Magnitude and phase spectrum in Signal Processing

Introduction

Magnitude and phase spectrum help us understand the strength and timing of different frequencies in a signal.

To analyze sound signals and find dominant tones.

To study vibrations in machines for fault detection.

To process images by looking at frequency components.

To filter signals by removing unwanted frequencies.

To understand communication signals and their properties.

Syntax

Signal Processing

magnitude = np.abs(fft_result)
phase = np.angle(fft_result)

fft_result is the output of the Fourier Transform on your signal.

np.abs() gives the magnitude (strength) of each frequency.

np.angle() gives the phase (timing) of each frequency.

Examples

Calculate magnitude and phase for a simple 4-point signal.

Signal Processing

import numpy as np
signal = np.array([1, 0, -1, 0])
fft_result = np.fft.fft(signal)
magnitude = np.abs(fft_result)
phase = np.angle(fft_result)

Find magnitude and phase of a 5 Hz sine wave sampled 100 times.

Signal Processing

import numpy as np
signal = np.sin(2 * np.pi * 5 * np.linspace(0, 1, 100, endpoint=False))
fft_result = np.fft.fft(signal)
magnitude = np.abs(fft_result)
phase = np.angle(fft_result)

Sample Program

This program creates a signal with two frequencies (10 Hz and 20 Hz), computes its FFT, and prints the magnitude and phase for the first 10 frequency components.

Signal Processing

import numpy as np
import matplotlib.pyplot as plt

# Create a signal with two frequencies
fs = 100  # Sampling frequency
t = np.linspace(0, 1, fs, endpoint=False)
signal = 3 * np.sin(2 * np.pi * 10 * t) + 2 * np.cos(2 * np.pi * 20 * t)

# Compute FFT
fft_result = np.fft.fft(signal)

# Calculate magnitude and phase
magnitude = np.abs(fft_result)
phase = np.angle(fft_result)

# Frequency axis
freq = np.fft.fftfreq(len(signal), 1/fs)

# Print first 10 values of magnitude and phase
for i in range(10):
    print(f"Freq: {freq[i]:.1f} Hz, Magnitude: {magnitude[i]:.2f}, Phase: {phase[i]:.2f} radians")

Important Notes

The magnitude shows how strong each frequency is in the signal.

The phase tells where each frequency wave starts in time.

Magnitude and phase together fully describe the frequency content of a signal.

Summary

Magnitude spectrum shows the strength of frequencies.

Phase spectrum shows the timing of frequencies.

Both are found by applying Fourier Transform and then using np.abs() and np.angle().

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