Oversampling Data (Explained with Audio) - Computerphile

In the previous discussion, we explored saturation, which generates higher-frequency harmonics from a low harmonic. While straightforward in the digital realm for lower frequencies, challenges arise when dealing with high-frequency signals, such as a 20,000 Hz signal, especially with a limited sample rate like 48 kHz. Nyquist theory dictates that the maximum frequency that can be accurately sampled is half the sample rate, meaning 24 kHz for a 48 kHz sample rate. This limitation poses a problem when attempting to create higher harmonics, as they would exceed the sampling capability. To illustrate, consider a 10,000 Hz sample rate, where the maximum producible frequency is 5,000 Hz. If a 6,000 Hz signal is introduced, it is 1,000 Hz above the limit. Instead of being accurately represented, it will produce a frequency at 4,000 Hz, effectively reflecting back from the upper limit. This phenomenon, known as aliasing, occurs when frequencies exceeding the Nyquist limit are sampled, causing them to fold back into the audible spectrum as lower, inharmonic frequencies. Such aliasing can result in undesirable, dissonant sounds, often described as "chirping" or "chattery." The sampling rate dictates how many times per second an analog-to-digital converter measures the voltage of a signal. According to Nyquist-Shannon theory, at least two sample points are required to accurately reconstruct a sine wave. However, if the sampling rate is insufficient to capture the full waveform, particularly with higher frequencies, significant data loss occurs, leading to an inaccurate representation of the original signal and a perceived change in pitch. For instance, if a sine wave is sampled only at two points that happen to align with the start and end of a cycle, the reconstructed wave would be a single large wave between those points, losing all intermediate data and resulting in a lower-pitched sound. Aliasing is not exclusive to audio; it can occur in various digital signal processing contexts. It is analogous to visual aliasing in images, where jagged edges appear due to insufficient resolution. To combat aliasing in audio, a technique called oversampling is employed. Oversampling works by artificially increasing the sample rate of the signal before processing, such as saturation, which generates new harmonics. This higher sample rate pushes any newly created high-frequency harmonics beyond the original Nyquist limit, preventing them from folding back into the audible range as aliased frequencies. After the processing, a low-pass filter is applied to remove these intentionally generated super-high frequencies, and the signal is then downsampled back to its original sample rate. One common method for oversampling is "zero-padding." This involves inserting zeros between each existing sample point in the digital signal, effectively doubling the number of samples and thus the sample rate. For example, an array of four numbers representing sample points can be expanded to eight numbers by inserting zeros between each original number and at the end. While this direct zero-padding initially creates a "crazier" looking wave and introduces mirrored frequencies in the upper spectrum, it's a crucial step. When the frequency spectrum of this zero-padded signal is analyzed, a mirrored version of the original spectrum appears at the higher frequency range. The magic of oversampling then comes into play. After zero-padding and any subsequent processing (like saturation), a low-pass filter is applied. This filter effectively removes the newly created mirrored frequencies and any other unwanted high-frequency content that would otherwise cause aliasing. The remaining signal, now at a higher sample rate but with the aliasing components removed, can then be downsampled back to the original sample rate. This process, often involving a moving average filter, smooths out the curve, transforming the inserted zeros into meaningful data points and resulting in a much cleaner, alias-free signal with double the original number of points. Demonstrations using audio plugins clearly show the effectiveness of oversampling. When a signal is saturated without oversampling, a messy pattern of inharmonic, mirrored frequencies appears, producing the undesirable "chirping" sound. However, when oversampling is activated, these aliased frequencies are significantly reduced or eliminated, resulting in a much cleaner and more accurate representation of the desired harmonics. The higher the quality setting for oversampling, the more effectively aliasing is suppressed. This technique is fundamental in achieving high-quality audio processing in digital environments, as well as in other disciplines where accurate signal representation is critical.