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**About Digital Signal Processing 1: Basic Concepts and Algorithms** **Course**

The subfield of engineering known as digital signal processing is responsible for the extraordinary levels of interpersonal communication and on-demand entertainment that have been made possible in the space of just a few decades. DSP, or digital signal processing, is at the core of the digital revolution that gave rise to CDs, DVDs, MP3 players, mobile phones, and countless other gadgets. It does this by rearranging the fundamental concepts of electronics, telecommunication, and computer science into a unifying paradigm.

You will start from nothing and work your way up to mastering the principles of digital signal processing with this four-part course series. Building a DSP toolkit that is comprehensive enough to analyze a real-world communication system in minute detail will require us to begin with the definition of a discrete-time signal at its most fundamental level. From there, we will move on to topics such as Fourier analysis, filter design, sampling, interpolation, and quantization. In order to bridge the gap that exists between theory and practice, we will routinely make use of hands-on examples and demonstrations.

You will be given several programming examples in the form of Python notebooks, but you are free to use any programming language you like to test the algorithms described in the course. It is recommended that you are proficient in basic calculus and linear algebra in order to get the most out of this class.

**WHAT YOU WILL LEARN**

- The nature of discrete-time signals
- Discrete-time signals are vectors in a vector space
- Discrete-time signals can be analyzed in the frequency domain via the Fourier transform

**Course Apply Link – Digital Signal Processing 1: Basic Concepts and Algorithms**

**Digital Signal Processing 1: Basic Concepts and Algorithms** **Quiz Answers**

### Week 1 Quiz Answers

#### Quiz 1: Homework for Module 1.1 Quiz Answers

Q1. (Difficulty: \starโ) What are the advantages of using digital signals over analog ones? Choose the correct answer(s).

- Digital signals are more robust to noise.
- Processing of digital signals can be easily implemented in modern computers.
- Digital signals can be easily stored.
- Digital signals contains more information than analog ones.

Q2. (Difficulty: \starโ) Amongst the signals listed below, select those that are in digital format.

- Music recorded on a CD.
- Music recorded on a vinyl record.
- A handwritten book manuscript.
- JPEG image on a website.

Q3. (Difficulty: \starโ \starโ) Consider the following finite support signal:

x[n]={(โ1)nn0for n=1,2,3otherwise

Consider also its periodic repetition

y[n] = \sum_{k=-\infty}^{\infty} x[n + 7k]y[n]=โ

k=โโ

โ

โ

x[n+7k].

Compute the energy of x[n]x[n].

For a guide to entering numerical results, see here; infinity is written as oo, i.e. a double lowercase letter โoโ.

Enter answer here

Q4. (Difficulty: \starโ \starโ) Consider the same signals as the previous question. Compute the power of x[n]x[n].

For a guide to entering numerical results, see here; infinity is written as oo, i.e. a double lowercase letter โoโ.

Enter answer here

Q5. (Difficulty: \starโ \starโ) Consider the same signals as the previous question. Compute the energy of y[n]y[n].

For a guide to entering numerical results, see here; infinity is written as oo, i.e. a double lowercase letter โoโ.

Enter answer here

Q6. (Difficulty: \starโ \starโ) Consider the same signals as the previous question. Compute the power of y[n]y[n].

Enter answer here

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Q7. (Difficulty: \starโ \starโ) Consider the signal

x[n] = \delta[n] + 2\delta[n โ 1] + 3\delta[n โ 2].x[n]=ฮด[n]+2ฮด[nโ1]+3ฮด[nโ2].

Consider now its moving average, i.e. the signal

- y[n]=\frac{1}{2}(x[n]+x[n-1])y[n]=
- (x[n]+x[nโ1])

Select the correct expressions from the options below.

y[n]=0.5\delta[n] + 1.5\delta[n โ 1] + 2.5\delta[n โ 2]+1.5\delta[n-3]y[n]=0.5ฮด[n]+1.5ฮด[nโ1]+2.5ฮด[nโ2]+1.5ฮด[nโ3].

The output for n\ge 0nโฅ0 is always zero.

Q8. (Difficulty: \starโ) A music song recorded in a studio is stored as a digital sequence on a CD. The analog signal representing the music is 2 minutes long and is sampled at a frequency f_s=44100\;s^{-1}f

s

โ

=44100s

โ1

. How many samples should be stored on the CD? Write the number of samples without commas or dots. Assume that the audio file is mono, or in other words, single channel.

Enter answer here

Q9. (Difficulty: \star \star \starโโโ) Given the following filter

What is the input-output relationship?

- y[n]=b(ax[n]+x[n-1])-(c+1)x[n-3]y[n]=b(ax[n]+x[nโ1])โ(c+1)x[nโ3].
- y[n]=abx[n]+bx[n-1]+cx[n-3]-x[n-4]y[n]=abx[n]+bx[nโ1]+cx[nโ3]โx[nโ4].
- y[n]=(bax[n]+x[n-1])+(cx[n-3]+x[n-4])y[n]=(bax[n]+x[nโ1])+(cx[nโ3]+x[nโ4]).
- y[n]=b(ax[n]+x[n-1])-(cx[n-3]+x[n-4])y[n]=b(ax[n]+x[nโ1])โ(cx[nโ3]+x[nโ4]).

Q10. (Difficulty: \star \starโโ) What is the minimum period PP (in samples) of the signal e^{j(M/N)2\pi n}e

j(M/N)2ฯn

for M=1,N=3M=1,N=3.

Enter answer here

Q11. (Difficulty: \star \starโโ) What is the minimum period PP (in samples) of the signal e^{j(M/N)2\pi n}e

j(M/N)2ฯn

for M=5, N=7M=5,N=7.

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Enter answer here

Q12. (Difficulty: \star \starโโ) What is the minimum period PP (in samples) of the signal e^{j(M/N)2\pi n}e

j(M/N)2ฯn

for M=35, N=15M=35,N=15.

Enter answer here

### Week 2 Quiz Answers

#### Quiz 1: Homework for Module 1.2 Quiz Answers

Q1. (Difficulty: \starโ) Write the value for the inner product \langle \mathbf{v}^{(0)}, \mathbf{v}^{(1)}\rangleโจv

(0)

,v

(1)

โฉ where

\mathbf{v}^{(0)}=

โกโฃโขโขโขโขโขโขโขโขโขโขโข12121212โคโฆโฅโฅโฅโฅโฅโฅโฅโฅโฅโฅโฅ

and \mathbf{v}^{(1)}=

โกโฃโขโขโขโขโขโขโขโขโขโขโข1212โ12โ12โคโฆโฅโฅโฅโฅโฅโฅโฅโฅโฅโฅโฅ

,

Enter answer here

Q2. (Difficulty: \starโ\starโ) Consider the following vectors in \mathbb R^4R

4

\mathbf{v}^{(0)}=

โกโฃโขโขโขโขโขโขโขโขโขโขโข12121212โคโฆโฅโฅโฅโฅโฅโฅโฅโฅโฅโฅโฅ

โ

, \mathbf{v}^{(1)}=

โกโฃโขโขโขโขโขโขโขโขโขโขโข1212โ12โ12โคโฆโฅโฅโฅโฅโฅโฅโฅโฅโฅโฅโฅ

โ

and \mathbf{v}^{(2)}=

โกโฃโขโขโขโขโขโขโขโขโขโขโข12โ1212โ12โคโฆโฅโฅโฅโฅโฅโฅโฅโฅโฅโฅโฅ

You can verify that the vectors are mutually orthogonal and have unit norm.

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How many different vectors \mathbf{v}^{(3)}v

(3)

could we find such that {\mathbf{v}^{(0)},\ \mathbf{v}^{(1)},\ \mathbf{v}^{(2)},\ \mathbf{v}^{(3)}}{v

(0)

, v

(1)

, v

(2)

, v

(3)

} is a full orthogonal basis in \mathbb R^4R

4

?

0

1

2

3

3

Q3. Let \mathbf{y}=

โกโฃโขโขโ0.50.50.51.5โคโฆโฅโฅ

,

what are the expansion coefficients of \mathbf{y}y in the basis {\mathbf{v}^{(0)}, \mathbf{v}^{(1)}, \mathbf{v}^{(2)}, \mathbf{v}^{(3)}}{v

where

\mathbf{v}^{(0)}=

โกโฃโขโขโขโขโขโขโขโขโขโขโข12121212โคโฆโฅโฅโฅโฅโฅโฅโฅโฅโฅโฅโฅ

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, \mathbf{v}^{(1)}=

โกโฃโขโขโขโขโขโขโขโขโขโขโข1212โ12โ12โคโฆโฅโฅโฅโฅโฅโฅโฅโฅโฅโฅโฅ

โ

, \mathbf{v}^{(2)}=

โกโฃโขโขโขโขโขโขโขโขโขโขโข12โ1212โ12โคโฆโฅโฅโฅโฅโฅโฅโฅโฅโฅโฅโฅ

โ

, \mathbf{v}^{(3)}=

โกโฃโขโขโขโขโขโขโขโขโขโขโข12โ12โ1212โคโฆโฅโฅโฅโฅโฅโฅโฅโฅโฅโฅโฅ

Important: Enter your answer as space separated floating point decimal numbers, e.g. the vector \mathbf{y}y would be entered as :

-0.5 0.5 0.5 1.5

Enter answer here

Q4. Which of the following sets form a basis of \mathbb R^4R?

- {\mathbf{y},\mathbf{v}_0,\mathbf{v}_1,\mathbf{v}_2}{y,v
- {\mathbf{y},\mathbf{v}_0,\mathbf{v}_2,\mathbf{v}_3}{y,v
- {\mathbf{y},\mathbf{v}_1+\mathbf{v}_2,\mathbf{v}_2,\mathbf{v}_3}{y,v
- {\mathbf{y},\mathbf{v}_0,\mathbf{v}_1,\mathbf{v}_3}{y,v

Q5. (Difficulty: \starโ) If we represent finite-length signals as vectors in Euclidean space, many operations on signals can be encoded as a matrix-vector multiplication. Consider for example a circular shift in \mathbb{C}

: a delay by one (i.e. a right shift) transforms the signal \mathbf{x} = [x_0 \, x_1 \, x_2]^Tx=[x

into \mathbf{x}โ = [x_2 \, x_0 \, x_1]^Tx

and it can be described by the matrix

D= โกโฃ010001100โคโฆ

so that \mathbf{x}โ = D\mathbf{x}x

โฒ

=Dx.

Determine the matrix FF that implements the one-step-difference operator in {C}

i.e. the operator that transforms a signal \mathbf{x}x into [(x_0-x_2) \,\, (x_1-x_0) \,\, (x_2-x_1)]^T[(x

Write the 9 integer matrix coefficients one after the other, row by row and separated by spaces.

Enter answer here

Q6. (Difficulty \starโ) Given the matrix

A= โกโฃโขโข0100001000011000โคโฆโฅโฅ

compute the matrix A^4A

(i.e. the fourth power of AA).

(Hint: there is a simple way to do that and, if youโve solved the previous question, it should be obvious).

Write the 16 integer matrix coefficients one after the other, row by row and separated by spaces.

Enter answer here

### Week 3 Quiz Answers

#### Quiz 1: Homework for Module 1.3 Quiz Answers

Q1. (Difficulty: \starโ) Write out the phase of the complex numbers a_1=1-\mathrm{j}a

1

โ

=1โj and a_2=-1-\mathrm{j}a

2

โ

=โ1โj.

Express the phase in degrees and separate the two phases by a single white space. Each phase should be a number in the range [-180,180][โ180,180].

Enter answer here

Q2. (Difficulty: \starโ) Let W_N^k=e^{-\mathrm{j}\frac{2\pi}{N}k}W

N

k

โ

=e

โj

N

2ฯ

โ

k

and N>1N>1. Then W^{N/2}_NW

N

N/2

โ

is equal toโฆ

-1

1

-\mathrm{j}โj

e^{-\mathrm{j}(2\pi/N)+N}e

โj(2ฯ/N)+N

Q3. (Difficulty: \starโ) Which of the following signals (continuous- and discrete-time) are periodic signals?

Note that t\in\mathbb{R}tโR and n\in\mathbb{Z}nโZ.

1 point

x[n]=1x[n]=1.

x(t)=\cos(2\pi f_0 t+\phi)x(t)=cos(2ฯf

0

โ

t+ฯ) with f_0\in\mathbb{R}f

0

โ

โR.

x(t)=t-\operatorname{floor}(t)x(t)=tโfloor(t).

x[n]=e^{-\mathrm{j} f_0 n}+e^{+\mathrm{j} f_0 n}x[n]=e

โjf

0

โ

n

+e

+jf

0

โ

n

, where f_0 = \sqrt{2}f

0

โ

=

2

โ

.

x[n] = \sin(n)x[n]=sin(n).

Q4. (Difficulty: \star\star\starโโโ) Choose the correct statements from the choices below.

Consider the length-NN signal x[n]=(-1)^nx[n]=(โ1)

n

with NN even. Then X[k]=0X[k]=0 for all kk except k=N/2k=N/2

If we apply the DFT twice to a signal x[n]x[n], we obtain the signal itself scaled by NN, i.e. Nx[n]Nx[n].

Consider the length-NN signal x[n]=\cos(\frac{2\pi}{N}Ln+\phi),x[n]=cos(

N

2ฯ

โ

Ln+ฯ), where NN is even and L = N/2L=N/2. Then X[k]=

โงโฉโจN2ejฯ0 for k=L otherwise

.X[k]={

2

N

โ

e

jฯ

0

โ

for k=L

otherwise

โ

.

Q5. (Difficulty: \starโ) Consider the Fourier basis {\mathbf{w}^k}_{k=0,\ldots,N-1}{w

k

}

k=0,โฆ,Nโ1

โ

, where \mathbf{w}^k[n]=e^{-j \frac{2\pi}{N}nk}w

k

[n]=e

โj

N

2ฯ

โ

nk

for 0\le n\le N-10โคnโคNโ1.

Select the correct statement below.

The elements of the basis are orthonormal:

\langle \mathbf{w}^i,\mathbf{w}^j\rangle=

{1 for i=j0 otherwise.

\quadโจw

i

,w

j

โฉ={

1 for i=j

0 otherwise.

โ

The orthogonality of the vectors depends on the length NN of the elements of the basis.

The elements of the basis are orthogonal:

\langle \mathbf{w}^i,\mathbf{w}^j\rangle=

{N for i=j0 otherwise.

\quadโจw

i

,w

j

โฉ={

N for i=j

0 otherwise.

โ

Q6. (Difficulty: \star\starโโ) Consider the three sinusoids of length N=64N=64 as illustrated in the above figure; note that the signal values are shown from n=0n=0 to n=63n=63.

Call y_1[n]y

1

โ

[n] the blue signal, y_2[n]y

2

โ

[n] the green and y_3[n]y

3

โ

[n] the red. Further, define x[n] = y_1[n] + y_2[n] + y_3[n]x[n]=y

1

โ

[n]+y

2

โ

[n]+y

3

โ

[n].

Choose the correct statements from the list below. Note that the capital letters indicate the DFT vectors.

Y_2[k]=

โงโฉโจ16j16j0 for k=8 for k=56 otherwise

Y

2

โ

[k]=

โฉ

โช

โช

โจ

โช

โช

โง

โ

16j

16j

0

โ

for k=8

for k=56

otherwise

โ

Y_1[k]=

{N0 for k=4,60 otherwise

Y

1

โ

[k]={

N

0

โ

for k=4,60

otherwise

โ

Y_3[k]=

โงโฉโจ32320 for k=0 for k=64 otherwise

Y

3

โ

[k]=

โฉ

โช

โช

โจ

โช

โช

โง

โ

32

32

0

โ

for k=0

for k=64

otherwise

โ

|x|_2^2=|X|_2^2=12800โฅxโฅ

2

2

โ

=โฅXโฅ

2

2

โ

=12800

Q7. (Difficulty: \star\star\starโโโ) Consider the length-NN signal

x[n]=\cos\left(2\pi\frac{L}{M}n\right)x[n]=cos(2ฯ

M

L

โ

n)

where MM and LL are integer parameter with 0 \lt L \le N-10<LโคNโ1, 0 \lt M \le N0<MโคN.

Choose the correct statements among the choices below.

If N=MN=M and N\neq 2LN

๎

โ

=2L, it is easier to compute the norm of the signal |\mathbf{x}|_2โฅxโฅ

2

โ

in the Fourier domain, using Parsevalโs Identity.

Consider the circularly shifted signal y[n]=x[(n-D)\mod N].y[n]=x[(nโD)modN]. In the Fourier domain, since the DFT operator is shift invariant, it is Y[k]=X[k]Y[k]=X[k].

The signal has always exactly LL periods 0 \le n \lt N0โคn<N

The DFT X[k]X[k] has two elements different from zero if N=MN=M and N\neq 2lN

๎

โ

=2l.

Q8. (Difficulty: \starโ) Consider an orthogonal basis {\phi_i}_{i=0,\dots,N-1}{ฯ

i

โ

}

i=0,โฆ,Nโ1

โ

for \mathbb{R}^NR

N

. Select the statements that hold for any vector \mathbf{x} \in \mathbb{R}^NxโR

N

.

\displaystyle \Vert \mathbf{x}\Vert_2^2 = \frac{1}{P}\sum_{i=0}^{N-1} \vert \langle x , \phi_i \rangle \vert^2โฅxโฅ

2

2

โ

=

P

1

โ

i=0

โ

Nโ1

โ

โฃโจx,ฯ

i

โ

โฉโฃ

2

if and only if |\phi_i|_2^2=Pโฅฯ

i

โ

โฅ

2

2

โ

=P \forall iโi.

\displaystyle \Vert \mathbf{x}\Vert_2^2 = \frac{1}{P}\sum_{i=0}^{N-1} \vert \langle x , \phi_i \rangle \vert^2โฅxโฅ

2

2

โ

=

P

1

โ

i=0

โ

Nโ1

โ

โฃโจx,ฯ

i

โ

โฉโฃ

2

if and only if |\phi_i|_2=Pโฅฯ

i

โ

โฅ

2

โ

=P \forall iโi.

\displaystyle \Vert \mathbf{x}\Vert_2^2 = \sum_{i=0}^{N-1} \vert \langle x , \phi_i \rangle \vert^2 . โฅxโฅ

2

2

โ

=

i=0

โ

Nโ1

โ

โฃโจx,ฯ

i

โ

โฉโฃ

2

.

\displaystyle \Vert \mathbf{x}\Vert_2^2 = \sum_{i=0}^{N-1} \vert \langle x , \phi_i \rangle \vert^2โฅxโฅ

2

2

โ

=

i=0

โ

Nโ1

โ

โฃโจx,ฯ

i

โ

โฉโฃ

2

if and only if |\phi_i|_2=1โฅฯ

i

โ

โฅ

2

โ

=1 \forall iโi.

Q9. (Difficulty: \star\starโโ) Pick the correct sentence(s) among the following ones regarding the DFT \mathbf{X}X of a signal \mathbf{x}x of length NN, where NN is odd.

Remember the following definitions for an arbitrary signal (asterisk denotes conjugation):

hermitian-symmetry: x[0]x[0] real and x[n]=x[N-n]^*x[n]=x[Nโn]

โ

for n=1,\dots,N-1n=1,โฆ,Nโ1.

hermitian-antisymmetry: x[0]=0x[0]=0 and x[n]=-x[N-n]^*x[n]=โx[Nโn]

โ

for n=1,\dots,N-1n=1,โฆ,Nโ1.

- If the signal \mathbf{x}x is hermitian antisymmetric, then its DFT \mathbf{X}X is purely imaginary.
- If the signal \mathbf{x}x is hermitian-symmetric, then the DFT \mathbf{X}X is also hermitian-symmetric.
- If the signal \mathbf{x}x is purely real, then the DFT \mathbf{X}X is purely imaginary.
- If the signal \mathbf{x}x is hermitian-symmetric, then its DFT is real.

### Week 4 Quiz Answers

#### Quiz 1: Homework for Module 1.4 Quiz Answers

Q1. (Difficulty: \starโ) Consider the magnitude DTFTs |X(e^{j\omega})|โฃX(e

jฯ

)โฃ and |Y(e^{j\omega})|โฃY(e

jฯ

)โฃ shown below, where vertical lines represent Dirac deltas:

Both underlying signals x[n]x[n] and y[n]y[n] are periodic. Find their periods and write them below, separated by a space. Please write the smallest period, i.e. a 5-periodic signal is also obviously 15-periodic but weโre interested in 5.

Enter the period of x[n]x[n] and y[n]y[n] with a unique white space in between.

Enter answer here

Q2. (Difficulty: \starโ) We will see in later lectures that communication systems must fulfill what is called the โbandwidth constraintโ, that is, the energy of the signals that they transmit must strictly fit into pre-defined frequency bands. In this problem we will look at the bandwidth constraint in the discrete-time domain.

The signal x[n]x[n] is real-valued and its spectrum is nonzero only over the [-\pi/8,\ \pi/8][โฯ/8, ฯ/8] interval. Due to the bandwidth constraint we need to โfitโ the signal over the bands indicated in green in the following figure

To this aim, we need to design a processing block \mathcal{H}H in order to convert x[n]x[n] into a sequence s[n]s[n] satisfying the following requirements:

The support of the DTFT of s[n] must be limited to [โ3ฯ/4, โฯ/2]โช[ฯ/2, 3ฯ/4]

The sequence s[n] must be real-valued (x[n] is real-valued)

Which of the following input/output relationships for \mathcal{H}H meet both requirements? (check all correct answers) :

s[n]=\sin(\frac{21\pi}{8}n)\cdot x[n]s[n]=sin(

8

21ฯ

โ

n)โ
x[n]

s[n] = e^{j\frac{5\pi}{8}n}\cdot x[n] s[n]=e

j

8

5ฯ

โ

n

โ
x[n]

s[n]=\cos(\frac{5\pi}{8}n)\cdot x[n]s[n]=cos(

8

5ฯ

โ

n)โ
x[n]

s[n]=\mathrm{IDTFT}{X(e^{j(\omega-5\pi/8)})}s[n]=IDTFT{X(e

j(ฯโ5ฯ/8)

)}

Q3. (Difficulty: \starโ) Consider the length-LL signal

x[n]=

{100โคnโคMโ1MโคnโคLโ1

\; ,x[n]={

1

0

โ

0โคnโคMโ1

MโคnโคLโ1

โ

,

Write out the closed-form analytical expression for its DFT coefficients X[k]X[k].

Be careful with your typing since the regular-expression parser can be a bit picky. Check Coursera help to enter math expression. In particular, remember that in the Coursera platform the symbols are different:

II (capital i) is used for the imaginary unit instead of jj

Eulerโs number is EE instead of ee

you can also use the exponential function \exp(\cdot)exp(โ )

\piฯ is defined as pipi

The only other symbols youโll need for the answer are the case-sensitive variables k, M, Lk,M,L.

Finally, do not forget to validate your syntax by clicking โPreviewโ before submitting your answer.

For instance, the expression e^{j(\pi L + 3\pi)}/(k + M)e

j(ฯL+3ฯ)

/(k+M) should be entered as

E^(I*(pi*L + 3 * pi))/(k + M)

Preview will appear hereโฆ

Enter math expression here

Q4. The real and imaginary parts of X(e^{j\omega})X(e

jฯ

) are:

After examining the plots, check all the correct statements below.

x[n]x[n] is Hermitian-symmetric x[n]=x^\ast[-n]x[n]=x

โ

[โn].

x[n]x[n] is 0-mean, i.e. \sum_{n\in\mathbb Z} x[n]=0โ

nโZ

โ

x[n]=0.

x[n]x[n] is real valued.

Q5. (Difficulty: \star \starโโ) Consider a signal x[n]x[n] and its DTFT X(e^{j\omega})X(e

jฯ

). Assume X(e^{j\omega})X(e

jฯ

) is differentiable. Compute the inverse DTFT of

j\frac{d}{d\omega}X(e^{j\omega})j

dฯ

d

โ

X(e

jฯ

).

You should write your answer in terms of x[n]x[n] and elementary functions and constants, for example \frac{\pi}{2}x[n]

2

ฯ

โ

x[n] would be written :

pi/2*x[n]

Enter answer here

Q6. (Difficulty: \starโ) Which property of the DTFT allows you to easily compute the inverse DTFT of 4X(e^{j\omega})/\pi -24X(e

jฯ

)/ฯโ2 once you know x[n]x[n]? Just type the name of the property.

Enter answer here

Q7. (Difficulty: \starโ) Take a length-NN signal x[n]x[n] and its DFT X[k]X[k], with 0 \le n,k, \le N-10โคn,k,โคNโ1. Next, consider its periodized version \tilde x[n]=x[n\ \mathrm{mod}\ N]

x

~

[n]=x[n mod N] with its DFS \tilde X[k]

X

~

[k] where now n,k \in \mathbb{Z}n,kโZ.

Which of the following statements are true?

\tilde X[l] = X[l\ \mathrm{mod}\ N]

X

~

[l]=X[l mod N], for all l\in\mathbb ZlโZ

\tilde X[-2] = X[2]

X

~

[โ2]=X[2] for all x[n]x[n] and N>2N>2

\tilde X[k+lN] = X[k]

X

~

[k+lN]=X[k], for all l\in\mathbb ZlโZ and k=0,โฆ,N-1k=0,โฆ,Nโ1.

Q8. (Difficulty: โ) In the class, we learned how the modulation theorem can help us tune a musical instrument. Martin showed us an example with a bass but of course the same works with a classical guitar. Listen carefully to these two samples (with earphones, if possible); each audio clip is the recording of two notes played together:

Audio clip A

Audio Clip B

Select the correct options below.

- The notes are in tune in audio clip A and out of tune in audio clip B
- The notes are in tune in audio clip B and out of tune in audio clip A
- The notes are out of tune in both audio clips
- The notes are in tune in both audio clips

Q9. (Difficulty: \star\starโโ) 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 while in the USA, the ringbback tone is the sum of 2 sinusoids with relatively close frequencies turned on and off periodically.

Here are two audio clips:

Sample A

Sample B

Just by listening to the clips, you should be able to identify the US ringback tone. Explain in the box below what helped you identify the US tone. Use the wording and concepts that appear in the lecture slides. No credit, without a proper explanation, e.g., โI live thereโ is not an answer.

(use only lowercase letters in your answer)

Enter answer here

Q10. (Difficulty: \starโ) As explained in previous question, the European and US ringback tones are composed of one or two sinusoids respectively, multiplied by a square wave switching between 00 and 11. This multiplication by a square wave periodically mutes the sinusoid(s).

Look at the following magnitude DTFT plots (each plot shows the interval [-\pi\ \pi][โฯ ฯ]) :

Select the correct statement amongst the choices below.

Spectrum (a) corresponds to the European ringback tone.

Spectrum (b) corresponds to the US ringback tone.

Spectrum (a) corresponds to the US ringback tone.

Spectrum (b) corresponds to the European ringback tone.

Q11. (Difficulty: \starโ) Consider the sequences

x_1[n]=\cos(2\pi n\cdot\sqrt{2}/30)x

1

โ

[n]=cos(2ฯnโ

2

โ

/30)

x_2[n]=\cos(2\pi n\cdot 1.41421356/30)x

2

โ

[n]=cos(2ฯnโ
1.41421356/30)

with n\in\mathbb ZnโZ

Select the correct statements below.

There exists N\in\mathbb NNโN for which x_2[n]x

2

โ

[n] has a DFS of size NN

There exists N\in\mathbb NNโN for which x_1[n]x

1

โ

[n] has a DFS of size NN

Q12. (Difficulty: \star\starโโ) Consider a signal x[n]x[n]; the only thing we know about the signal is that its DTFT is strictly bandlimited between -\frac{\pi}{10}โ

10

ฯ

โ

and \frac{\pi}{10}

10

ฯ

โ

. We now modulate the signal to obtain y[n] = x[n]\cos(\omega_c n)y[n]=x[n]cos(ฯ

c

โ

n).

Among the possibilities below, select all the values for \omega_cฯ

c

โ

that allow us to perfectly demodulate y[n]y[n].

\frac{11\pi}{12}

12

11ฯ

โ

\frac{9\pi}{10}

10

9ฯ

โ

\frac{\pi}{3}

3

ฯ

โ

\frac{\pi}{20}

20

ฯ

**Conclusion**

Hopefully, this article will be useful for you to find all theย **Week, final assessment, and Peer Graded Assessment Answers of Digital Signal Processing 1: Basic Concepts and Algorithms** **Quiz of Coursera**ย and grab some premium knowledge with less effort. If this article really helped you in any way then make sure to share it with your friends on social media and let them also know about this amazing training. You can also check out our other courseย Answers.ย So, be with us guys we will share a lot more free courses and their exam/quiz solutions also, and follow ourย Techno-RJย **Blog**ย for more updates.

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