The Spectral Centroid is a measure of the "center of mass" of the spectrum, representing the perceived brightness of a sound. It is defined as:

\[ C = \frac{\sum_{k} f_k A_k}{\sum_{k} A_k} \]

Where:

• \(f_k\): Frequency of the k-th bin (in Hz)
• \(A_k\): Amplitude of the k-th frequency bin

The spectral centroid is often used in music and audio processing to describe the timbral texture of sounds, with higher values corresponding to brighter sounds.

The Spectral Flatness metric indicates how "noisy" or "tonal" a signal is. Higher values suggest a flatter spectrum (more noise-like), while lower values suggest a more tonal quality.

\[ SF = \frac{(\sum_{k} A_k)^2}{\sum_{k} A_k^2} \]

Where:

• \(A_k\): Amplitude of the k-th frequency bin

Spectral flatness is used in audio signal processing for classification and timbre analysis.

Spectral Bandwidth measures the width of the spectrum around the spectral centroid, representing the spread of frequencies in the signal. It is defined as:

\[ BW = \sqrt{\frac{\sum_{k} (f_k - C)^2 A_k}{\sum_{k} A_k}} \]

Where:

• \(f_k\): Frequency of the k-th bin (in Hz)
• \(C\): Spectral centroid
• \(A_k\): Amplitude of the k-th frequency bin

This metric is useful for analyzing the "sharpness" or "dullness" of a sound.

Spectral Roll-off is the frequency below which a specified percentage (e.g., 85%) of the total spectral energy lies. It is often used to identify the cutoff frequency of a signal.

\[ SR = \frac{\sum_{k} A_k \cdot \mathbb{1}(f_k \geq f_{\text{roll}})}{\sum_{k} A_k} \]

Where:

• \(A_k\): Amplitude of the k-th frequency bin
• \(f_{\text{roll}}\): Roll-off frequency threshold

Spectral Contrast measures the difference in energy between peaks and valleys in the spectrum. It highlights the texture and richness of a signal.

\[ SC = \sum_{i} \left| E_{i} - E_{i+1} \right| \]

Where:

• \(E_i\): Energy of the i-th spectral band

Spectral Energy quantifies the total energy of the spectrum. It is defined as:

\[ E = \sum_{k} A_k^2 \]

Where:

• \(A_k\): Amplitude of the k-th frequency bin

This metric is commonly used in audio analysis to describe the power or loudness of a signal.

Spectral Flux measures the rate of change in the spectrum over time, capturing variations in amplitude between consecutive frames. It is defined as:

\[ SF = \sum_{k} (A_k^{(n)} - A_k^{(n-1)})^2 \]

Where:

• \(A_k^{(n)}\): Amplitude of the k-th frequency bin in the current frame
• \(A_k^{(n-1)}\): Amplitude of the k-th frequency bin in the previous frame

This metric is often used to detect onset and rhythm changes in audio signals.

Spectral Entropy measures the disorder or randomness in the spectrum, often used as a feature for classification tasks. It is defined as:

\[ H = - \sum_{k} P_k \log P_k \]

Where:

• \(P_k\): Normalized power of the k-th frequency bin

Lower entropy suggests a more tonal signal, while higher entropy indicates a noise-like signal.

The RMS represents the Root Mean Square energy of the signal, commonly used for loudness estimation in audio processing. It is defined as:

\[ RMS = \sqrt{\frac{1}{N} \sum_{n=1}^N x_n^2} \]

Where:

• \(x_n\): Sample amplitude
• \(N\): Number of samples

Peak Amplitude measures the maximum absolute value of the signal's amplitude. It is defined as:

\[ A_{peak} = \max(|x_n|) \]

Where:

• \(x_n\): Sample amplitude

Zero Crossing Rate counts the number of sign changes in the waveform, indicative of the signal's noisiness or tonality. It is defined as:

\[ ZCR = \sum_{n=1}^{N-1} \mathbb{1}(x_n x_{n+1} < 0) \]

Where:

• \(x_n, x_{n+1}\): Consecutive samples

Low Energy Ratio measures the proportion of low-energy frames in the signal, useful for speech and music segmentation. It is defined as:

\[ LER = \frac{\sum_{i \in \text{Low}} E_i}{\sum_{i} E_i} \]

Where:

• \(E_i\): Energy of the i-th frame

Attack Time measures how quickly a sound reaches its peak amplitude, useful for characterizing percussive sounds. It is defined as:

\[ T_{attack} = \frac{t_{90} - t_{10}}{2} \]

Where:

• \(t_{90}, t_{10}\): Times at 90% and 10% of peak amplitude

Decay Time measures how quickly the sound amplitude decreases after the initial attack, often used in reverb analysis. It is defined as:

\[ T_{decay} = t_{90} - t_{10} \]

Where:

• \(t_{90}, t_{10}\): Times at 90% and 10% of amplitude decay

Sustain Level measures the average amplitude during the sustain phase of a sound. It is defined as:

\[ S_{level} = \frac{\sum_{t \in \text{sustain}} A_t}{\sum_{t \in \text{full}} A_t} \]

Where:

• \(A_t\): Amplitude at time \(t\)

Release Time measures how quickly a sound's amplitude falls to silence after the sustain phase. It is defined as:

\[ T_{release} = t_0 - t_f \]

Where:

• \(t_0, t_f\): Start and end times of release phase

Temporal Centroid represents the "center of gravity" of a sound in time, often related to the perceptual timing of its energy. It is defined as:

\[ TC = \frac{\sum_{t} t \cdot A_t}{\sum_{t} A_t} \]

Where:

• \(t\): Time
• \(A_t\): Amplitude at time \(t\)

Frequency Peaks are the dominant frequencies in the spectrum, calculated as the peak values of the spectral magnitude:

\[ \text{Peaks} = \text{argmax}(X(f)) \]

Where:

• \(X(f)\): Spectral magnitude at frequency \(f\)

Octave Bands calculate energy within specific frequency bands, often used in audio equalization:

\[ \text{Octave Band Energy} = 10 \log_{10} \left(\sum_{f \in \text{band}} |X(f)|^2 \right) \]

Where:

• \(X(f)\): Magnitude at frequency \(f\)

The Harmonic Ratio measures the ratio of harmonic energy to total energy in the spectrum:

\[ \text{HR} = \frac{\text{Energy}_{\text{Harmonic}}}{\text{Total Energy}} \]

Where:

• \(\text{Energy}_{\text{Harmonic}}\): Energy in harmonic frequencies
• \(\text{Total Energy}\): Total spectral energy

The Dominant Frequency identifies the most prominent frequency in the spectrum:

\[ f_{\text{Dominant}} = \text{argmax}(|X(f)|) \]

Where:

• \(|X(f)|\): Spectral magnitude at frequency \(f\)

The Fundamental Frequency represents the lowest frequency of a periodic waveform, corresponding to the perceived pitch:

\[ f_0 = \frac{1}{T} \]

Where:

• \(T\): Period of the waveform

Total Harmonic Distortion (THD) quantifies the presence of harmonics relative to the fundamental frequency:

\[ \text{THD} = \frac{\sum_{n=2}^N |H_n|^2}{|H_1|^2} \]

Where:

• \(H_n\): Amplitude of the n-th harmonic
• \(H_1\): Amplitude of the fundamental frequency

Spectral Slope measures the rate of change in amplitude with respect to frequency, indicating brightness or dullness:

\[ S = \frac{\Delta A}{\Delta f} \]

Where:

• \(\Delta A\): Change in amplitude
• \(\Delta f\): Change in frequency

Inharmonicity quantifies the deviation of partials from a harmonic series:

\[ I = \frac{\sum_{n=1}^N (f_n - nf_0)^2}{\sum_{n=1}^N f_n^2} \]

Where:

• \(f_n\): Frequency of the n-th partial
• \(f_0\): Fundamental frequency

Energy per Octave calculates the energy distribution across octave bands, useful in analyzing sound system response:

\[ E_{\text{Octave}} = \frac{\text{Energy in Octave}}{\text{Total Energy}} \]

Where:

• \(E_{\text{Octave}}\): Energy in a specific octave band
• \(\text{Total Energy}\): Total spectral energy

Spectral Centroid represents the "center of mass" of the spectrum, often linked to the brightness of the sound:

\[ C = \frac{\sum f |X(f)|}{\sum |X(f)|} \]

Where:

• \(f\): Frequency
• \(|X(f)|\): Magnitude of the spectrum at frequency \(f\)

Spectral Flatness measures how noise-like a spectrum is, distinguishing between tonal and noise-like sounds:

\[ \text{Flatness} = \frac{\text{Geometric Mean}(|X(f)|)}{\text{Arithmetic Mean}(|X(f)|)} \]

Where:

• \(|X(f)|\): Spectral magnitude at frequency \(f\)

Spectral Kurtosis measures the sharpness or peakiness of the spectrum:

\[ K = \frac{\mu_4}{\sigma^4} \]

Where:

• \(\mu_4\): Fourth central moment
• \(\sigma\): Standard deviation of the spectrum

Spectral Skewness quantifies the asymmetry of the spectral distribution, identifying whether energy is concentrated at higher or lower frequencies:

\[ \text{Skewness} = \frac{\mu_3}{\sigma^3} \]

Where:

• \(\mu_3\): Third central moment
• \(\sigma\): Standard deviation of the spectrum

Noise-to-Harmonic Ratio (NHR) measures the balance between noise and harmonic energy in a signal:

\[ \text{NHR} = \frac{\text{Energy}_{\text{Noise}}}{\text{Energy}_{\text{Harmonic}}} \]

Where:

• \(\text{Energy}_{\text{Noise}}\): Energy in noise-like components
• \(\text{Energy}_{\text{Harmonic}}\): Energy in harmonic components

Total Energy represents the cumulative energy of the signal over time:

\[ E_{\text{total}} = \int P(t) \, dt \]

Where:

• \(P(t)\): Power of the signal at time \(t\)

Mean Energy calculates the average energy over a given time interval:

\[ E_{\text{mean}} = \frac{1}{T} \int_{0}^{T} P(t) \, dt \]

Where:

• \(T\): Duration of the time interval
• \(P(t)\): Power of the signal at time \(t\)

Bass Energy Distribution represents the proportion of total energy contained in the bass frequency range:

\[ \frac{E_{\text{bass}}}{E_{\text{total}}} \times 100 \]

Where:

• \(E_{\text{bass}}\): Energy in the bass frequency range
• \(E_{\text{total}}\): Total energy of the signal

Treble Energy Distribution represents the proportion of total energy contained in the treble frequency range:

\[ \frac{E_{\text{treble}}}{E_{\text{total}}} \times 100 \]

Where:

• \(E_{\text{treble}}\): Energy in the treble frequency range
• \(E_{\text{total}}): Total energy of the signal

Dynamic Range measures the ratio between the loudest and quietest parts of the signal:

\[ \text{DR} = 20 \log_{10} \frac{P_{\text{max}}}{P_{\text{min}}} \]

Where:

• \(P_{\text{max}}\): Maximum power of the signal
• \(P_{\text{min}}\): Minimum power of the signal

Crest Factor measures the ratio of the peak signal power to the root mean square (RMS) power:

\[ \text{CF} = \frac{P_{\text{peak}}}{P_{\text{rms}}} \]

Where:

• \(P_{\text{peak}}\): Peak power of the signal
• \(P_{\text{rms}}\): Root mean square power of the signal

Energy Envelope Variance quantifies the variability of energy over time:

\[ \sigma^2 = \frac{1}{N} \sum_{i=1}^N (E_i - \mu)^2 \]

Where:

• \(E_i\): Energy at time \(i\)
• \(\mu\): Mean energy

Normalized Energy is the ratio of the current energy level to the maximum possible energy:

\[ E_{\text{norm}} = \frac{E_{\text{current}}}{E_{\text{max}}} \]

Where:

• \(E_{\text{current}}\): Current energy level
• \(E_{\text{max}}\): Maximum possible energy

Transient Energy represents the percentage of total energy attributed to transient components:

\[ \frac{E_{\text{transient}}}{E_{\text{total}}} \times 100 \]

Where:

• \(E_{\text{transient}}\): Energy of transient components
• \(E_{\text{total}}\): Total energy of the signal

Sustain Energy represents the percentage of total energy contained during the sustain phase of a signal:

\[ \frac{E_{\text{sustain}}}{E_{\text{total}}} \times 100 \]

Where:

• \(E_{\text{sustain}}\): Energy during the sustain phase
• \(E_{\text{total}}\): Total energy of the signal

Peak-to-Average Ratio (PAR) is the ratio of the peak power to the average power of the signal:

\[ \text{PAR} = \frac{P_{\text{peak}}}{P_{\text{average}}} \]

Where:

• \(P_{\text{peak}}\): Peak power of the signal
• \(P_{\text{average}}\): Average power of the signal