HelpHomework for Day 4

The hard deadline for this quiz is Sat 2 Nov 2013 9:00 AM PDT (UTC -0700).

Every question is rated with its difficulty, that is indicated with the number of stars.

: Easy questions, there should be enough to get the 40% of the maximum grade.

: Medium difficulty.

: Hard questions

The scores and the explanations for each questions will be available after the hard deadline.

You can repeat the test 5 times, your best score will be considered.

Questions with multiple possible correct answers may have also zero correct answers due to

the randomization.

In accordance with the Coursera Honor Code, I (Arbaaz Khan) certify that the

answers here are my own work.

Question 1

(Difficulty: ) Let

be a DTFT transform-pair. Assume to be differentiable, compute the inverse DTFT of

.

Hint : use integration by parts

You should write your answer in term of and elementary functions and constants, for

example would be written :

pi/2*x[n]

⋆⋆ ⋆⋆ ⋆ ⋆

⋆ ⋆

x[n] ↔ X( )ejω

X

j X( )ddω

ejω

x[n]

x[n]π2

Question 2

(Difficulty: ) Which property of the DTFT allows you to compute easily the inverse DTFT of

?

Remember the result you obtained in the previous question.

Question 3

(Difficulty: ) A discrete sequence has for DTFT .

The real and imaginary parts of are:

By visual inspection of the plots, tick all the true statements about the sequence :

is Hermitian-symmetric .

is real valued.

is 0-mean, i.e. .

⋆

X( )/π − 2ddω

ejω

⋆ x[n] X( )ejω

X( )ejω

x[n]

x[n] x[n] = [−n]x∗

x[n]

x[n] x[n] = 0∑n∈Z

Preview

Question 4

(Difficulty: ) Consider the following signal

and its DFT defined as .

Compute the mathematical expression for

You can find here a short guide for answering quizzes requiring an equation as answer. In

particular we underline that in the Coursera platform the symbol (capital i) is used for the

imaginary unit instead of , that is the symbol used in class. Moreover, you can use both the

exponential function and the exponentiation of the Euler's number (instead of the

classic ). Pi number is defined as . Apart from the usual mathematical constants and

functions you should only use the (case-sensitive) variable names

k M L

(do not forget to validate your syntax by clicking "Preview")

Help

Question 5

(Difficulty: ) Consider a finite-length discrete signal of length ( )

and its DFT .

Consider its periodization and with its DFS .

Which of the following statements are true?

,

for all and .

(assume )

,

for all

⋆ x[l] = { ,10

0 ≤ l ≤ M − 1M ≤ l ≤ L − 1

X[k] = x[l]∑L−1l=0 e−j lk

2π

L

X[k]

I

j

exp(⋅) E

e pi

⋆ x[n] N n = 0, . . . , N − 1

X[k]

[n] = x[n mod N]x~ [k]X~

[k + lN] = X[k]X~

l ∈ Z k = 0, . . . , N − 1

[−2] = X[2]X~

N > 2

[l] = X[l mod N]X~

l ∈ Z

In accordance with the Coursera Honor Code, I (Arbaaz Khan) certify that the

answers here are my own work.

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