Magnitude and phase spectrum in Signal Processing - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the magnitude spectrum of a signal?

The magnitude spectrum shows how strong each frequency component is in a signal. It tells us the size or amplitude of each frequency.

beginner

What does the phase spectrum represent in a signal?

The phase spectrum shows the shift or delay of each frequency component in the signal. It tells us how each frequency is shifted in time.

intermediate

How do you calculate the magnitude spectrum from a complex Fourier transform value?

You calculate the magnitude by taking the square root of the sum of squares of the real and imaginary parts: magnitude = sqrt(real² + imag²).

intermediate

How is the phase spectrum calculated from a complex Fourier transform value?

The phase is calculated using the arctangent of the imaginary part divided by the real part: phase = arctan2(imag, real).

beginner

Why are both magnitude and phase spectra important in signal analysis?

Magnitude shows how much of each frequency is present, while phase shows how those frequencies align in time. Together, they fully describe the signal's frequency content.

What does the magnitude spectrum tell us about a signal?

A The total energy of the signal

B The time delay of the signal

C The noise level in the signal

D The strength of each frequency component

The magnitude spectrum shows how strong or large each frequency component is in the signal.

Which mathematical function is used to find the phase from the real and imaginary parts?

A Square root

B Arctangent

C Logarithm

D Exponential

The phase is calculated using the arctangent of the imaginary part divided by the real part.

If the Fourier transform of a signal at a frequency is 3 + 4i, what is the magnitude?

A 12

B 7

C 5

D 1

Magnitude = sqrt(3² + 4²) = sqrt(9 + 16) = sqrt(25) = 5.

What does the phase spectrum help us understand about a signal?

A Time alignment of frequencies

B Frequency strength

C Signal amplitude

D Signal noise

Phase spectrum shows how frequency components are shifted or aligned in time.

Which two spectra together fully describe a signal's frequency content?

A Magnitude and phase

B Power and noise

C Amplitude and energy

D Frequency and time

Magnitude and phase spectra together describe both strength and timing of frequency components.

Explain in your own words what the magnitude and phase spectra represent in signal processing.

Think about how a signal can be broken into parts that tell us how loud and how shifted each frequency is.

Describe how you would calculate the magnitude and phase from a complex number representing a frequency component.

Remember the formulas for magnitude and angle of a complex number.

Practice

1. What does the magnitude spectrum represent in signal processing?

easy

A. The strength or amplitude of different frequency components

B. The timing or phase shift of frequency components

C. The original time-domain signal values

D. The noise level in the signal

Solution

1. Step 1: Understand magnitude spectrum meaning

   The magnitude spectrum shows how strong or intense each frequency is in a signal.

2. Step 2: Differentiate from phase spectrum

   The phase spectrum shows timing, not strength, so it is not the magnitude.

3. Final Answer:

   The strength or amplitude of different frequency components -> Option A

4. Quick Check:

   Magnitude spectrum = strength [OK]

Hint: Magnitude = strength of frequencies, phase = timing [OK]

Common Mistakes:

• Confusing magnitude with phase
• Thinking magnitude shows time values
• Mixing magnitude with noise level

2. Which Python function is used to get the phase spectrum from a Fourier Transform result stored in variable F?

easy

A. np.abs(F)

B. np.angle(F)

C. np.fft(F)

D. np.real(F)

Solution

1. Step 1: Recall phase extraction function

   In Python's NumPy, np.angle() returns the phase angles of complex numbers.

2. Step 2: Differentiate from magnitude and other functions

   np.abs() returns magnitude, np.fft() is not a valid function, and np.real() returns real parts only.

3. Final Answer:

   np.angle(F) -> Option B

4. Quick Check:

   Phase spectrum = np.angle() [OK]

Hint: Use np.angle() for phase, np.abs() for magnitude [OK]

Common Mistakes:

• Using np.abs() instead of np.angle() for phase
• Trying to use np.fft() which is invalid
• Confusing real part with phase

3. Given the code:

import numpy as np
x = np.array([1, 0, -1, 0])
F = np.fft.fft(x)
magnitude = np.abs(F)
phase = np.angle(F)
print(np.round(magnitude, 2))
print(np.round(phase, 2))

What is the printed magnitude and phase arrays?

medium

A. [4.00 0.00 0.00 0.00]
[0.00 0.00 0.00 0.00]

B. [0.00 2.00 0.00 2.00]
[0.00 0.00 0.00 0.00]

C. [2.00 2.00 2.00 2.00]
[ 0.00 -1.57 3.14 1.57]

D. [2.00 0.00 2.00 0.00]
[0.00 1.57 3.14 -1.57]

Solution

1. Step 1: Compute FFT of x

   The FFT of [1, 0, -1, 0] is [0+0j, 2+0j, 0+0j, 2+0j].

2. Step 2: Calculate magnitude and phase arrays

   Magnitude is [2.00 2.00 2.00 2.00]. Phase angles are [0.00 -1.57 3.14 1.57].

3. Final Answer:

   [2.00 2.00 2.00 2.00] [ 0.00 -1.57 3.14 1.57] -> Option C

4. Quick Check:

   Magnitude = [2 2 2 2], Phase = approx [-pi/2, pi, pi/2] [OK]

Hint: FFT magnitude is abs(), phase is angle() in radians [OK]

Common Mistakes:

• Confusing magnitude with real part
• Not rounding phase correctly
• Mixing up signs of phase angles

4. The following code attempts to compute magnitude and phase but has an error:

import numpy as np
x = np.array([1, 2, 3, 4])
F = np.fft.fft(x)
magnitude = np.angle(F)
phase = np.abs(F)
print(magnitude)
print(phase)

What is the error?

medium

A. No error; code runs correctly

B. np.fft.fft() is used incorrectly; should be np.fft.ifft()

C. The input array x must be complex numbers

D. Magnitude and phase are swapped; np.angle() should be for phase, np.abs() for magnitude

Solution

1. Step 1: Check function usage for magnitude and phase

   Magnitude is the absolute value, so np.abs() should be used for magnitude, not np.angle(). Phase is angle, so np.angle() is for phase.

2. Step 2: Identify swapped assignments

   The code assigns magnitude = np.angle(F) and phase = np.abs(F), which is reversed.

3. Final Answer:

   Magnitude and phase are swapped; np.angle() should be for phase, np.abs() for magnitude -> Option D

4. Quick Check:

   Magnitude = abs(), Phase = angle() [OK]

Hint: Remember: abs() for magnitude, angle() for phase [OK]

Common Mistakes:

• Swapping magnitude and phase functions
• Thinking FFT input must be complex
• Confusing FFT and IFFT usage

5. You have a noisy signal and want to keep only frequencies with magnitude above 5. Which steps correctly apply this filter using FFT results stored in F?

hard

A. Calculate magnitude = np.abs(F), create mask = magnitude > 5, then set F_filtered = F * mask

B. Calculate phase = np.angle(F), create mask = phase > 5, then set F_filtered = F * mask

C. Calculate magnitude = np.abs(F), create mask = magnitude < 5, then set F_filtered = F * mask

D. Calculate magnitude = np.abs(F), create mask = magnitude > 5, then set F_filtered = np.abs(F) * mask

Solution

1. Step 1: Compute magnitude and create mask

   Calculate magnitude with np.abs(F). Create a boolean mask where magnitude > 5 to keep strong frequencies.

2. Step 2: Apply mask to original FFT data

   Multiply the complex FFT array F by the mask to zero out weak frequencies, preserving phase and magnitude of strong ones.

3. Final Answer:

   Calculate magnitude = np.abs(F), create mask = magnitude > 5, then set F_filtered = F * mask -> Option A

4. Quick Check:

   Filter by magnitude > 5, multiply mask with F [OK]

Hint: Filter frequencies by magnitude mask, multiply mask with FFT [OK]

Common Mistakes:

• Filtering by phase instead of magnitude
• Using magnitude < 5 to keep frequencies
• Multiplying mask with magnitude only, losing phase info