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Homework for Day 6

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Warning: The hard deadline has passed. You can attempt it, but you will not get credit for it. You are welcome to try it as a learning exercise.

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

In accordance with the Coursera Honor Code, I (Juan Carlos Vega Oliver) certify that the answers here are my own work.

Question 1 (Difficulty: ⋆ ⋆) A ringback tone is the sound you hear in your landline telephone when the remote phone you are trying to call is ringing. In most European countries, the ringback tone is a single sinusoid turned on and off periodically. In the USA, the ringbback tone is the sum of 2 sinusoids with relatively close frequencies turned on and off periodically. Here are two samples A and B (.wav audio file) : Sample A Sample B

You should be able to tell which one is the american tone just by listening (you don't need to know the pitch of the sinusoids nor the on/off timings). Explain which effect helped you find the answer (use the wording of module 4.7)

No credit, without a proper explanation ("I live there" is not a proper explanation), and try to use the terminology used in the lecture slides

Question 2 In the setup of the previous question. (Difficulty: ⋆) As seen in previous question, the ring-back tones of USA and Europe are respectively two or a single sinusoid multiplied in time by a square-wave taking alternatively the values 0 and 1. This multiplication with a square wave makes the "beep" alternative. Given the following two DTFT spectrum plots (each plot shows the interval [−π

π] )

:

Tick the single correct assertion : (Hints: how many "peaks" does the spectrum of a sinusoid have? What does multiplication by a square wave in the time domain do in the frequency domain?) Spectrum (a) corresponds to the USA ringback tone Spectrum (b) corresponds to the USA ringback tone Spectrum (a) corresponds to the European ringback tone Spectrum (b) corresponds to the European ringback tone

Question 3 (Difficulty: ⋆)Consider the sequences

x[n] = cos(2πn ⋅ √2/30)

y[n] = cos(2πn ⋅ 1.41421356/30) n ∈ Z

Tick all the correct assertions

There exists N

∈ N

for which

x[n]

has a DFS of size

N

There exists N

∈ N

for which

y[n]

has a DFS of size

N

Question 4 (Difficulty: ⋆ ⋆) Consider a general signal x[n] bandlimited between

For which value(s) of ωc , can we demodulate

x[n]

from y[n]

π

−

10

and

π 10

.

= x[n] cos(ωc n) ?

(Note that the answer should be valid for any signal x[n] bandlimited to

[−

π 10

,

π 10

] ).

π 3 π 20 11π 12 9π 10

Question 5 (Difficulty: ⋆) We want to compute the DFT of a signal with length 2048. We use the divide-andconquer method until we reach size-2 DFT problems. How many size-2 DFTs do we need to compute?

Question 6 (Difficulty: ⋆ ⋆) We saw in the lecture that finding the DFT of a finite sequence is equivalent to multiplying it with the DFT matrix W from left. X = Wx.

n

Suppose the length of the signal is 2

and we are interested in finding

X.

Assume

the complexity order when we use simple matrix multiplication as above, and

O(K1 )

O(K2 )

is the

complexity order when we use divide and conquer (until we reach DFT-2 size problems).

is

What is the ratio

K1 K2

?

Apart from the usual mathematical constants and functions you should only use the (casesensitive) variable names n

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In accordance with the Coursera Honor Code, I (Juan Carlos Vega Oliver) certify that the answers here are my own work.

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