# Spectral Descriptors

Signal Toolbox™ and Audio Toolbox™ provide a suite of functions that describe the shape, sometimes referred to as *timbre*, of audio. This example defines the equations used to determine the spectral features, cites common uses of each feature, and provides examples so that you can gain intuition about what the spectral descriptors are describing.

Spectral descriptors are widely used in machine and deep learning applications, and perceptual analysis. Spectral descriptors have been applied to a range of applications, including:

`Spectral Centroid`

`Spectral Centroid`

The spectral centroid (`spectralCentroid`

) is the frequency-weighted sum normalized by the unweighted sum [1]:

$${\mu}_{1}=\frac{\sum _{\mathit{k}={\mathit{b}}_{1}}^{{\mathit{b}}_{2}}{\mathit{f}}_{\mathit{k}\text{\hspace{0.17em}}}{\mathit{s}}_{\mathit{k}}}{\sum _{\mathit{k}={\mathit{b}}_{1}}^{{\mathit{b}}_{2}}{\mathit{s}}_{\mathit{k}}}$$

where

${\mathit{f}}_{\mathit{k}\text{\hspace{0.17em}}}$ is the frequency in Hz corresponding to bin $\mathit{k}$.

${\mathit{s}}_{\mathit{k}}$ is the spectral value at bin $\mathit{k}$. The magnitude spectrum and power spectrum are both commonly used.

${\mathit{b}}_{1}$ and ${\mathit{b}}_{2}$ are the band edges, in bins, over which to calculate the spectral centroid.

The spectral centroid represents the "center of gravity" of the spectrum. It is used as an indication of *brightness *[2] and is commonly used in music analysis and genre classification. For example, observe the jumps in the centroid corresponding to high hat hits in the audio file.

[audio,fs] = audioread('FunkyDrums-44p1-stereo-25secs.mp3'); audio = sum(audio,2)/2; centroid = spectralCentroid(audio,fs); subplot(2,1,1) t = linspace(0,size(audio,1)/fs,size(audio,1)); plot(t,audio) ylabel('Amplitude') subplot(2,1,2) t = linspace(0,size(audio,1)/fs,size(centroid,1)); plot(t,centroid) xlabel('Time (s)') ylabel('Centroid (Hz)')

The spectral centroid is also commonly used to classify speech as voiced or unvoiced [3]. For example, the centroid jumps in regions of unvoiced speech.

[audio,fs] = audioread('Counting-16-44p1-mono-15secs.wav'); centroid = spectralCentroid(audio,fs); subplot(2,1,1) t = linspace(0,size(audio,1)/fs,size(audio,1)); plot(t,audio) ylabel('Amplitude') subplot(2,1,2) t = linspace(0,size(audio,1)/fs,size(centroid,1)); plot(t,centroid) xlabel('Time (s)') ylabel('Centroid (Hz)')

`Spectral Spread`

`Spectral Spread`

Spectral spread (`spectralSpread`

) is the standard deviation around the spectral centroid [1]:

$${\mu}_{2}=\sqrt{\frac{\sum _{\mathit{k}={\mathit{b}}_{1}}^{{\mathit{b}}_{2}}{\left({\mathit{f}}_{\mathit{k}}-{\mu}_{1}\right)}^{2}{\mathit{s}}_{\mathit{k}}}{\sum _{\mathit{k}={\mathit{b}}_{1}}^{{\mathit{b}}_{2}}{\mathit{s}}_{\mathit{k}}}}$$

where

${\mathit{f}}_{\mathit{k}\text{\hspace{0.17em}}}$ is the frequency in Hz corresponding to bin $\mathit{k}$.

${\mathit{s}}_{\mathit{k}}$ is the spectral value at bin $\mathit{k}$. The magnitude spectrum and power spectrum are both commonly used.

${\mathit{b}}_{1}$ and ${\mathit{b}}_{2}$ are the band edges, in bins, over which to calculate the spectral spread.

${\mu}_{1}$ is the spectral centroid.

The spectral spread represents the "instantaneous bandwidth" of the spectrum. It is used as an indication of the dominance of a tone. For example, the spread increases as the tones diverge and decreases as the tones converge.

fs = 16e3; tone = audioOscillator('SampleRate',fs,'NumTones',2,'SamplesPerFrame',512,'Frequency',[2000,100]); duration = 5; numLoops = floor(duration*fs/tone.SamplesPerFrame); signal = []; for i = 1:numLoops signal = [signal;tone()]; if i<numLoops/2 tone.Frequency = tone.Frequency + [0,50]; else tone.Frequency = tone.Frequency - [0,50]; end end spread = spectralSpread(signal,fs); subplot(2,1,1) spectrogram(signal,round(fs*0.05),round(fs*0.04),2048,fs,'yaxis') subplot(2,1,2) t = linspace(0,size(signal,1)/fs,size(spread,1)); plot(t,spread) xlabel('Time (s)') ylabel('Spread')

`Spectral Skewness`

`Spectral Skewness`

Spectral skewness (`spectralSkewness`

) is computed from the third order moment [1]:

$${\mu}_{3}=\frac{\sum _{\mathit{k}={\mathit{b}}_{1}}^{{\mathit{b}}_{2}}{\left({\mathit{f}}_{\mathit{k}}-{\mu}_{1\text{\hspace{0.17em}}}\right)}^{3}{\mathit{s}}_{\mathit{k}}}{{\left({\mu}_{2}\right)}^{3}\sum _{\mathit{k}={\mathit{b}}_{1}}^{{\mathit{b}}_{2}}{\mathit{s}}_{\mathit{k}}}$$

where

${\mathit{f}}_{\mathit{k}\text{\hspace{0.17em}}}$ is the frequency in Hz corresponding to bin $\mathit{k}$.

${\mathit{s}}_{\mathit{k}}$ is the spectral value at bin $\mathit{k}$. The magnitude spectrum and power spectrum are both commonly used.

${\mathit{b}}_{1}$ and ${\mathit{b}}_{2}$ are the band edges, in bins, over which to calculate the spectral skewness.

${\mu}_{1}$ is the spectral centroid.

${\mu}_{2}$ is the spectral spread.

The spectral skewness measures symmetry around the centroid. In phonetics, spectral skewness is often referred to as *spectral tilt* and is used with other spectral moments to distinguish the place of articulation [4]. For harmonic signals, it indicates the relative strength of higher and lower harmonics. For example, in the four-tone signal, there is a positive skew when the lower tone is dominant and a negative skew when the upper tone is dominant.

fs = 16e3; duration = 99; tone = audioOscillator('SampleRate',fs,'NumTones',4,'SamplesPerFrame',fs,'Frequency',[500,2000,2500,4000],'Amplitude',[0,0.4,0.6,1]); signal = []; for i = 1:duration signal = [signal;tone()]; tone.Amplitude = tone.Amplitude + [0.01,0,0,-0.01]; end skewness = spectralSkewness(signal,fs); t = linspace(0,size(signal,1)/fs,size(skewness,1))/60; subplot(2,1,1) spectrogram(signal,round(fs*0.05),round(fs*0.04),round(fs*0.05),fs,'yaxis','power') view([-58 33]) subplot(2,1,2) plot(t,skewness) xlabel('Time (minutes)') ylabel('Skewness')

`Spectral Kurtosis`

`Spectral Kurtosis`

Spectral kurtosis (`spectralKurtosis`

) is computed from the fourth order moment [1]:

$${\mu}_{4}=\frac{\sum _{\mathit{k}={\mathit{b}}_{1}}^{{\mathit{b}}_{2}}{\left({\mathit{f}}_{\mathit{k}}-{\mu}_{1\text{\hspace{0.17em}}}\right)}^{4}{\mathit{s}}_{\mathit{k}}}{{\left({\mu}_{2}\right)}^{4}\sum _{\mathit{k}={\mathit{b}}_{1}}^{{\mathit{b}}_{2}}{\mathit{s}}_{\mathit{k}}}$$

where

${\mathit{b}}_{1}$ and ${\mathit{b}}_{2}$ are the band edges, in bins, over which to calculate the spectral kurtosis.

${\mu}_{1}$ is the spectral centroid.

${\mu}_{2}$ is the spectral spread.

The spectral kurtosis measures the flatness, or non-Gaussianity, of the spectrum around its centroid. Conversely, it is used to indicate the peakiness of a spectrum. For example, as the white noise is increased on the speech signal, the kurtosis decreases, indicating a less peaky spectrum.

[audioIn,fs] = audioread('Counting-16-44p1-mono-15secs.wav'); noiseGenerator = dsp.ColoredNoise('Color','white','SamplesPerFrame',size(audioIn,1)); noise = noiseGenerator(); noise = noise/max(abs(noise)); ramp = linspace(0,.25,numel(noise))'; noise = noise.*ramp; audioIn = audioIn + noise; kurtosis = spectralKurtosis(audioIn,fs); t = linspace(0,size(audioIn,1)/fs,size(audioIn,1)); subplot(2,1,1) plot(t,audioIn) ylabel('Amplitude') t = linspace(0,size(audioIn,1)/fs,size(kurtosis,1)); subplot(2,1,2) plot(t,kurtosis) xlabel('Time (s)') ylabel('Kurtosis')

`Spectral Entropy`

`Spectral Entropy`

Spectral entropy (`spectralEntropy`

) measures the peakiness of the spectrum [6]:

$$\mathrm{entropy}=\frac{-\sum _{\mathit{k}={\mathit{b}}_{1}}^{{\mathit{b}}_{2}}{\mathit{s}}_{\mathit{k}}\mathrm{log}\left({\mathit{s}}_{\mathit{k}}\right)}{\mathrm{log}\left({\mathit{b}}_{2}-{\mathit{b}}_{1}\right)}$$

where

${\mathit{b}}_{1}$ and ${\mathit{b}}_{2}$ are the band edges, in bins, over which to calculate the spectral entropy.

Spectral entropy has been used successfully in voiced/unvoiced decisions for automatic speech recognition [6]. Because entropy is a measure of disorder, regions of voiced speech have lower entropy compared to regions of unvoiced speech.

[audioIn,fs] = audioread('Counting-16-44p1-mono-15secs.wav'); entropy = spectralEntropy(audioIn,fs); t = linspace(0,size(audioIn,1)/fs,size(audioIn,1)); subplot(2,1,1) plot(t,audioIn) ylabel('Amplitude') t = linspace(0,size(audioIn,1)/fs,size(entropy,1)); subplot(2,1,2) plot(t,entropy) xlabel('Time (s)') ylabel('Entropy')

Spectral entropy has also been used to discriminate between speech and music [7] [8]. For example, compare histograms of entropy for speech, music, and background audio files.

fs = 8000; [speech,speechFs] = audioread('Rainbow-16-8-mono-114secs.wav'); speech = resample(speech,fs,speechFs); speech = speech./max(speech); [music,musicFs] = audioread('RockGuitar-16-96-stereo-72secs.flac'); music = sum(music,2)/2; music = resample(music,fs,musicFs); music = music./max(music); [background,backgroundFs] = audioread('Ambiance-16-44p1-mono-12secs.wav'); background = resample(background,fs,backgroundFs); background = background./max(background); speechEntropy = spectralEntropy(speech,fs); musicEntropy = spectralEntropy(music,fs); backgroundEntropy = spectralEntropy(background,fs); figure h1 = histogram(speechEntropy); hold on h2 = histogram(musicEntropy); h3 = histogram(backgroundEntropy); h1.Normalization = 'probability'; h2.Normalization = 'probability'; h3.Normalization = 'probability'; h1.BinWidth = 0.01; h2.BinWidth = 0.01; h3.BinWidth = 0.01; title('Spectral Entropy') legend('Speech','Music','Background','Location',"northwest") xlabel('Entropy') ylabel('Probability') hold off

`Spectral Flatness`

`Spectral Flatness`

Spectral flatness (`spectralFlatness`

) measures the ratio of the geometric mean of the spectrum to the arithmetic mean of the spectrum [9]:

$$\mathrm{flatness}=\frac{{\left(\prod _{\mathit{k}={\mathit{b}}_{1}}^{{\mathit{b}}_{2}}{\mathit{s}}_{\mathit{k}}\right)}^{\frac{1}{{\mathit{b}}_{2}-{\mathit{b}}_{1}}}}{\frac{1}{{\mathit{b}}_{2}-{\mathit{b}}_{1}}\sum _{\mathit{k}={\mathit{b}}_{1}}^{{\mathit{b}}_{2}}{\mathit{s}}_{\mathit{k}}}$$

where

${\mathit{b}}_{1}$ and ${\mathit{b}}_{2}$ are the band edges, in bins, over which to calculate the spectral flatness.

Spectral flatness is an indication of the peakiness of the spectrum. A higher spectral flatness indicates noise, while a lower spectral flatness indicates tonality.

[audio,fs] = audioread('WaveGuideLoopOne-24-96-stereo-10secs.aif'); audio = sum(audio,2)/2; noise = (2*rand(numel(audio),1)-1).*linspace(0,0.05,numel(audio))'; audio = audio + noise; flatness = spectralFlatness(audio,fs); subplot(2,1,1) t = linspace(0,size(audio,1)/fs,size(audio,1)); plot(t,audio) ylabel('Amplitude') subplot(2,1,2) t = linspace(0,size(audio,1)/fs,size(flatness,1)); plot(t,flatness) ylabel('Flatness') xlabel('Time (s)')

Spectral flatness has also been applied successfully to singing voice detection [10] and to audio scene recognition [11].

`Spectral Crest`

`Spectral Crest`

Spectral crest (`spectralCrest`

) measures the ratio of the maximum of the spectrum to the arithmetic mean of the spectrum [1]:

$$\mathrm{crest}=\frac{\mathrm{max}\left({\mathit{s}}_{\mathit{k}\u03f5\left[{\mathit{b}}_{1},{\mathit{b}}_{2}\right]}\right)}{\frac{1}{{\mathit{b}}_{2}-{\mathit{b}}_{1}}\sum _{\mathit{k}={\mathit{b}}_{1}}^{{\mathit{b}}_{2}}{\mathit{s}}_{\mathit{k}}}$$

where

${\mathit{b}}_{1}$ and ${\mathit{b}}_{2}$ are the band edges, in bins, over which to calculate the spectral crest.

Spectral crest is an indication of the peakiness of the spectrum. A higher spectral crest indicates more tonality, while a lower spectral crest indicates more noise.

[audio,fs] = audioread('WaveGuideLoopOne-24-96-stereo-10secs.aif'); audio = sum(audio,2)/2; noise = (2*rand(numel(audio),1)-1).*linspace(0,0.2,numel(audio))'; audio = audio + noise; crest = spectralCrest(audio,fs); subplot(2,1,1) t = linspace(0,size(audio,1)/fs,size(audio,1)); plot(t,audio) ylabel('Amplitude') subplot(2,1,2) t = linspace(0,size(audio,1)/fs,size(crest,1)); plot(t,crest) ylabel('Crest') xlabel('Time (s)')

`Spectral Flux`

`Spectral Flux`

Spectral flux (`spectralFlux`

) is a measure of the variability of the spectrum over time [12]:

$$\mathrm{flux}\left(\mathit{t}\right)={\left(\sum _{\mathit{k}={\mathit{b}}_{1}}^{{\mathit{b}}_{2}}{|\mathit{s}}_{\mathit{k}}\left(\mathit{t}\right)-{\mathit{s}}_{\mathit{k}}\left(\mathit{t}-1\right){|}^{\mathit{p}}\right)}^{\frac{1}{\mathit{p}}}$$

where

${\mathit{b}}_{1}$ and ${\mathit{b}}_{2}$ are the band edges, in bins, over which to calculate the spectral flux.

$\mathit{p}$ is the norm type.

Spectral flux is popularly used in onset detection [13] and audio segmentation [14]. For example, the beats in the drum track correspond to high spectral flux.

[audio,fs] = audioread('FunkyDrums-48-stereo-25secs.mp3'); audio = sum(audio,2)/2; flux = spectralFlux(audio,fs); subplot(2,1,1) t = linspace(0,size(audio,1)/fs,size(audio,1)); plot(t,audio) ylabel('Amplitude') subplot(2,1,2) t = linspace(0,size(audio,1)/fs,size(flux,1)); plot(t,flux) ylabel('Flux') xlabel('Time (s)')

`Spectral Slope`

`Spectral Slope`

Spectral slope (`spectralSlope`

) measures the amount of decrease of the spectrum [15]:

$$\mathrm{slope}=\frac{\sum _{\mathit{k}={\mathit{b}}_{1}}^{{\mathit{b}}_{2}}\left({\mathit{f}}_{\mathit{k}}-{\mu}_{\mathit{f}}\right)\left({\mathit{s}}_{\mathit{k}}-{\mu}_{\mathit{s}}\right)}{\sum _{\mathit{k}={\mathit{b}}_{1}}^{{\mathit{b}}_{2}}{\left({\mathit{f}}_{\mathit{k}}-{\mu}_{\mathit{f}}\right)}^{2}}$$

where

${\mu}_{\mathit{f}}$ is the mean frequency.

${\mathit{s}}_{\mathit{k}}$ is the spectral value at bin $\mathit{k}$. The magnitude spectrum is commonly used.

${\mu}_{\mathit{s}}$ is the mean spectral value.

${\mathit{b}}_{1}$ and ${\mathit{b}}_{2}$ are the band edges, in bins, over which to calculate the spectral slope.

Spectral slope has been used extensively in speech analysis, particularly in modeling speaker stress [19]. The slope is directly related to the resonant characteristics of the vocal folds and has also been applied to speaker identification [21]. Spectral slope is a socially important aspect of timbre. Spectral slope discrimination has been shown to occur in early childhood development [20]. Spectral slope is most pronounced when the energy in the lower formants is much greater than the energy in the higher formants.

[female,femaleFs] = audioread('FemaleSpeech-16-8-mono-3secs.wav'); female = female./max(female); femaleSlope = spectralSlope(female,femaleFs); t = linspace(0,size(female,1)/femaleFs,size(femaleSlope,1)); subplot(2,1,1) spectrogram(female,round(femaleFs*0.05),round(femaleFs*0.04),round(femaleFs*0.05),femaleFs,'yaxis','power') subplot(2,1,2) plot(t,femaleSlope) title('Female Speaker') ylabel('Slope') xlabel('Time (s)')

`Spectral Decrease`

`Spectral Decrease`

Spectral decrease (`spectralDecrease`

) represents the amount of decrease of the spectrum, while emphasizing the slopes of the lower frequencies [1]:

$$\mathrm{decrease}=\frac{\sum _{\mathit{k}={\mathit{b}}_{1}+1}^{{\mathit{b}}_{2}}\frac{{\mathit{s}}_{\mathit{k}}-{\mathit{s}}_{{\mathit{b}}_{1}}}{\mathit{k}-1}}{\sum _{\mathit{k}={\mathit{b}}_{1}+1}^{{\mathit{b}}_{2}}{\mathit{s}}_{\mathit{k}}}$$

where

${\mathit{s}}_{\mathit{k}}$ is the spectral value at bin $\mathit{k}$. The magnitude spectrum is commonly used.

${\mathit{b}}_{1}$ and ${\mathit{b}}_{2}$ are the band edges, in bins, over which to calculate the spectral decrease.

Spectral decrease is used less frequently than spectral slope in the speech literature, but it is commonly used, along with slope, in the analysis of music. In particular, spectral decrease has been shown to perform well as a feature in instrument recognition [22].

[guitar,guitarFs] = audioread('RockGuitar-16-44p1-stereo-72secs.wav'); guitar = mean(guitar,2); [drums,drumsFs] = audioread('RockDrums-44p1-stereo-11secs.mp3'); drums = mean(drums,2); guitarDecrease = spectralDecrease(guitar,guitarFs); drumsDecrease = spectralDecrease(drums,drumsFs); t1 = linspace(0,size(guitar,1)/guitarFs,size(guitarDecrease,1)); t2 = linspace(0,size(drums,1)/drumsFs,size(drumsDecrease,1)); subplot(2,1,1) plot(t1,guitarDecrease) title('Guitar') ylabel('Decrease') axis([0 10 -0.3 0.3]) subplot(2,1,2) plot(t2,drumsDecrease) title('Drums') ylabel('Decrease') xlabel('Time (s)') axis([0 10 -0.3 0.3])

`Spectral Rolloff Point`

`Spectral Rolloff Point`

The spectral rolloff point (`spectralRolloffPoint`

) measures the bandwidth of the audio signal by determining the frequency bin under which a given percentage of the total energy exists [12]:

$$\mathrm{Rolloff}\text{\hspace{0.17em}}\mathrm{Point}=\mathit{i}\text{\hspace{0.17em}\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{such}\text{\hspace{0.17em}}\mathrm{that}\text{\hspace{0.17em}}\text{\hspace{0.17em}\hspace{0.17em}}\sum _{\mathit{k}={\mathit{b}}_{1}}^{\mathit{i}}\left|{\mathit{s}}_{\mathit{k}}\right|=\kappa \text{\hspace{0.17em}}\sum _{\mathit{k}={\mathit{b}}_{1}}^{{\mathit{b}}_{2}}{\mathit{s}}_{\mathit{k}}$$

where

${\mathit{b}}_{1}$ and ${\mathit{b}}_{2}$ are the band edges, in bins, over which to calculate the spectral rolloff point.

$\kappa $ is the specified energy threshold, usually 95% or 85%.

$\mathit{i}$ is converted to Hz before it is returned by `spectralRolloffPoint`

.

The spectral rolloff point has been used to distinguish between voiced and unvoiced speech, speech/music discrimination [12], music genre classification [16], acoustic scene recognition [17], and music mood classification [18]. For example, observe the different mean and variance of the rolloff point for speech, rock guitar, acoustic guitar, and an acoustic scene.

dur = 5; % Clip out 5 seconds from each file. [speech,fs1] = audioread('SpeechDFT-16-8-mono-5secs.wav'); speech = speech(1:min(end,fs1*dur)); [electricGuitar,fs2] = audioread('RockGuitar-16-44p1-stereo-72secs.wav'); electricGuitar = mean(electricGuitar,2); % Convert to mono for comparison. electricGuitar = electricGuitar(1:fs2*dur); [acousticGuitar,fs3] = audioread('SoftGuitar-44p1_mono-10mins.ogg'); acousticGuitar = acousticGuitar(1:fs3*dur); [acousticScene,fs4] = audioread('MainStreetOne-16-16-mono-12secs.wav'); acousticScene = acousticScene(1:fs4*dur); r1 = spectralRolloffPoint(speech,fs1); r2 = spectralRolloffPoint(electricGuitar,fs2); r3 = spectralRolloffPoint(acousticGuitar,fs3); r4 = spectralRolloffPoint(acousticScene,fs4); t1 = linspace(0,size(speech,1)/fs1,size(r1,1)); t2 = linspace(0,size(electricGuitar,1)/fs2,size(r2,1)); t3 = linspace(0,size(acousticGuitar,1)/fs3,size(r3,1)); t4 = linspace(0,size(acousticScene,1)/fs4,size(r4,1)); figure plot(t1,r1) title('Speech') ylabel('Rolloff Point (Hz)') xlabel('Time (s)') axis([0 5 0 4000])

figure plot(t2,r2) title('Rock Guitar') ylabel('Rolloff Point (Hz)') xlabel('Time (s)') axis([0 5 0 4000])

figure plot(t3,r3) title('Acoustic Guitar') ylabel('Rolloff Point (Hz)') xlabel('Time (s)') axis([0 5 0 4000])

figure plot(t4,r4) title('Acoustic Scene') ylabel('Rolloff Point (Hz)') xlabel('Time (s)') axis([0 5 0 4000])

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