Digital modulation (ASK, FSK, PSK) in Signal Processing - Practice Problems & Coding Challenges

Challenge - 5 Problems

🧠 Conceptual

intermediate

Understanding ASK modulation characteristics

Which of the following statements correctly describes Amplitude Shift Keying (ASK) in digital modulation?

A ASK varies both frequency and phase of the carrier signal simultaneously to represent digital data.

B ASK varies the amplitude of the carrier signal to represent digital data while keeping frequency and phase constant.

C ASK varies the phase of the carrier signal to represent digital data while keeping amplitude and frequency constant.

D ASK varies the frequency of the carrier signal to represent digital data while keeping amplitude and phase constant.

💡 Hint

Think about which property of the carrier wave changes in ASK.

ASK changes the amplitude of the carrier signal to encode digital bits, while frequency and phase remain unchanged.

❓ Predict Output

intermediate

Output of FSK signal generation code snippet

What is the shape of the output signal array generated by the following Python code simulating Frequency Shift Keying (FSK) for a 2-bit sequence?

Signal Processing

import numpy as np

bit_seq = [0, 1]
fs = 1000  # sampling frequency
bit_duration = 0.01  # seconds
f0 = 100  # frequency for bit 0
f1 = 200  # frequency for bit 1

signal = np.array([])
for bit in bit_seq:
    t = np.linspace(0, bit_duration, int(fs * bit_duration), endpoint=False)
    freq = f0 if bit == 0 else f1
    wave = np.sin(2 * np.pi * freq * t)
    signal = np.concatenate((signal, wave))

output_shape = signal.shape

A (100,)

B (10,)

C (200,)

D (20,)

💡 Hint

Calculate samples per bit: sampling frequency times bit duration.

Each bit lasts 0.01 seconds with 1000 samples per second, so each bit has 10 samples. Two bits produce 20 samples total. But code uses int(fs * bit_duration) = 10 samples per bit, concatenated twice, total 20 samples. The shape is (20,).

❓ data_output

advanced

PSK constellation points for binary data

Given a Binary Phase Shift Keying (BPSK) modulation with symbols mapped as 0 -> +1 and 1 -> -1 on the real axis, what is the resulting constellation array for the bit sequence [0, 1, 0, 1]?

Signal Processing

import numpy as np

bits = np.array([0, 1, 0, 1])
constellation = np.where(bits == 0, 1+0j, -1+0j)

A [1.+0.j, -1.+0.j, 1.+0.j, -1.+0.j]

B [0.+1.j, 0.-1.j, 0.+1.j, 0.-1.j]

C [1.+1.j, -1.-1.j, 1.+1.j, -1.-1.j]

D [-1.+0.j, 1.+0.j, -1.+0.j, 1.+0.j]

💡 Hint

Recall BPSK maps bits to points on the real axis.

BPSK maps bit 0 to +1 and bit 1 to -1 on the real axis, so the constellation array matches the bit sequence accordingly.

❓ visualization

advanced

Visualizing FSK signal for a 3-bit sequence

Which plot correctly shows the Frequency Shift Keying (FSK) signal for the bit sequence [1, 0, 1] with frequencies 150 Hz for bit 1 and 75 Hz for bit 0 over 0.02 seconds per bit?

Signal Processing

import numpy as np
import matplotlib.pyplot as plt

bit_seq = [1, 0, 1]
fs = 1000
bit_duration = 0.02
f1 = 150
f0 = 75
signal = np.array([])
time = np.array([])
for i, bit in enumerate(bit_seq):
    t = np.linspace(i*bit_duration, (i+1)*bit_duration, int(fs*bit_duration), endpoint=False)
    freq = f1 if bit == 1 else f0
    wave = np.sin(2 * np.pi * freq * (t - i*bit_duration))
    signal = np.concatenate((signal, wave))
    time = np.concatenate((time, t))
plt.plot(time, signal)
plt.xlabel('Time (s)')
plt.ylabel('Amplitude')
plt.title('FSK Signal')
plt.grid(True)
plt.show()

A A plot with three segments: high frequency sine wave, low frequency sine wave, high frequency sine wave.

B A plot with constant frequency sine wave throughout the time.

C A plot with three segments: low frequency sine wave, high frequency sine wave, low frequency sine wave.

D A plot with random noise instead of sine waves.

💡 Hint

FSK changes frequency based on bit value.

The signal switches frequency according to bits: 1 maps to 150 Hz (high freq), 0 maps to 75 Hz (low freq), so the plot shows high-low-high frequency segments.

🔧 Debug

expert

Identify error in PSK signal generation code

What error will the following Python code raise when generating a Phase Shift Keying (PSK) signal for bits [0, 1] with phases 0 and pi respectively? import numpy as np bits = [0, 1] fs = 1000 bit_duration = 0.01 phases = {0: 0, 1: np.pi} signal = np.array([]) for bit in bits: t = np.linspace(0, bit_duration, fs * bit_duration) wave = np.cos(2 * np.pi * 100 * t + phases[bit]) signal = np.concatenate(signal, wave)

Signal Processing

import numpy as np

bits = [0, 1]
fs = 1000
bit_duration = 0.01
phases = {0: 0, 1: np.pi}
signal = np.array([])
for bit in bits:
    t = np.linspace(0, bit_duration, int(fs * bit_duration))
    wave = np.cos(2 * np.pi * 100 * t + phases[bit])
    signal = np.concatenate((signal, wave))

A NameError: name 'np' is not defined

B ValueError: operands could not be broadcast together

C TypeError: concatenate() takes 1 positional argument but 2 were given

D SyntaxError: invalid syntax

💡 Hint

Check how numpy.concatenate expects its arguments.

numpy.concatenate expects a single sequence (like a list or tuple) of arrays as its first argument, but the code passes two separate arguments, causing a TypeError.

Practice

1. Which digital modulation technique changes the amplitude of the carrier wave to represent data bits?

easy

A. Phase Shift Keying (PSK)

B. Amplitude Shift Keying (ASK)

C. Frequency Shift Keying (FSK)

D. Pulse Code Modulation (PCM)

Solution

1. Step 1: Understand modulation types

   ASK changes the amplitude of the carrier wave to represent data bits, while FSK changes frequency and PSK changes phase.

2. Step 2: Match modulation to description

   The question asks for the technique that changes amplitude, which is ASK.

3. Final Answer:

   Amplitude Shift Keying (ASK) -> Option B

4. Quick Check:

   Amplitude change = ASK [OK]

Hint: ASK changes amplitude, FSK frequency, PSK phase [OK]

Common Mistakes:

• Confusing ASK with FSK or PSK
• Choosing PCM which is not a modulation technique
• Mixing phase and amplitude changes

2. Which of the following is the correct Python syntax to generate a Frequency Shift Keying (FSK) signal using numpy arrays for two frequencies f1 and f2?

easy

A. signal = np.sin(2 * np.pi * f1 * t) + np.sin(2 * np.pi * f2 * t)

B. signal = np.cos(2 * np.pi * f1 * t) + np.cos(2 * np.pi * f2 * t)

C. signal = np.sin(2 * np.pi * f1 * t) * np.sin(2 * np.pi * f2 * t)

D. signal = np.sin(2 * np.pi * (f1 if bit == 0 else f2) * t)

Solution

1. Step 1: Understand FSK signal generation

   FSK uses one frequency for bit 0 and another for bit 1, so the signal switches frequency based on the bit.

2. Step 2: Check syntax for conditional frequency

   signal = np.sin(2 * np.pi * (f1 if bit == 0 else f2) * t) uses a conditional expression to select frequency based on bit, which is correct for FSK.

3. Final Answer:

   signal = np.sin(2 * np.pi * (f1 if bit == 0 else f2) * t) -> Option D

4. Quick Check:

   Conditional frequency selection = signal = np.sin(2 * np.pi * (f1 if bit == 0 else f2) * t) [OK]

Hint: FSK switches frequency based on bit using conditional [OK]

Common Mistakes:

• Adding two sine waves instead of switching frequency
• Multiplying sine waves which is incorrect
• Using cosine instead of sine without reason

3. Given the following Python code snippet for Phase Shift Keying (PSK), what will be the output signal array for bits = [0, 1] and phase0 = 0, phase1 = pi?

import numpy as np
t = np.linspace(0, 1, 5)
bits = [0, 1]
phase0 = 0
phase1 = np.pi
signal = np.array([np.cos(2 * np.pi * 1 * t + (phase0 if bit == 0 else phase1)) for bit in bits])
print(signal)

medium

A. [[ 1. 0. -1. 0. 1.]
[-1. 0. 1. 0. -1.]]

B. [[ 0. 0. 0. 0. 0.]
[ 1. 1. 1. 1. 1.]]

C. [[ 1. 0. -1. 0. 1.]
[ 1. 0. -1. 0. 1.]]

D. [[ 1. 1. 1. 1. 1.]
[ 1. 1. 1. 1. 1.]]

Solution

1. Step 1: Calculate cosine values for bit 0

   For bit 0, phase = 0, t = np.linspace(0,1,5) = [0, 0.25, 0.5, 0.75, 1], angles = 2πt = [0, π/2, π, 3π/2, 2π], cos = [1, 0, -1, 0, 1].

2. Step 2: Calculate cosine values for bit 1

   For bit 1, phase = π, angles = 2πt + π, cos(θ + π) = -cos(θ) = [-1, 0, 1, 0, -1].

3. Final Answer:

   [[ 1. 0. -1. 0. 1.] [-1. 0. 1. 0. -1.]] -> Option A

4. Quick Check:

   PSK phase shift: cos(θ + π) = -cos(θ) [OK]

Hint: PSK phase π: cos(θ + π) = -cos(θ) [OK]

Common Mistakes:

• Confusing phase shift effect on cosine
• Misinterpreting array shape or values
• Ignoring phase addition in cosine

4. The following code attempts to generate an ASK signal but produces an error. What is the error and how to fix it?

import numpy as np
bit = 0
t = np.linspace(0, 1, 100)
f = 5
signal = np.sin(2 * np.pi * f * t) if bit == 1 else 0
print(signal)

medium

A. Error: signal is assigned a scalar 0 instead of an array; fix by replacing 0 with np.zeros_like(t)

B. Error: np.sin requires integer input; fix by casting t to int

C. Error: Missing parentheses in np.sin; fix by adding parentheses

D. No error; code runs correctly

Solution

1. Step 1: Identify the error in conditional assignment

   If bit == 1, signal is an array from np.sin, else signal is scalar 0, causing inconsistent types.

2. Step 2: Fix by using array of zeros

   Replace 0 with np.zeros_like(t) to keep signal as an array in both cases.

3. Final Answer:

   Error: signal is assigned a scalar 0 instead of an array; fix by replacing 0 with np.zeros_like(t) -> Option A

4. Quick Check:

   Consistent array types needed = Error: signal is assigned a scalar 0 instead of an array; fix by replacing 0 with np.zeros_like(t) [OK]

Hint: Use np.zeros_like(t) for zero arrays, not scalar 0 [OK]

Common Mistakes:

• Using scalar 0 causing type mismatch
• Casting t to int which is unnecessary
• Ignoring conditional expression structure

5. You want to design a digital communication system that must be robust to amplitude noise but can tolerate frequency variations. Which modulation technique should you choose and why?

hard

A. ASK, because it changes amplitude which is sensitive to noise

B. ASK, because it changes phase which is robust to amplitude noise

C. FSK, because it changes frequency which is less affected by amplitude noise

D. PSK, because it changes phase which is sensitive to frequency noise

Solution

1. Step 1: Analyze noise tolerance requirements

   The system must be robust to amplitude noise, so modulation should not rely on amplitude changes.

2. Step 2: Match modulation to noise tolerance

   FSK changes frequency, which is less affected by amplitude noise, making it suitable. ASK changes amplitude and is sensitive to amplitude noise. PSK changes phase and can be sensitive to frequency noise.

3. Final Answer:

   FSK, because it changes frequency which is less affected by amplitude noise -> Option C

4. Quick Check:

   Amplitude noise robustness = FSK [OK]

Hint: Choose FSK to avoid amplitude noise sensitivity [OK]

Common Mistakes:

• Choosing ASK despite amplitude noise sensitivity
• Confusing phase noise sensitivity with frequency noise
• Misunderstanding modulation noise robustness