Signal energy and power in Signal Processing - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Calculate energy of a discrete signal

Given the discrete signal x = [2, -1, 3, 0], what is the energy of the signal computed as the sum of squares of its samples?

Signal Processing

import numpy as np
x = np.array([2, -1, 3, 0])
energy = np.sum(x**2)
print(energy)

A 10

B 14

C 6

D 9

💡 Hint

Energy is sum of squares of each sample value.

Energy = 2^2 + (-1)^2 + 3^2 + 0^2 = 4 + 1 + 9 + 0 = 14.

❓ data_output

intermediate

Calculate power of a discrete signal

For the discrete signal x = [1, 2, 3, 4], what is the average power computed as the mean of the squared samples?

Signal Processing

import numpy as np
x = np.array([1, 2, 3, 4])
power = np.mean(x**2)
print(power)

A 7.5

B 5.0

C 3.5

D 10.0

💡 Hint

Power is average of squared values.

Power = (1^2 + 2^2 + 3^2 + 4^2) / 4 = (1 + 4 + 9 + 16) / 4 = 30 / 4 = 7.5.

❓ visualization

advanced

Visualize energy distribution of a signal

Which plot correctly shows the energy contribution of each sample in the signal x = [1, -2, 2, -1] as bars representing squared values?

Signal Processing

import matplotlib.pyplot as plt
import numpy as np
x = np.array([1, -2, 2, -1])
energy_per_sample = x**2
plt.bar(range(len(x)), energy_per_sample)
plt.xlabel('Sample Index')
plt.ylabel('Energy Contribution')
plt.title('Energy per Sample')
plt.show()

A Bar heights: [1, 4, 4, 1]

B Bar heights: [1, -2, 2, -1]

C Bar heights: [1, 2, 2, 1]

D Bar heights: [1, 8, 8, 1]

💡 Hint

Energy per sample is square of sample value, so negative values become positive.

Energy per sample = [1^2, (-2)^2, 2^2, (-1)^2] = [1, 4, 4, 1].

🧠 Conceptual

advanced

Energy vs Power of signals

Which statement correctly describes the difference between energy and power signals?

A Energy signals have zero energy and zero power; power signals have infinite energy and power.

B Energy signals have infinite energy and finite average power; power signals have finite energy and zero average power.

C Energy signals and power signals both have finite energy and power.

D Energy signals have finite energy and zero average power; power signals have infinite energy and finite average power.

💡 Hint

Energy signals have total energy finite but power zero; power signals have finite power but infinite energy.

Energy signals: total energy finite, average power zero. Power signals: total energy infinite, average power finite.

🔧 Debug

expert

Identify error in power calculation code

What error will occur when running this code to calculate power of signal x = [1, 2, 3]? import numpy as np x = [1, 2, 3] power = np.mean(x**2) print(power)

Signal Processing

import numpy as np
x = [1, 2, 3]
power = np.mean(np.array(x)**2)
print(power)

A ValueError: operands could not be broadcast together

B NameError: name 'np' is not defined

C TypeError: unsupported operand type(s) for ** or pow(): 'list' and 'int'

D No error, prints 4.666666666666667

💡 Hint

Check the data type of x and if it supports element-wise power operation.

x is a list, and lists do not support the ** operator element-wise; numpy arrays do.

Practice

1. What does the energy of a signal represent in signal processing?

easy

A. The frequency content of the signal

B. The average strength of the signal at a single point

C. The total strength of the signal over all time

D. The duration of the signal

Solution

1. Step 1: Understand the definition of signal energy

   Energy is the sum of the squared signal values over time, representing total strength.

2. Step 2: Compare with other options

   Average strength relates to power, frequency content is different, and duration is time length.

3. Final Answer:

   The total strength of the signal over all time -> Option C

4. Quick Check:

   Energy = total signal strength [OK]

Hint: Energy sums signal strength over time, not average [OK]

Common Mistakes:

• Confusing energy with power
• Thinking energy measures frequency
• Assuming energy is signal duration

2. Which of the following formulas correctly calculates the energy \(E\) of a discrete signal \(x[n]\) with \(N\) samples?

easy

A. \(E = \frac{1}{N} \sum_{n=0}^{N-1} |x[n]|\)

B. \(E = \max(|x[n]|)\)

C. \(E = \sum_{n=0}^{N-1} x[n]\)

D. \(E = \sum_{n=0}^{N-1} |x[n]|^2\)

Solution

1. Step 1: Recall energy formula for discrete signals

   Energy is the sum of squared magnitudes of signal samples over all N points.

2. Step 2: Evaluate each option

   \(E = \sum_{n=0}^{N-1} |x[n]|^2\) matches the energy formula. \(E = \frac{1}{N} \sum_{n=0}^{N-1} |x[n]|\) is average magnitude, C is sum without squaring, D is max value.

3. Final Answer:

   \(E = \sum_{n=0}^{N-1} |x[n]|^2\) -> Option D

4. Quick Check:

   Energy = sum of squares [OK]

Hint: Energy sums squared values, not averages or max [OK]

Common Mistakes:

• Using average instead of sum
• Not squaring the signal values
• Taking max instead of sum

3. Consider the discrete signal \(x = [2, -1, 3, 0]\). What is the energy of this signal?

medium

A. 14

B. 6

C. 4

D. 10

Solution

1. Step 1: Square each element of the signal

   Squares: \(2^2=4\), \((-1)^2=1\), \(3^2=9\), \(0^2=0\).

2. Step 2: Sum the squared values

   Sum = 4 + 1 + 9 + 0 = 14.

3. Final Answer:

   14 -> Option A

4. Quick Check:

   Energy = 4+1+9+0 = 14 [OK]

Hint: Square each value, then add all [OK]

Common Mistakes:

• Forgetting to square negative values
• Adding values without squaring
• Miscounting elements

4. The following Python code attempts to calculate the power of a signal \(x = [1, 2, 3, 4]\). What is the error?

import numpy as np
x = [1, 2, 3, 4]
power = np.sum(x**2) / len(x)
print(power)

medium

A. Cannot use ** operator on list directly

B. Division by zero error

C. np.sum is not a function

D. len(x) returns wrong length

Solution

1. Step 1: Identify the data type of x

   x is a Python list, which does not support element-wise power with ** operator.

2. Step 2: Understand numpy operations on lists

   To use ** on elements, x must be a numpy array, not a list.

3. Step 3: Correct code would convert list to array first

   Use x = np.array(x) before power calculation.

4. Final Answer:

   Cannot use ** operator on list directly -> Option A

5. Quick Check:

   List ** 2 causes error [OK]

Hint: Convert list to numpy array before power ops [OK]

Common Mistakes:

• Trying to square list directly
• Ignoring data type errors
• Assuming len(x) causes error

5. A continuous-time signal \(x(t)\) has infinite duration but finite average power. Which statement is true about its energy and power?

hard

A. Energy is finite, power is infinite

B. Energy is infinite, power is finite

C. Both energy and power are infinite

D. Both energy and power are zero

Solution

1. Step 1: Recall definitions for infinite duration signals

   Signals with infinite duration and finite average power have infinite energy because energy sums over all time.

2. Step 2: Understand power for such signals

   Power is average energy per unit time, so it can be finite even if total energy is infinite.

3. Step 3: Match statements with definitions

   Energy is infinite, power is finite correctly states energy infinite and power finite.

4. Final Answer:

   Energy is infinite, power is finite -> Option B

5. Quick Check:

   Infinite duration + finite power = infinite energy [OK]

Hint: Infinite duration signals have infinite energy, finite power [OK]

Common Mistakes:

• Confusing energy and power for infinite signals
• Assuming finite energy for infinite duration
• Mixing definitions of energy and power