### Get All Weeks Digital Signal Processing 4: Applications Coursera Quiz Answers

Digital Signal Processing is the branch of engineering that, in the space of just a few decades, has enabled unprecedented levels of interpersonal communication and of on-demand entertainment. By reworking the principles of electronics, telecommunication, and computer science into a unifying paradigm, DSP is the heart of the digital revolution that brought us CDs, DVDs, MP3 players, mobile phones, and countless other devices.

The goal, for students of this course, will be to learn the fundamentals of Digital Signal Processing from the ground up. Starting from the basic definition of a discrete-time signal, we will work our way through Fourier analysis, filter design, sampling, interpolation, and quantization to build a complete DSP toolset to analyze a practical communication system in detail. Hands-on examples and demonstrations will be routinely used to close the gap between theory and practice.

To make the best of this class, it is recommended that you are proficient in basic calculus and linear algebra; several programming examples will be provided in the form of Python notebooks but you can use your favorite programming language to test the algorithms described in the course.

### Digital Signal Processing 4: Applications Coursera Quiz Answers

### Week 1 Quiz Answers

#### Quiz 1: Homework for Module 4.1

Q1. (Difficulty \star⋆) A raster scanning is obtained by reading out a 2D signal row-by-row, starting from (n_1,\,n_2) = (0,\,0)(n

1

,n

2

)=(0,0). Each row is scanned in the same direction — from left to right.

Consider the raster scans s_1,\ldots,~s_6s

1

,…, s

6

of different 2D images :

Since each scan has a repetitive pattern, it can be characterised by three parameters: N_1N

1

, N_2N

2

and N_3N

3

:

Parameter N_1N

1

denotes the distance from the origin of the first non-zero sample, N_2N

2

is the length of non-zero segments, and N_3N

3

is the distance between the non-zero segments.

Associate the 2D signals s_1,\ldots,~s_6s

1

,…, s

6

to the appropriate raster scans

Type the 2D signals number separated by a single space, from the one corresponding to the raster scan s_1s

to the one corresponding to s_6s

, e.g. 1 2 3 4 5 6.

Enter answer here

Q2. (Difficulty: \star\star⋆⋆) Consider a 2D signal with a rectangular support :

and a set of filters with impulse responses :

Associate to each filter the result of its convolution with the given 2D signal

Type the number of the resulting convolution outputs separated by a space, from f_1f

to f_6f

6

, e.g. 1 2 3 4 5 6.

Enter answer here

Q3. (Difficulty: \star⋆) Which of the following transforms preserve the distance between two points? Select all that apply.

- Scaling
- Rotation
- Shear
- Flips
- Translation
- Affine transform

Q4. (Difficulty: \star⋆) Which of the following transforms preserve the angle between two lines? Select all that apply.

- Rotation
- Scaling
- Translation
- Affine transform
- Flips
- Shear

Q5. (Difficulty: \star\star⋆⋆) Let AA be an image with resolution equal to 256×256 pixels. We only know that the values of AA’s upper left and lower right pixels are equal to 255 (white colour) and those of AA′s upper right and lower left pixels are equal to 0 (black colour).

Which one of the following images is obtained by bilinear interpolation of the missing values?

Select all the answers that apply.

- b
- a
- e
- f
- c
- d

Q6. (Difficulty: \star⋆) Consider 2D filters with the following impulse responses:

Using the separability property where appropriate, write the minimum total number of arithmetic operations (number of additions and multiplications summed together) per sample needed to filter a 2D image with each of the given filters.

Assume that the size of the filtered 2D image is N\times NN×N, where NN is very large.

Write the minimum number of arithmetic operations separated by a space, from the one corresponding to the filter f_1f

to the one corresponding to f_4f

For example, if f_1f

required 1 addition and 1 multiplication per sample, f_2f

2 additions and 2 multiplications per sample, f_3f

3 additions and 3 multiplications per sample, and f_4f

4 additions and 4 multiplications per sample, the correct answer would be:

Enter answer here

Q7. (Difficulty: \star\star⋆⋆) Consider the 2D signal II of size 16\times 1616×16:

Full (blue) circles correspond to points where the signal has the value 1, otherwise its value is 0.

From the 2D signals I_1,\ldots,\,I_6I

1

,…,I

6

given below, check those that have the same magnitude 2D-DFT spectrum as the 2D signal II.

Select all the answers that apply.

Q8. (Difficulty: \star\star⋆⋆) Which of the following filters would you use to reduce the noise in images taken under low light conditions? Select all the answers that apply.

Q9. (Difficulty: \star\star⋆⋆) Which of the following filters can be used for edge detection? Select all the answers that apply.

Q10. (Difficulty: \star\star⋆⋆) Consider the image given below:

Give the right correspondence between filters f_1,\ldots,\,f_6f

and the filtered images shown below. Please note that in the images below, black pixels indicate a negative value, white pixels a positive value and gray pixels a value of zero.

Type the filtered images’ numbers separated by a space, from the one corresponding to the filter f_1f

to the one corresponding to f_6f

, e.g. 1 2 3 4 5 6.

Enter answer here

Q11. (Difficulty: \star⋆) Which of the following codes are prefix-free codes for four symbols A, B, C, and D? Select all the answers that apply.

- A: 111, B: 110, C: 10, D: 0
- A: 1, B: 010, C: 101, D: 01
- A: 11, B: 101, C: 01, D: 111
- A: 0, B: 10, C: 101, D: 11
- A: 00, B: 01, C: 10, D: 11
- A: 01, B: 11, C: 00, D: 10

Q12. (Difficulty: \star\star⋆⋆) Five symbols, AA, BB, CC, DD, and EE, have the following probabilities:

- p(A) = 0.1p(A)=0.1
- p(B) = 0.23p(B)=0.23
- p(C) = 0.2p(C)=0.2
- p(D) = 0.15p(D)=0.15
- p(E) = 0.32p(E)=0.32

Which of the following codes can be the encoding result from the Huffman coding? Select all the answers that apply.

- A: 00, B: 01, C: 111, D: 110, E: 10
- A: 111, B: 00, C: 01, D: 110, E: 10
- A: 001, B: 10, C: 101, D:00, E: 11
- A: 11, B: 010, C: 011, D: 10, E: 00
- A: 010, B: 11, C: 10, D: 011, E: 00
- A: 011, B: 10, C: 11, D: 010, E: 00

Q13. (Difficulty: \star\star⋆⋆) You are given the following 8\times 88×8 data matrix:

Which of the following choices is the correct answer if we implement zigzag scan and runlength encoding on matrix \mathbf{A}A?

Note: the first number in the parentheses denotes the run length, i.e., the number of zeros before the current value, and the second number is the actual value. Select all the answers that apply.

- (0, 80), (1,-10), (3, 2), (0, 1), (1, -2), (10, 1), (0,0)
- (0, 80), (2, 2), (4, 10), (1, 1), (13, -2), (15, 1), (0, 0)
- (0, 80), (1,-10), (3, 2), (0, 1), (1, -2), (10, 1), (42,0)
- (0, 80), (0, -10), (1, -2), (1, 1), (11, 1), (6, 2), (0, 0)
- (0, 80), (0, -10), (1, -2), (1, 1), (11, 1), (13, 2), (0, 0)
- (0, 80), (2, 2), (9, 1), (1, -10), (8, -2), (15, 1), (0,0)

Q14. (Difficulty: \star⋆) Which of the steps in the JPEG image compression algorithm are lossy, i.e. which steps introduce loss of information.

Note: assume that the DCT transform is computed with infinite precision. Select all the answers that apply.

- Splitting the image into 8\times 88×8 blocks
- DCT transform on 8\times 88×8 blocks
- Quantization
- Zig-zag scan
- Run-length encoding
- Hufman coding

Q15. (Difficulty: \star\star⋆⋆) Consider the definition of the two-dimesional Discrete Cosine Transform:

C[k1,k2]=∑n1=0N−1∑n2=0N−1x[n1,n2]cos[πN(n1+12)k1]cos[πN(n2+12)k2].

Assume that the pixel values x[n_1,n_2]x[n

take on integer values between 00 and 255255 and that the 2D DCT is performed on blocks of size 8\times 88×8 (N=8N=8).

Determine the minimum possible value that a DCT coefficient can have.

Hint: This question requires programming.

Enter answer here

- Using the same setup as the previous question, determine the maximum possible value that a DCT coefficient can have.

Enter answer here

### Week 2 Quiz Answers

#### Quiz 1: Homework for Module 4.2

Q1. (Difficulty \star⋆) You need to design a data transmission system where the data symbols come from an alphabet A with cardinality 32; all symbols are equiprobable. The bandwidth constraint is F_{\min}=250~F

min

=250 MHz, F_{\max}=500~F

max

=500 MHz.

To meet the bandwidth constraint, the signal is upsampled by a factor of 4 and interpolated at F_s=1~F

s

=1 GHz before being converted to the analog domain.

Determine the Baud rate (in symbols/s) and throughput (in bits/s) of the system.

Type the values of Baud rate (in symbols/s) and throughput (in bits/s) separated by a space; write the values as integers (i.e. no exponential notation). For example, if the Baud rate is 10^{6}10

6

symbols/s and throughput 2\cdot 10^62⋅10

6

bits/s, the answer should be written in the following form:

1000000 2000000

Enter answer here

Q2. (Difficulty \star⋆) Consider a QAM system designed to transmit over a bandwidth of 3~3 kHz. The channel’s power constraint imposes a maximum SNR of 30~30 dB. The system can tolerate a probability of error of 10^{-6}10

−6

.

Determine the maximum throughput of the system in bits per second.

Enter the bit rate in bits/s as an integer (i.e. no exponential notation)

Enter answer here

Q3. (Difficulty \star\star⋆⋆) In the specifications for a data transmission system, you are given the bandwidth constraint F_{\min}=400~F

min

=400 MHz, F_{\max}=600~F

max

=600 MHz. Assume the sequence of digital symbols to transmit is i.i.d.

From the options below, choose the combinations of upsampling factor KK and interpolation frequency F_sF

s

that allow you to build an analog transmitted signal meeting the bandwidth constraint.

Select all the answers that apply.

- F_s=1.5~F

s

=1.5 GHz, K=5K=5 - F_s=1.5~F

s

=1.5 GHz, K=10K=10 - F_s=2.4~F

s

=2.4 GHz, K=12K=12 - F_s=1.9~F

s

=1.9 GHz, K=9K=9 - F_s=1~F

s

=1 GHz, K=5K=5

Q4. (Difficulty \star⋆) Consider the raised cosine spectra given below:

Associate the impulse responses shown in random order below with the associated raised cosine spectra.

Type the impulse response numbers separated by a space, starting from the one corresponding to the raised cosine spectrum with \beta = 0β=0 to the one corresponding to \beta = 0.75β=0.75.

Enter answer here

Q5. (Difficulty \star\star\star⋆⋆⋆) Consider a 32-PAM transmission system, where the signaling symbols are placed on the real line like so:

Assume the transmitted symbols are uniformly distributed and independent. The transmission channel is affected by white noise, whose sample distribution is uniform over the interval [-100,\,100][−100,100].

Find the minimum value for GG that guarantees an error probability less than or equal to 10^{-2}10

−2

.

Type the computed value of GG without the use of exponents.

Example: if you found that G=10.52G=10.52, your answer should be:

10.52

Enter answer here

Q6. (Difficulty \star⋆) A transmission channel has a bandwidth of 6~6 MHz and a SNR of 20~20 dB.

Check the throughputs below that are theoretically possible for the given channel.

Select all the answers that apply.

- 36 Mbit/s
- 42 Mbit/s
- 50 Mbit/s
- 6.5 Mbit/s
- 12 Mbit/s

Q7. (Difficulty \star⋆) Assume that we are using a QAM signalling scheme to communicate over a given channel; the system is designed to meet the bandwidth and power constraints. If we want to decrease the error rate, which of the following steps can we take?

Select all the answers that apply.

- Increase the baud rate
- Increase the constellation size MM
- Decrease signal power
- Decrease the constellation size MM
- Decrease the bit rate

Q8. (Difficulty \star\star\star⋆⋆⋆) With respect to the block diagram of a QAM receiver given below, assume \hat{s}[n]

s

^

[n] is a real-valued bandpass signal occupying the interval [ωmin,ωmax] on the positive frequency axis (and symmetric in magnitude around \omega=0ω=0). Assume also that the modulation frequency ωc=(ωmin+ωmax)/2 is much larger than the bandwidth, i.e. \omega_c \gg \omega_{\max}-\omega_{\min}ω

Q9. (Difficulty \star⋆) Consider a simplified ADSL signaling scheme where there are only 8 sub-channels and where the power constraint is the same for all sub-channels. All subchannels have equal width and each sub-channel CHk is centered at \omega_k = \frac{\pi k}{N}ω

k

=

N

πk

, N=8N=8. Assume further that we are allowed to send only on the last six sub-channels,

CH2 to CH7.

We use QAM signalling on each of the allowed sub-channels, and the sub-channels’ SNRs in dBs are shown here:

Which sub-channel will have the smallest throughput (in bits/second), and which will have the largest?

Type the indices of the channel with the lowest and the channel with the highest throughput, separated by a space. Note that the baseband channel (of which we see the [0, pi/16][0,pi/16] portion in the plot) is channel number zero.

Enter answer here

Q10. (Difficulty \star\star⋆⋆) We are still working with the simplified ADSL specification from the previous problem (N=8)(N=8). The sub-channels’ SNRs are slightly different, and they are given below.

To simplify the problem and avoid making calculations,

you are given the error rate curves for different QAM signalling schemes with square constellations (like the ones you saw in the lecture) in the figure below.

Based on the sub-channels’ SNRs shown in the first figure, number of channels N=8N=8, sampling frequency F_s=2~F

s

=2 MHz and the used signalling scheme, determine the maximum throughputs of channels CH3, CH4 and CH7.

Assume that we are not willing to accept the error probabilities higher than P_{\it err}=10^{-6}P

err

=10

−6

on any of the sub-channels.

Type the maximum throughputs on channels CH3, CH4 and CH7 as integers separated by a space.

Enter answer here

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