Modulation concept (AM, FM, PM) in Signal Processing - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to define the amplitude modulation (AM) signal.

Signal Processing

am_signal = carrier * (1 + [1] * message)

Drag options to blanks, or click blank then click option'

Afrequency

Bphase

Cmodulation_index

Damplitude

Attempts:

3 left

💡 Hint

Common Mistakes

Using frequency or phase instead of modulation index.

Confusing amplitude with modulation index.

The modulation index controls the strength of the amplitude modulation in AM signals.

2fill in blank

medium

Complete the code to generate a frequency modulated (FM) signal.

Signal Processing

fm_signal = carrier_amplitude * np.cos(2 * np.pi * carrier_frequency * t + [1] * np.sin(2 * np.pi * message_frequency * t))

Drag options to blanks, or click blank then click option'

Amodulation_index

Bcarrier_frequency

Cmessage_amplitude

Dphase_shift

Attempts:

3 left

💡 Hint

Common Mistakes

Using carrier frequency instead of modulation index.

Confusing phase shift with modulation index.

The modulation index in FM controls the frequency deviation caused by the message signal.

3fill in blank

hard

Fix the error in the phase modulation (PM) signal generation code.

Signal Processing

pm_signal = carrier_amplitude * np.cos(2 * np.pi * carrier_frequency * t + [1] * message)

Drag options to blanks, or click blank then click option'

Aamplitude_modulation_index

Bfrequency_deviation

Cmodulation_index

Dphase_deviation

Attempts:

3 left

💡 Hint

Common Mistakes

Using frequency deviation instead of phase deviation.

Using amplitude modulation index in PM.

Phase deviation is the correct parameter to scale the message signal in phase modulation.

4fill in blank

hard

Fill both blanks to create a dictionary with modulation types and their key parameters.

Signal Processing

modulation_params = {'AM': '[1]', 'FM': '[2]'}

Drag options to blanks, or click blank then click option'

Amodulation_index

Bphase_deviation

Cfrequency_deviation

Damplitude

Attempts:

3 left

💡 Hint

Common Mistakes

Mixing phase deviation with FM.

Using amplitude instead of modulation index for AM.

AM uses modulation index for amplitude changes, FM uses frequency deviation for frequency changes.

5fill in blank

hard

Fill all three blanks to create a dictionary comprehension filtering modulation types with index > 0.5.

Signal Processing

filtered_modulations = {k: v for k, v in modulation_indices.items() if v [1] [2] [3]

Drag options to blanks, or click blank then click option'

A>

B0.5

Cand

D<

Attempts:

3 left

💡 Hint

Common Mistakes

Using '<' instead of '>'.

Omitting the logical operator 'and'.

The code filters modulation indices greater than 0.5 using the 'and' logical operator.

Practice

1. Which type of modulation changes the amplitude of the carrier wave to transmit information?

easy

A. Frequency Modulation (FM)

B. Amplitude Modulation (AM)

C. Phase Modulation (PM)

D. Pulse Modulation

Solution

1. Step 1: Understand modulation types

   Amplitude Modulation (AM) changes the amplitude of the carrier wave to carry information.

2. Step 2: Compare with other modulation types

   Frequency Modulation changes frequency, Phase Modulation changes phase, so they are not correct here.

3. Final Answer:

   Amplitude Modulation (AM) -> Option B

4. Quick Check:

   Amplitude change = AM [OK]

Hint: Amplitude change means AM modulation [OK]

Common Mistakes:

• Confusing amplitude change with frequency or phase
• Choosing FM or PM instead of AM
• Thinking pulse modulation is amplitude modulation

2. Which of the following is the correct way to represent Frequency Modulation (FM) mathematically for a carrier wave c(t) = A_c cos(2πf_c t) with message signal m(t)?

easy

A. s(t) = A_c cos(2πf_c t + k_f m(t))

B. s(t) = A_c (1 + k_a m(t)) cos(2πf_c t)

C. s(t) = A_c cos(2π(f_c + k_f m(t)) t)

D. s(t) = A_c cos(2πf_c t) + m(t)

Solution

1. Step 1: Recall FM formula

   Frequency Modulation changes the frequency of the carrier. The instantaneous frequency is f_c + k_f m(t).

2. Step 2: Match formula to options

   s(t) = A_c cos(2π(f_c + k_f m(t)) t) shows s(t) = A_c cos(2π(f_c + k_f m(t)) t), which correctly represents FM.

3. Final Answer:

   s(t) = A_c cos(2π(f_c + k_f m(t)) t) -> Option C

4. Quick Check:

   Frequency inside cosine = FM [OK]

Hint: FM changes frequency inside cosine argument [OK]

Common Mistakes:

• Choosing amplitude modulation formula instead
• Ignoring the time variable in frequency term
• Adding message signal outside cosine

3. Given the Python code below simulating a simple AM signal, what will be the output array values?

import numpy as np

A_c = 1
k_a = 0.5
f_c = 5
fs = 50
t = np.arange(0, 1, 1/fs)
m_t = np.sin(2 * np.pi * 1 * t)
s_t = A_c * (1 + k_a * m_t) * np.cos(2 * np.pi * f_c * t)
print(np.round(s_t[:5], 3))

medium

A. [1.0, 0.309, -0.809, -1.191, -0.691]

B. [1.0, 0.809, -0.309, -1.191, -0.691]

C. [1.0, 0.309, 0.809, 1.191, 0.691]

D. [0.0, 0.309, -0.809, -1.191, -0.691]

Solution

1. Step 1: Calculate message and carrier values

   Message m_t is a 1 Hz sine wave sampled at 50 Hz. Carrier frequency is 5 Hz. Calculate first 5 samples of s_t using AM formula.

2. Step 2: Compute first 5 samples

   Using formula: s_t = 1 * (1 + 0.5 * m_t) * cos(2π * 5 * t), the first 5 rounded values are approximately [1.0, 0.309, -0.809, -1.191, -0.691].

3. Final Answer:

   [1.0, 0.309, -0.809, -1.191, -0.691] -> Option A

4. Quick Check:

   AM signal first samples = [1.0, 0.309, -0.809, -1.191, -0.691] [OK]

Hint: AM output = carrier * (1 + k_a * message) [OK]

Common Mistakes:

• Mixing up frequency and amplitude terms
• Not rounding values correctly
• Using wrong sampling frequency

4. The following code attempts to generate a Phase Modulated (PM) signal but has an error. What is the error?

import numpy as np

A_c = 1
k_p = 0.5
f_c = 10
fs = 100

t = np.arange(0, 1, 1/fs)
m_t = np.sin(2 * np.pi * 2 * t)
s_t = A_c * np.cos(2 * np.pi * f_c * t + k_p * np.sin(m_t))
print(np.round(s_t[:5], 3))

medium

A. The time vector t is incorrectly defined

B. The carrier frequency f_c is too high for sampling frequency fs

C. The amplitude A_c should be multiplied inside the cosine argument

D. The modulation term should be k_p * m_t, not k_p * np.sin(m_t)

Solution

1. Step 1: Analyze modulation term in PM formula

   Phase modulation uses k_p * m_t inside the cosine argument, not k_p * np.sin(m_t).

2. Step 2: Identify the error in code

   The code incorrectly applies np.sin(m_t) instead of directly using m_t as the phase modulation signal.

3. Final Answer:

   The modulation term should be k_p * m_t, not k_p * np.sin(m_t) -> Option D

4. Quick Check:

   PM uses message signal directly in phase [OK]

Hint: PM uses message signal directly inside cosine phase [OK]

Common Mistakes:

• Applying sine twice on message signal
• Confusing amplitude with phase modulation
• Incorrect time vector or sampling frequency

5. You want to transmit a message signal m(t) that has sudden jumps (discontinuities). Which modulation technique among AM, FM, and PM is best to minimize distortion caused by these jumps, and why?

hard

A. FM, because frequency changes are less affected by amplitude noise

B. PM, because phase changes handle discontinuities smoothly

C. AM, because amplitude changes are less sensitive to sudden jumps

D. None, all modulation types distort sudden jumps equally

Solution

1. Step 1: Understand effect of sudden jumps on modulation

   Sudden jumps cause rapid changes in signal. AM changes amplitude, which is sensitive to noise and distortion.

2. Step 2: Compare FM and PM for handling jumps

   FM changes frequency, which is more robust to amplitude noise and sudden jumps than AM. PM also changes phase but can be more sensitive to rapid phase shifts.

3. Final Answer:

   FM, because frequency changes are less affected by amplitude noise -> Option A

4. Quick Check:

   FM resists distortion from sudden jumps [OK]

Hint: FM resists noise and sudden jumps better than AM or PM [OK]

Common Mistakes:

• Choosing AM thinking amplitude is always best
• Assuming PM handles jumps better than FM
• Ignoring noise sensitivity differences