Signal energy and power in Signal Processing - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the energy of a signal?

Energy of a signal is the total sum of the squared values of the signal over all time. It tells us how much 'work' the signal can do, like how much strength it has in total.

beginner

How do you calculate the power of a signal?

Power is the average energy per unit time. You find it by taking the energy over a time period and dividing by that time length. It shows how strong the signal is on average.

intermediate

What is the difference between an energy signal and a power signal?

An energy signal has finite energy but zero average power. A power signal has finite power but infinite energy. Energy signals are like short bursts, power signals last forever with steady strength.

beginner

Write the formula for energy of a continuous-time signal x(t).

Energy E = ∫ from -∞ to ∞ of |x(t)|² dt. This means you square the signal values, then add them up over all time.

intermediate

Write the formula for power of a continuous-time signal x(t).

Power P = lim (T→∞) (1/2T) ∫ from -T to T of |x(t)|² dt. This means average the squared signal over a very long time.

Which of the following best describes an energy signal?

A Signal with zero energy and zero power

B Signal with infinite energy and finite power

C Signal with infinite energy and infinite power

D Signal with finite energy and zero power

Energy signals have finite total energy but their average power is zero.

How is power of a signal calculated?

A Sum of signal values over time

B Maximum value of the signal

C Average of squared signal values over time

D Integral of signal values

Power is the average of the squared signal values over a time interval.

What does the energy of a signal represent?

A The total strength or work the signal can do

B The maximum amplitude of the signal

C The frequency of the signal

D The duration of the signal

Energy measures the total strength or work the signal can perform.

Which formula represents the energy of a continuous-time signal x(t)?

A E = ∫ |x(t)| dt

B E = ∫ |x(t)|² dt

C E = lim (T→∞) (1/2T) ∫ |x(t)|² dt

D E = max |x(t)|

Energy is the integral of the squared magnitude of the signal over all time.

What type of signal has finite power but infinite energy?

A Power signal

B Energy signal

C Zero signal

D Impulse signal

Power signals last forever and have finite average power but infinite total energy.

Explain in your own words the difference between signal energy and signal power.

Think about a short burst of sound versus a steady hum.

Describe how you would calculate the energy and power of a continuous-time signal using integrals.

Remember energy sums all squared values, power averages them.

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