Aliasing and Nyquist-Shannon sampling theorem

 Signal processing is an integral part of various engineering disciplines, including mechanical, electronic, electrical, mechatronics, chemical, and even civil engineering. Taking my own field, mechanical engineering, as an example, we find ourselves immersed in a world driven by automation. Our focus now extends to even more advanced levels of automation, incorporating technologies like machine learning and self-driving vehicles. Achieving these goals demands a diverse range of technologies. However, it's crucial that future advancements stem not just from the allure of cutting-edge technology, but from a solid foundation of established and reliable innovations.

Currently, much research centers around enhancing the sensory perception of robots, involving new sensors, LiDAR, and cameras, as well as fortifying the computational "brain" with advanced semiconductors or GPUs. Amidst these advancements, it's essential to acknowledge the existing technologies such as sensors tailored for processing electrical signals. This is where the significance of signal processing emerges as a pivotal discipline in modern engineering. The essence of signal processing lies in the need to quantify elements for the purpose of validation, inspection, or prediction. Consequently, signal processing assumes a paramount role in these processes.

Based on my experience, contemporary mechanical engineers employ sensors for diverse purposes within the domains of control, robotics, and design. While the utilization of sensors in robotics and control scenarios is intuitive, the significance of incorporating electrical signals in the design and manufacturing processes might be less commonly recognized.

Let's take the example of designing a chamber. Throughout this process, we formulate the design and refine it through Finite Element Analysis (FEA), leveraging Computer Aided Design (CAD) software tools. Furthermore, to achieve the desired quality, the design must undergo validation procedures, including Computer Aided Machining (CAM). This ensures that the design translates accurately into the manufacturing process. Following the manufacturing stage, it becomes imperative to conduct mode analysis, particularly for rotating equipment or structures that undergo frequent internal changes due to factors like pressure or temperature fluctuations—examples include chambers or pipelines.

To validate these modes effectively, we employ sensors to facilitate modal analysis. This integration of sensors into the analysis process empowers us to comprehensively understand the structural behavior and ensure its intended performance.

You can explore the in-depth information on Signal Processing in my previous blog post available below.

Signal Processing for Mechanical Engineers

Let's talk about today's topic, aliasing. At its core, aliasing is a phenomenon that occurs when a higher-frequency signal appears to be a lower-frequency signal. In other words, it's a distortion that happens when a signal is sampled at too low a rate to accurately represent its true frequency content. This can lead to incorrect interpretations of the signal's characteristics and potentially introduce artifacts that weren't present in the original signal.

Original Vibration Signal (top) and Undersampled Vibration Signal (bottom). We can see significant signal loss due to aliasing.

The Nyquist-Shannon sampling theorem is a fundamental concept in signal processing that provides insights into avoiding aliasing. According to this theorem, in order to accurately reconstruct a continuous signal from its samples, the sampling rate must be at least twice the highest frequency present in the signal. This principle is crucial in preventing the loss of information due to aliasing. Aliasing can be caused by a variety of factors. One common cause is undersampling, where the signal is not sampled frequently enough to capture its higher-frequency components. Another cause is improper filtering, which can allow high-frequency components to fold back into the lower-frequency range. Understanding these causes is essential in developing strategies to mitigate aliasing effects.

Let's consider a scenario where there are three distinct vibration patterns occurring at 50Hz, 250Hz, and 400Hz. As depicted in the figure above, when observing these vibrations over time, it's challenging to pinpoint the specific mode occurring at a given frequency. To overcome this limitation, I utilize the Fast Fourier Transform (FFT) to analyze the frequency domain, as illustrated in the lower part of the figure. Here, you can clearly observe peaks corresponding to our specified frequencies. I'll provide a detailed explanation of the Fourier Transform and its practical applications in the upcoming chance.

Now, let's delve into the impact of aliasing on the distortion of the vibration signal. The original signal is sampled at a rate of 1,000Hz. Here, we're examining a signal with a frequency of 300Hz, which is significantly less than half of the original sampling rate (500Hz).

The provided code below demonstrates how we can leverage the Fast Fourier Transform (FFT) for signal processing in the frequency domain. You can access the Python Notebook containing the code via this link.

Code

In the figure, the top portion represents the original signal in the frequency domain, as observed previously. However, in the bottom portion, we can clearly observe both magnitude and frequency distortion. This illustrates how aliasing disrupts the accurate detection of the signal and underscores the critical importance of setting an appropriate sampling rate. The below is code for them.

import numpy as np

import matplotlib.pyplot as plt

# Parameters for the vibration signal

sampling_rate = 1000 # Sampling rate in Hz

duration = 1.0 # Duration of the signal in seconds

frequencies = [50, 250, 400] # Frequencies of the vibration components in Hz

# Generate the vibration signal

t = np.linspace(0, duration, int(sampling_rate * duration), endpoint=False)

vibration_signal = np.sum([np.sin(2 * np.pi * freq * t) for freq in frequencies], axis=0)

# Compute the FFT of the vibration signal

fft_freqs = np.fft.fftfreq(len(vibration_signal), d=1/sampling_rate)

fft_values = np.fft.fft(vibration_signal)

fft_magnitude = np.abs(fft_values)

# Simulate undersampling to demonstrate aliasing

undersampled_rate = 300 # Undersampled rate in Hz

undersampled_indices = np.arange(0, len(t), int(sampling_rate / undersampled_rate))

undersampled_signal = vibration_signal[undersampled_indices]

# Compute the FFT of the undersampled signal

fft_freqs_undersampled = np.fft.fftfreq(len(undersampled_signal), d=1/sampling_rate)

fft_values_undersampled = np.fft.fft(undersampled_signal)

fft_magnitude_undersampled = np.abs(fft_values_undersampled)

# Plot the original and undersampled FFT plots

plt.figure(figsize=(10, 10))

# Original FFT plot

plt.subplot(2, 1, 1)

plt.plot(fft_freqs, fft_magnitude)

plt.title("Frequency Domain Representation (Original Signal)")

plt.xlabel("Frequency (Hz)")

plt.ylabel("Magnitude")

plt.xlim(0, sampling_rate / 2) # Display only positive frequencies

# Undersampled FFT plot

plt.subplot(2, 1, 2)

plt.plot(fft_freqs_undersampled, fft_magnitude_undersampled, 'r')

plt.title("Frequency Domain Representation (Undersampled Signal)")

plt.xlabel("Frequency (Hz)")

plt.ylabel("Magnitude")

plt.xlim(0, sampling_rate / 2) # Display only positive frequencies

plt.tight_layout()

plt.show()

In our exploration today, we delved into the concept of aliasing and learned strategies to mitigate its effects. I sincerely appreciate your time and attention in reading this blog.