Fundamental Terms of Signal Processing
Welcome aboard! In this article, i’ll share some depictions and friendly describe of some important signal processing concepts. It may be challenging when try to understanding some primary terms in signal processing. So in this article, you’ll
- Learn what they are
- Understand why do we use them
- See and Aware their subtle differences each other with their depictions
Key terms:
Fourier Transform, Inverse Fourier Transform (IFT), Discrete Fourier Transform (DFT), Fast Fourier Transform (FFT), Spectral analysis, Spectral Bandwidth, Spectral Rolloff, MFCC
We will begin with Fourier transform and its popular subtypes, than we shall continue to other methods. The topics will consist 3 parts:
- What is it, Why we do use it, Summary.
First off:
1- Fourier Transform
What is it?
Fourier Transform is a technique used to analyze signals and data in the frequency domain. It transforms a signal from its original time or spatial domain into the frequency domain, representing it as a sum of sinusoidal components, each with its own amplitude and phase. Briefly we would say:
“Fourier transforms allows us to understand the frequency content of the signal”
Or more artistically, Consider it as our goal is extracting harmonious scream of signals.
If there is needed an analogy to liken the Fourier Transform to a tool in everyday life, it could be a prism. A prism separates white light into its constituent colors, the Fourier Transform separates a signal into its constituent frequencies. The prism allows us to see the different colors that make up white light, and similarly, the Fourier Transform allows us to see the different frequencies that make up a signal. Thus we can use it with change depending on the problem. The output of Fourier transform can be used in CNN’s or any model which capable of feature extraction.
Why we do use it?
- Signal Analysis: The Fourier Transform is extensively used to analyze various types of signals, such as audio, images, and time-series datas. It helps identify the underlying frequencies, harmonics, and periodicities present in the signal.
- Filtering: it can be used for filtering out unwanted frequencies from a signal. Imagine it like, the signal that we got, have already contains different waves like Figure 1. Therefore if we separate these waves from it, we can also filter it.
- Spectral Analysis: The Fourier Transform is employed to analyze the frequency content of a signal. Similar to filtering but instead we’re just analysing the waves this time.
Summary
It takes a sequence of numbers representing a time-domain signal and calculates the amplitudes and phases of the various sinusoidal components (frequencies) present in that signal. The result is a spectrum that shows the frequency content of the original signal.
2- Inverse Fourier Transform
What is it?
There is no magic, It’s in the name. The Inverse Fourier Transform is the mathematical operation that exactly reverses the process of the Fourier Transform. It takes a signal from the frequency domain back to its original time or spatial domain. In other words, if a signal is transformed into the frequency domain using the Fourier Transform, the Inverse Fourier Transform allows us to reconstruct the original signal from its frequency representation.
Unfortunately, there is no image that can exactly depict the inverse Fourier transform, yet you can imagine it like:
Time domain representation -> Fourier transform -> Frequency domain representation -> Inverse Fourier transform -> Time domain representation
Why we do use it?
It can be used where main Fourier Transform is used. And also:
- Image processing: Yes, the inverse Fourier Transform can also be used in image processing. Where it is used to convert images from the frequency domain to the spatial domain. This allows for image enhancement and restoration, as well as the removal of noise and other unwanted artifacts.
- Filtering: In signal processing, Fourier Transform is used to filter out unwanted frequencies from a signal. Assume that you have a noisy audio signal that you want to clean up. Solution would be taking the Fourier Transform of the noisy audio signal to analyze its frequency components. Identify the noise frequencies and attenuate or remove them in the frequency domain. Finally, apply the Inverse Fourier Transform to get back the cleaned audio signal.
- Solving Partial Differential Equations: When you need to solve a partial differential equation, IFT can be used. Interesting right? Convert the partial differential equation into the frequency domain using the Fourier Transform. This might help simplify the equation or make it easier to solve. Once you find the solution in the frequency domain, use the Inverse Fourier Transform to transform it back into the time domain.
Summary
It enables us to manipulate and process signals in the frequency domain and then convert them back to the time domain for further analysis or use.
3- Discrete Fourier Transform (DFT)
What is it?
Convert a sequence of discrete data points in the time domain into its frequency domain representation. In simpler terms, it allows us to analyze a signal or a sequence of numbers and determine the different frequencies present in that signal.
Speaking of which, like the Inverse fourier transform, there is also “Inverse DFT”. Once we have the DFT representation, we can also use the Inverse Discrete Fourier Transform (IDFT) to convert the data back from the frequency domain to the time domain. This allows us to reconstruct the original signal from its frequency components.
Do not forget, the DFT is a sampled version of the Fourier Transform.
It’s operates on discrete-time signals with a limited frequency range conversely to Fourier Transform. That’s the key difference between them
Why we do use it?
- Fourier Transforms of Signals with Finite Duration: For signals with finite duration, the continuous Fourier Transform is not directly applicable. The DFT provides a practical and efficient way to compute the frequency domain representation of such signals.
- Pitch Detection: DFT can be used to determine audio signal containing a musical note or human voice, the fundamental frequency (pitch) of the note being played or sung.
DFT can help us to identify different musical notes in a piece of music.
Summary
The Discrete Fourier Transform (DFT) is a mathematical algorithm used to transform a sequence of discrete data points from its original time domain representation into its frequency domain representation
4- Fast Fourier Transform (FFT)
What is it?
FFT rapidly computes such transformations by factorizing the DFT matrix into a product of sparse (mostly zero) factors. As a result, it manages to reduce the complexity of computing the DFT
FFT algorithm is commonly used to compute the DFT or IDFT more efficiently. The FFT algorithm reduces the computational complexity of the DFT from O(n²) to O(n log n), where n is the number of data points. If it still didn’t fit in, consider it like Fast Discrete Fourier Transform.
- It should be underlined, Fourier Transform is a function, Fast Fourier Transform is an algorithm.
Why we do use it?
It can be used where DFT can be used, additionally:
- Audio Signal Analysis: Apply FFT to convert the audio signal into the frequency domain. Analyze the magnitude and phase information of the frequency components to identify relevant features or patterns.
- Speech Recognition: Fast Fourier Transform can be used for analyze the audio signal of the speech. After extracting relevant features from the frequency domain, such as Mel-Frequency Cepstral Coefficients (MFCCs), which are commonly used in speech recognition systems and they are mentioned too at below.
In some problems, both DFT and FFT can be used but, if we consider data size, computation cost and complexity, than it’s better to use FFT instead of DFT.
Summary
It is an algorithm which used to transform a signal or a set of data from its original domain (like time) to the frequency domain. Key power of FFT is its speed.
Now let’s look Spectral crew
5- Spectral Centroid
What is it?
The spectral centroid value indicates the balance or center of mass of the spectrum. If the centroid is towards the lower frequencies, it indicates a darker or bass-heavy sounds.
Why we do use it?
- Feature Extraction: It is one such feature that can be used along with other features to describe the characteristics of the sound for tasks like classification, recognition, and emotion analysis.
- Signal Separation: Utilized in some audio separation techniques to distinguish different sound sources based on their tonal characteristics.
- Music Information Retrieval: In music information retrieval systems, spectral centroid is used to index and search music based on tonal features, allowing users to find similar or related songs.
- Audio Compression: In audio coding and compression algorithms, spectral centroid information can be used to efficiently represent and compress audio signals, helping to reduce the file size while preserving essential characteristics of the sound.
Summary
In simpler terms, the spectral centroid gives an idea of the “brightness” or “tonal center” of a sound. It is a measure that represents the “center of mass” or the average frequency of a sound spectrum.
6- Spectral Bandwidth
What is it?
The spectral bandwidth value provides an indication of how spread out the frequencies are in the spectrum. A larger bandwidth suggests a wider range of frequencies, while a smaller bandwidth indicates a narrower range.
Why we do use it
- Noise and Distortion Analysis: Spectral bandwidth may be useful in identifying and analyzing noise and distortions in audio signals. Wide spectral bandwidth in certain frequency regions can indicate the presence of noise, while narrow bandwidths may indicate the presence of harmonic components. So we can use this difference.
- Speech Processing: In speech analysis, spectral bandwidth can provide information about the spectral characteristics of different phonemes or speech sounds.
- Sound Source Separation: Spectral bandwidth can be employed in sound source separation algorithms to distinguish between sound sources based on their frequency spread besides other powerful audio features.
Summary
It is a feature in audio and signal processing that measures the spread or width of frequencies present in an audio signal’s spectrum. It complements the information provided by the spectral centroid.
7- Spectral Rolloff
What is it?
The spectral rolloff provides information about the overall shape and spread of energy in the spectrum. If the spectral rolloff is at a lower frequency, it suggests that a larger portion of the energy is concentrated in the lower frequencies. Conversely, if the spectral rolloff is at a higher frequency, it indicates that the energy is spread across higher frequencies.
Why we do use it?
- Recognition: Spectral rolloff can be utilized in speech and speaker recognition tasks.
- Audio Effects: Spectral rolloff can be employed in audio effects processing to control the high-frequency content of a sound.
- Speech and Speaker Recognition: Different speakers and speech sounds may exhibit distinct rolloff characteristics, aiding and identifying different speaker voice textures.
Summary
It’s a measure of the frequency content of a signal. It is calculated by finding the frequency below which a specified percentage of the total spectral energy lies. For example, a spectral rolloff of 85% means that 85% of the total spectral energy lies below a certain frequency.
8- MFCC
What it is
MFCC stands for Mel-Frequency Cepstral Coefficients. MFCCs can be thought of as signal features that help us comprehend the underlying characteristics of the signals. MFCCs capture essential information about the spectral characteristics of audio signals while reducing the dimensionality of the data.
The MFCC algorithm involves several steps:
- Pre-emphasis: Enhance higher frequencies in the signal to improve the signal-to-noise ratio.
- Framing: Divide the audio signal into short overlapping frames.
- Windowing: Apply a window function (e.g., Hamming window) to each frame to reduce spectral leakage.
- Fast Fourier Transform (FFT): Compute the power spectrum of each frame.
- Mel Filterbank: Apply a set of triangular filters spaced on the Mel scale to the power spectrum.
- Logarithm: Take the logarithm of the filterbank energies.
- Discrete Cosine Transform (DCT): Apply DCT to the log filterbank energies to decorrelate the coefficients.
- MFCCs: Select a subset of the resulting DCT coefficients (typically the first few) to represent the MFCCs.
After obtaining the features, we can apply various machine learning or statistical techniques to analyze, classify, or process the data.
Why we do use it?
- Speech Processing: MFCCs are based on the Mel scale, which mimics how humans perceive sound frequencies. So it can be used in speech or speaker recognition.
- Music processing: It can be used for mood, Music genre classification, Automatic tagging.
- Machine Learning Compatibility: The compact and informative nature of MFCCs makes them suitable for feeding into machine learning algorithms.
Summary
MFCCs provide a tool for extracting relevant information from audio signals particularly in the context of human auditory perception.
