Signal properties (amplitude, frequency, phase) in Signal Processing - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of a sine wave with given amplitude and frequency

What is the output array of the following code that generates a sine wave with amplitude 2 and frequency 1 Hz sampled at 4 points?

Signal Processing

import numpy as np

fs = 4  # sampling frequency
f = 1   # frequency of sine wave
A = 2   # amplitude

t = np.arange(4) / fs
signal = A * np.sin(2 * np.pi * f * t)
print(signal)

A [0.0, 1.414, 2.0, 1.414]

B [0.0, 2.0, 0.0, -2.0]

C [0.0, 1.0, 0.0, -1.0]

D [0.0, -2.0, 0.0, 2.0]

💡 Hint

Recall that sine of 0 is 0, sine of pi/2 is 1, sine of pi is 0, sine of 3pi/2 is -1.

The sine wave values at t=0, 0.25, 0.5, 0.75 seconds with frequency 1 Hz and amplitude 2 are 0, 2, 0, -2 respectively.

❓ data_output

intermediate

Phase shift effect on sine wave values

Given a sine wave with amplitude 1, frequency 1 Hz, sampled at 4 points, what is the output array if the phase is shifted by pi/2?

Signal Processing

import numpy as np

fs = 4
f = 1
A = 1
phase = np.pi / 2

t = np.arange(4) / fs
signal = A * np.sin(2 * np.pi * f * t + phase)
print(signal)

A [-1.0, 0.0, 1.0, 0.0]

B [0.0, 1.0, 0.0, -1.0]

C [1.0, 0.0, -1.0, 0.0]

D [0.0, -1.0, 0.0, 1.0]

💡 Hint

Phase shift by pi/2 turns sine into cosine.

Adding a phase of pi/2 shifts sine wave to cosine wave, so values at t=0,0.25,0.5,0.75 are cos(0)=1, cos(pi/2)=0, cos(pi)=-1, cos(3pi/2)=0.

❓ visualization

advanced

Visualizing amplitude change on sine wave

Which plot correctly shows the effect of increasing amplitude from 1 to 3 on a sine wave with frequency 1 Hz?

Signal Processing

import numpy as np
import matplotlib.pyplot as plt

fs = 100
f = 1

t = np.linspace(0, 1, fs, endpoint=False)
signal1 = 1 * np.sin(2 * np.pi * f * t)
signal2 = 3 * np.sin(2 * np.pi * f * t)

plt.plot(t, signal1, label='Amplitude 1')
plt.plot(t, signal2, label='Amplitude 3')
plt.legend()
plt.show()

A Plot with two sine waves, second wave shifted in phase but same amplitude

B Plot with two sine waves, both with same peak heights

C Plot with two sine waves, second wave peaks one third of first

D Plot with two sine waves, second wave peaks three times higher than first

💡 Hint

Amplitude controls the height of the wave peaks.

Increasing amplitude scales the wave vertically, so the second wave peaks are three times higher than the first.

🧠 Conceptual

advanced

Effect of frequency on sine wave period

If the frequency of a sine wave doubles, what happens to its period?

A The period halves

B The period doubles

C The period stays the same

D The period becomes zero

💡 Hint

Period is the inverse of frequency.

Period is 1 divided by frequency, so doubling frequency halves the period.

🔧 Debug

expert

Identify error in phase shift calculation

What error does the following code produce when trying to apply a phase shift of 90 degrees to a sine wave? import numpy as np fs = 10 f = 1 A = 1 phase = 90 t = np.arange(fs) / fs signal = A * np.sin(2 * np.pi * f * t + phase) print(signal)

Signal Processing

import numpy as np
fs = 10
f = 1
A = 1
phase = 90

t = np.arange(fs) / fs
signal = A * np.sin(2 * np.pi * f * t + phase)
print(signal)

A The output is incorrect because phase should be in radians, not degrees

B SyntaxError due to missing colon

C TypeError because phase is an integer

D No error, output is correct

💡 Hint

Check units of phase in trigonometric functions.

Numpy sine function expects phase in radians; 90 degrees must be converted to pi/2 radians.

Practice

1. Which property of a signal describes how tall or strong the wave is?

easy

A. Frequency

B. Amplitude

C. Phase

D. Wavelength

Solution

1. Step 1: Understand amplitude meaning

   Amplitude measures the height or strength of a wave, showing how strong the signal is.

2. Step 2: Compare with other properties

   Frequency is how often the wave repeats, phase shifts the start, wavelength is distance, not height.

3. Final Answer:

   Amplitude -> Option B

4. Quick Check:

   Amplitude = wave height [OK]

Hint: Amplitude means wave height or strength [OK]

Common Mistakes:

• Confusing amplitude with frequency
• Thinking phase affects height
• Mixing wavelength with amplitude

2. Which of the following is the correct way to represent a sine wave with amplitude 3 and frequency 5 Hz in Python using numpy?
t = np.linspace(0, 1, 500)
y = ?

easy

A. 3 + np.sin(2 * np.pi * 5 * t)

B. np.sin(3 * 2 * np.pi * 5 * t)

C. 3 * np.sin(2 * np.pi * 5 * t)

D. 5 * np.sin(2 * np.pi * 3 * t)

Solution

1. Step 1: Recall sine wave formula

   The sine wave formula is amplitude * sin(2π * frequency * time).

2. Step 2: Match formula to options

   3 * np.sin(2 * np.pi * 5 * t) matches amplitude 3 and frequency 5 correctly as 3 * sin(2π * 5 * t).

3. Final Answer:

   3 * np.sin(2 * np.pi * 5 * t) -> Option C

4. Quick Check:

   Amplitude * sin(2π * frequency * t) [OK]

Hint: Amplitude multiplies sine, frequency inside sine argument [OK]

Common Mistakes:

• Multiplying frequency by amplitude inside sine
• Adding amplitude instead of multiplying
• Swapping amplitude and frequency

3. Given the code:
import numpy as np
t = np.linspace(0, 1, 5)
y = 2 * np.sin(2 * np.pi * 1 * t + np.pi/2)
print(np.round(y, 2))
What is the output?

medium

A. [ 2. 0. -2. 0. 2. ]

B. [ 2. 1.41 0. -1.41 -2. ]

C. [ 0. 1.41 2. 1.41 0. ]

D. [ 1.41 2. 1.41 0. -1.41]

Solution

1. Step 1: Calculate time points

   t = [0, 0.25, 0.5, 0.75, 1]

2. Step 2: Compute y values

   y = 2 * sin(2π * 1 * t + π/2) = 2 * cos(2π * t) due to phase shift π/2.
   Values: 2*cos(0)=2, 2*cos(π/2)=0, 2*cos(π)=-2, 2*cos(3π/2)=0, 2*cos(2π)=2
   np.round(y, 2) prints [ 2. 0. -2. 0. 2. ]

3. Final Answer:

   [ 2. 0. -2. 0. 2. ] -> Option A

4. Quick Check:

   Phase π/2 shifts sine to cosine [OK]

Hint: Phase π/2 shifts sine to cosine wave [OK]

Common Mistakes:

• Ignoring phase shift
• Rounding errors
• Mixing sine and cosine values

4. Identify the error in this code snippet that tries to create a sine wave with amplitude 4 and frequency 2 Hz:
import numpy as np
t = np.linspace(0, 1, 100)
y = 4 * np.sin(2 * np.pi * t) * 2

medium

A. The time vector t should be integers, not floats

B. Amplitude should be added, not multiplied

C. np.linspace should use 0 to 2 instead of 0 to 1

D. The frequency multiplier is in the wrong place; it should be inside the sine argument as 2 * np.pi * 2 * t

Solution

1. Step 1: Analyze sine argument order

   The sine argument is 2π * t (frequency 1 Hz), but *2 is outside sin(), incorrectly scaling the amplitude instead of frequency.

2. Step 2: Correct placement for frequency

   Frequency multiplier must be inside sine: 2 * np.pi * 2 * t.

3. Final Answer:

   The frequency multiplier is in the wrong place; it should be inside the sine argument as 2 * np.pi * 2 * t -> Option D

4. Quick Check:

   Frequency must multiply time inside sine [OK]

Hint: Write frequency right after 2π inside sine [OK]

Common Mistakes:

• Multiplying amplitude incorrectly
• Using wrong time vector type
• Changing linspace range unnecessarily

5. You have a signal defined as y = 5 * np.sin(2 * np.pi * 3 * t + π/4). If you want to double the frequency and halve the amplitude while keeping the phase shift the same, which expression is correct?

hard

A. y = 2.5 * np.sin(2 * np.pi * 6 * t + π/4)

B. y = 10 * np.sin(2 * np.pi * 1.5 * t + π/4)

C. y = 2.5 * np.sin(2 * np.pi * 3 * t + π/2)

D. y = 5 * np.sin(2 * np.pi * 6 * t)

Solution

1. Step 1: Adjust amplitude and frequency

   Original amplitude is 5, halved is 2.5. Original frequency is 3, doubled is 6.

2. Step 2: Keep phase shift same

   Phase shift π/4 remains unchanged in the new expression.

3. Final Answer:

   y = 2.5 * np.sin(2 * np.pi * 6 * t + π/4) -> Option A

4. Quick Check:

   Amplitude halved, frequency doubled, phase same [OK]

Hint: Multiply frequency by 2, amplitude by 0.5, keep phase [OK]

Common Mistakes:

• Changing phase shift incorrectly
• Multiplying amplitude instead of halving
• Swapping frequency and amplitude values