ_i=1

ν

i

( a

^T_i

x − ) b

i

x x

0

−

^∑ⁿⁱ⁼¹₂^νⁱ^a^Tⁱ

−

∑ⁿ_i=1νibi νia^T_i 2

−

^∑ⁿⁱ⁼¹₂^νⁱ^aⁱ

−

^∑ⁿⁱ⁼¹₂^νⁱ^aⁱ

[1 1 , [2 − 2 , [1 4 and [0 1 ]

^T

]

^T

]

^T

]

^T

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7)

2 points

8) 1.5 points

matrix ?

Score: 0

Consider the covariance matrix given below obtained from random vectors of length .

We intend to reduce the dimension of the vectors while retaining % of the energy. To what dimension can we reduce the vectors?

Score: 0

(Type: Numeric) 2

Consider the vectors , , , and . With

reference to lecture 76, which of the following represents the covariance matrix of these vectors?

Score: 0

C = [ 4 ] 1 1

4

[ 3 −3 ]

^T

[ −1 −1 ]

^T

[ 4 −1 ]

^T

[ 1 1 ]

^T

[ −1 −1 ]

^T

[ 1 1 ]

^T

C 5 3

C = ⎡

⎣ ⎢ 3

−1 2

−1 6 2

2 2 3

⎤

⎦ ⎥

95

[ 0 2 ]

^T

[ 0 −6 ]

^T

[ 6 0 ]

^T

[ −2 0 ]

^T

[ 36 ]

4 4

36

[ 12 ]

1.33 1.33 12 [ 9 ]

1 1 9 [ 1 ]

0 0 1

Mathematical Methods and Techniques in Signa... https://onlinecourses-archive.nptel.ac.in/noc19_...

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9)

2.5 points

10) 2.5 points

Using the KL transform obtained using the vectors in Question 8, let be the KL transform of . Let the lower energy correspond to the second coordinate. Consider the eigenvectors in obtaining the KL transform to be normalized. What is ?

Score: 0

(Type: Numeric) 8

(True/False): As converges

to in , it also converges uniformly to in . True

False

Score: 0

False

[ 36 ]

4 4

36

[a b]

^T

[3 − 5]

^T

|a||b|

(x) = {

f

n

1

0

³₅

−

_n¹2

≤ x ≤ +

3

5 1

n²

0 L

²

[0, 1] else 0 [0, 1]

Week 12 - Fourier Series - II and KL Transform - Nptel

Courses » Mathematical Methods and Techniques in Signal Processing

Register for Register for Certification exam Certification exam

Course outline

How to access the portal

Due on 2019-04-24, 23:59 IST.

1.5 points

Assignment 12

The due date for submitting this assignment has passed.

No, the answer is incorrect.

(x) = { where 100 − |k| > 0

Multirate Systems - IV

No, the answer is incorrect.

No, the answer is incorrect.

(x) = { where 100 − |k| > 0 f

0

2 points

Score: 0

Mathematical Methods and Techniques in Signa... https://onlinecourses-archive.nptel.ac.in/noc19_...

2.5 points

Score: 0

0

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