Short-Time Fourier Transform and Chroma Features

Lab Course
International Audio Laboratories Erlangen
Prof. Dr. Meinard Müller
Winter Term 2019/2020


Short-Time Fourier Transform and Chroma Features, © October 2019

The Fourier transform, which is used to convert a time-dependent signal to a frequency-dependent signal, is one of the most important mathematical tools in audio signal processing. Applying the Fourier transform to local sections of an audio signal, one obtains the short-time Fourier transform (STFT). In this lab course, we study a discrete version of the STFT. To work with the discrete STFT in practice, one needs to correctly interpret the discrete time and frequency parameters. Using Python, we compute a discrete STFT and visualize its magnitude in form of a spectrogram representation. Then, we derive from the STFT various audio features that are useful for analyzing music signals. In particular, we develop a log-frequency spectrogram, where the frequency axis is converted into an axis corresponding to musical pitches. From this, we derive a chroma representation, which is a useful tool for capturing harmonic information of music.

Audio signals can be complex mixtures consisting of a multitude of different sound components. A first step in better understanding a given signal is to decompose it into building blocks that are better accessible for the subsequent processing steps. In the case that these building blocks consist of complex-valued sinusoidal functions, such a process is also called Fourier analysis. The Fourier transform maps a time-dependent signal to a frequency-dependent function which reveals the spectrum of frequency components that compose the original signal. Loosely speaking, a signal and its Fourier transform are two sides of the same coin. On the one side, the signal displays the time information and hides the information about frequencies. On the other side, the Fourier transform reveals information about frequencies and hides the time information.

To obtain back the hidden time information, Dennis Gabor introduced in the year 1946 the modified Fourier transform, now known as short-time Fourier transform or simply STFT. This transform is a compromise between a time- and a frequency-based representation by determining the sinusoidal magnitude and phase content of local sections of a signal as it changes over time. In this way, the STFT does not only tell which frequencies are "contained" in the signal but also at which points of times or, to be more precise, in which time intervals these frequencies appear.

The figure sows various representations for a piano recording of the chromatic scale ranging from A0 ($p=21$) to C8 ($p=108$). (a) Piano keys representing the chromatic scale. (b) Spectrogram representation. (c) Pitch-based log-frequency spectrogram. (d) Chromagram representation. For visualization purposes the values are color-coded using a logarithmic scale. The C3 ($p=48$) played at time $t=30~{\mathrm{sec}}$ has been highlighted by the rectangular frames.

The main objective of this lab course is to acquire a good understanding of the STFT. To this end, we study a discrete version of the STFT using the discrete Fourier transform (DFT), which can be efficiently computed using the fast Fourier transform (FFT). The discrete STFT yields a discrete set of Fourier coefficients that are indexed by time and frequency parameters. The correct physical interpretation of these parameters in terms of units such as seconds and Hertz depends on the sampling rate, the window size, and the hop size used in the STFT computation. In this lab course, we will compute a discrete STFT using Python and then visualize its magnitude by a spectrogram representation, see the STFT-section. By applying the STFT to different audio examples and by modifying the various parameters, one should get a better understanding on how the STFT works in practice.

To make music data comparable and algorithmically accessible, the first step in basically all music processing tasks is to extract suitable features that capture relevant aspects while suppressing irrelevant details. In the second part of this lab course, we study audio features and mid-level representations that are particularly useful for capturing pitch information of music signals. Assuming that we are dealing with music that is based on the equal-tempered scale (the scale that corresponds to the keys of a piano keyboard), we will convert an audio recording into a feature representation that reveals the distribution of the signal's energy across the different pitches, see the Log-Frequency-Spectrogram-section. Technically, these features are obtained from a spectrogram by converting the linear frequency axis (measured in Hertz) into a logarithmic axis (measured in pitches). From this log-frequency spectrogram, we then derive a time-chroma representation by suitably combining pitch bands that correspond to the same chroma, see the Chroma-Features-section. The resulting chroma features show a high degree of robustness to variations in timbre and instrumentation.

The Fourier transform and in particular the discrete STFT serve as front-end transform, the first computing step, for deriving a large number of different musically relevant audio features. We now recall the definition of the discrete STFT while fixing some notation. Let $x:[0:L-1]:=\{0,1,\ldots,L-1\}\to{\mathbb R}$ be a real-valued discrete-time signal of length $L$ obtained by equidistant sampling with respect to a fixed sampling rate $F_\mathrm{s}$ given in Hertz ($\mathrm{Hz}$). Furthermore, let $w:[0:N-1]:=\{0,1,\ldots,N-1\}\to{\mathbb R}$ be a discrete-time window of length $N\in{\mathbb N}$ (usually a power of two) and let $H\in{\mathbb N}$ be a hop size parameter. With regards to these parameters, the discrete STFT ${\mathcal X}$ of the signal $x$ is given by

\begin{eqnarray} {\mathcal X}(m,k):= \sum_{n=0}^{N-1} x(n+mH)w(n)\exp(-2\pi ikn/N) \end{eqnarray}

with $m\in[0:\lfloor \frac{L-N}{H} \rfloor]$ and $k\in[0:K]$. The complex number ${\mathcal X}(m,k)$ denotes the $k^{\mathrm{th}}$ Fourier coefficient for the $m^{\mathrm{th}}$ time frame, where $K=N/2$ is the frequency index corresponding to the Nyquist frequency. Each Fourier coefficient ${\mathcal X}(m,k)$ is associated with the physical time position (using the start position of the window as reference point)

\begin{equation} {T_{\mathrm{coef}}(m)} := \frac{m\cdot H}{F_\mathrm{s}} \end{equation}

given in seconds (${\mathrm{sec}}$) and with the physical frequency

\begin{equation} F_{\mathrm{coef}}(k) := \frac{k\cdot F_\mathrm{s}}{N} \end{equation}

given in Hertz ($\mathrm{Hz}$). For example, using $F_\mathrm{s}=44100~\mathrm{Hz}$ as for a CD recording, a window length of $N=4096$, and a hop size of $H=N/2$, we obtain a time resolution of $H/F_\mathrm{s}\approx 46.4~\mathrm{ms}$ and frequency resolution of $F_\mathrm{s}/N\approx 10.8~\mathrm{Hz}$.

Installation
Before starting with programming, you should install some Python packages, needed later. Please install them by executing the following cell:
In [ ]:
!pip install pysoundfile==0.9.0.post1 librosa==0.6.1
Homework Exercise 1 (to be submitted in handwritten form and correct mathematical notation)
  1. Compute the time and frequency resolution of the resulting STFT when using the following parameters of $F_\mathrm{s}$, $N$ and $H$. What are the Nyquist frequencies?
    • $F_\mathrm{s} = 22050$, $N = 1024$, $H = 512$
    • $F_\mathrm{s} = 48000$, $N = 1024$, $H = 256$
    • $F_\mathrm{s} = 4000$, $N = 4096$, $H = 1024$
  2. Using $F_\mathrm{s}=44100$, $N=2048$ and $H=1024$, what is the physical meaning of the Fourier coefficients ${\mathcal X}(1000,1000)$, ${\mathcal X}(17,0)$, and ${\mathcal X}(56,1024)$?
In [ ]:
# write the functions T_coef and F_coef...

Fs, N, H = 22050, 1024, 512
print('Fs = %5d, N = %d, H = %4d: Tcoef = %6.2f msec, Fcoef = %5.2f Hz, Nyquist = %.2f Hz' % (Fs, N, H, T_coef(1, H, Fs)*1000, F_coef(1, N, Fs), Fs/2))

Fs, N, H = 48000, 1024, 256
print('Fs = %5d, N = %d, H = %4d: Tcoef = %6.2f msec, Fcoef = %5.2f Hz, Nyquist = %.2f Hz' % (Fs, N, H, T_coef(1, H, Fs)*1000, F_coef(1, N, Fs), Fs/2))

Fs, N, H = 4000, 4096, 1024
print('Fs = %5d, N = %d, H = %4d: Tcoef = %6.2f msec, Fcoef = %5.2f Hz, Nyquist = %.2f Hz' % (Fs, N, H, T_coef(1, H, Fs)*1000, F_coef(1, N, Fs), Fs/2))
In [ ]:
# write a function ex1_2 that returns the physical meaning of the given coefficients

Fs, N, H = 44100, 2048, 1024

m, k = 1000, 1000
ex1_2(Fs, N, H, k, m)

m, k = 17, 0
ex1_2(Fs, N, H, k, m)

m, k = 56, 1024
ex1_2(Fs, N, H, k, m)

The STFT is often visualized by means of a spectrogram, which is a two-dimensional representation of the squared magnitude:

\begin{equation} {\mathcal Y}(m,k) = |{\mathcal X}(m,k)|^2. \end{equation}

When generating an image of a spectrogram, the horizontal axis represents time, the vertical axis is frequency, and the dimension indicating the spectrogram value of a particular frequency at a particular time is represented by the intensity or color in the image.

Lab Experiment 1
Use the function sf.read to read the file Sound_TwoSineTwoImpulse.wav. This defines a signal $x$ as well as the sampling rate $F_\mathrm{s}$. In the case that the signal is stereo, only use the first channel.
In [ ]:
import soundfile as sf
from IPython.display import Audio

# your code here...

Audio(x, rate=Fs)
Initialize a length parameter $N=4096$ and a hop size parameter $H=2048$.
In [ ]:
# your code here...
Define a hann window function $w$ of length $N$ (using scipy.signal.get_window).
In [ ]:
from scipy import signal

# your code here...
Compute ${\mathcal X}$ using the function librosa.stft. The resulting matrix contains the complex-valued Fourier coefficients ${\mathcal X}(m,k)$. Make yourself familiar with the parameters of the function. What is the meaning of the parameters n_fft and win_length? Use zero padding at the edges of the signal (pad_mode='constant') and use centered windows (center=True).
In [ ]:
import librosa

# your code here...
Compute the spectrogram ${\mathcal Y}(m,k)$.
In [ ]:
import numpy as np

# your code here...
Compte the vector t containing the physical time positions (in seconds) of the time indices.
In [ ]:
# your code here, compute t...
Compute vector f containing the frequency values (in Hertz) of the frequency indices.
In [ ]:
# your code here, compute f...
Visualize the spectrogram in various ways with the axis given in form of indices. Use an appropriate figure size with the figsize keyword of plt.figure. For visualizing $\mathcal{Y}$, use the function plt.imshow. Explore its parameters aspect, origin, cmap. Furthermore, use the functions plt.colorbar(), plt.xlabel() and plt.ylabel(). Doing so, also get familiar with the various visualization parameters and tools offered by Python.
In [ ]:
from matplotlib import pyplot as plt
%matplotlib inline

# your code here...
Plot the spectrogram with the axis given in seconds and Hertz. This can be done with the extent keyword, using t and f. Furthermore, do only visualize the lowest 2 kHz by using plt.ylim.
In [ ]:
# your code here...
Next, use a logarithmic decibel-scale for visualizing the values ${\mathcal Y}(m,k)$. (Recall that, given a value $v \in {\mathbb R}$, the decibel value is $10 \log_{10}(v)$.)
In [ ]:
# your code here...
Compute spectrograms using different window sizes (for example, $N\in\{256,1024,4096,8192\}$) and different hop sizes (for example, $H\in\{1,N/4,N/2\}$). Do only visualize the lowest 2 kHz. Discuss the trade-off between time resolution and frequency resolution.
In [ ]:
# your code here...
Try out other audio files.
In [ ]:
# your code here...

The human sensation of the intensity of a sound is logarithmic in nature. In practice, sounds that have an extremely small intensity may still be relevant for human listeners. Therefore, one often uses a decibel scale, which is a logarithmic unit expressing the ratio between two values. As alternative of using a decibel scale, one often applies in audio processing a step also referred to as logarithmic compression, which works as follows. Let $\gamma\in{\mathbb R}_{>0}$ be a positive constant and $\Gamma_\gamma:{\mathbb R}_{>0} \to {\mathbb R}_{>0}$ a function defined by

\begin{equation} \Gamma_\gamma(v):=\log(1+ \gamma \cdot v). \end{equation}

for $v\in{\mathbb R}_{>0}$, where we use the natural logarithm. Note that the function $\Gamma_\gamma$ yields a positive value $\Gamma_\gamma(v)$ for any positive value $v\in{\mathbb R}_{>0}$. Now, for a representation with positive values such as a spectrogram, one obtains a compressed version by applying the function $\Gamma_\gamma$ to each of the values:

\begin{equation} (\Gamma_\gamma\circ {\mathcal Y})(m,k):=\log(1+ \gamma \cdot {\mathcal Y}(m,k)). \end{equation}

Why is this operation called compression and what is the role of the constant $\gamma$? The problem with representations such as a spectrogram is that its values possess a large dynamic range. As a result, small, but still relevant values may be dominated by large values. Therefore, the idea of compression is to balance out this discrepancy by reducing the difference between large and small values with the effect to enhance the small values. This exactly is done by the function $\Gamma_\gamma$, where the degree of compression can be adjusted by the constant $\gamma$. The larger $\gamma$, the larger the resulting compression

Homework Exercise 2 (to be submitted in handwritten form and correct mathematical notation)
Sketch the function $\Gamma_\gamma$ for the parameters $\gamma\in\{1,10,100\}$.
Lab Experiment 2
Use the file Tone_C4_Piano.wav to define a signal $x$.
In [ ]:
# your code here...
Compute the STFT and the spectrogram ${\mathcal Y}$ as above using a Hann window of size $N=4096$ and a hop size $H=2048$.
In [ ]:
# your code here...
Compute the compressed version $\Gamma_\gamma\circ {\mathcal Y}$ of the spectrogram using different constants $\gamma\in\{1,10,100,1000,10000\}$. Visualize the original spectrogram and its compressed versions. What do you see? Discuss the results.
In [ ]:
# your code here...
Try out other audio files.
In [ ]:
# your code here...

We now derive some audio features from the STFT by converting the frequency axis (given in Hertz) into an axis that corresponds to musical pitches. In Western music, the equal-tempered scale is most often used, where the pitches of the scale correspond to the keys of a piano keyboard. In this scale, each octave (the interval between two tones, whose fundamental frequencies differ by a factor of two) is split up into twelve logarithmically spaced units. In MIDI notation, one considers $128$ pitches, which are serially numbered starting with $0$ and ending with $127$. The MIDI pitch $p=69$ corresponds to the pitch ${A}{4}$ (having a center frequency of $440~\mathrm{Hz}$), which is often used as standard for tuning musical instruments. In general, the center frequency $F_{\mathrm{pitch}}(p)$ of a pitch $p\in[0:127]$ is given by the formula

\begin{equation} F_{\mathrm{pitch}}(p) = 2^{(p-69)/12} \cdot 440. \end{equation}

The logarithmic perception of frequency motivates the use of a time-frequency representation with a logarithmic frequency axis labeled by the pitches of the equal-tempered scale. To derive such a representation from a given spectrogram representation, the basic idea is to assign each spectral coefficient ${\mathcal X}(m,k)$ to the pitch with center frequency that is closest to the frequency $F_{\mathrm{coef}}(k)$. More precisely, we define for each pitch $p\in[0:127]$ the set

\begin{equation} P(p) := \{k\in[0:K]:F_{\mathrm{pitch}}(p - 0.5) \leq F_{\mathrm{coef}}(k) < F_{\mathrm{pitch}}(p + 0.5)\}. \end{equation}

From this, we obtain a log-frequency spectrogram ${\mathcal Y}_\mathrm{LF}:{\mathbb Z}\times [0:127]\to{\mathbb R}_{\geq 0}$ defined by

\begin{equation} {\mathcal Y}_\mathrm{LF}(m,p) := \sum_{k \in P(p)}{|{\mathcal X}(m,k)|^2}. \end{equation}

By this definition, the frequency axis is partitioned logarithmically and labeled linearly according to MIDI pitches.

Homework Exercise 3 (to be submitted in handwritten form and correct mathematical notation)
  1. Compute the center frequencies $F_{\mathrm{pitch}}(p)$ for $p=68$, $p=69$, and $p=70$.
  2. Compute the cutoff frequencies $F_{\mathrm{pitch}}(p - 0.5)$ and $F_{\mathrm{pitch}}(p + 0.5)$ of the frequency band corresponding to pitch $p=69$.
  3. Using $F_\mathrm{s}=22050$ and $N=4096$, determine the set $P(p) \subseteq [0:K]$ for $p=69$.
  4. Also compute $P(p)$ for $p=57$, $p=45$, and $p=33$.
  5. Please explain with your own words why the definition of ${\mathcal Y}_\mathrm{LF}(m,p)$ may be problematic for small values of pitch $p$? How is the size of the set $P(p)$ influenced? Support your answer with a brief example.
In [ ]:
# write the function F_pitch(p)

print('Fpitch(%d) = %.2f Hz' % (68, F_pitch(68)))
print('Fpitch(%d) = %.2f Hz' % (69, F_pitch(69)))
print('Fpitch(%d) = %.2f Hz' % (70, F_pitch(70)))
In [ ]:
# compute the cutoff frequencies
In [ ]:
# compute the set of coefficients for p=69 using a function

Fs, N = 22050, 4096
print('P(%d) = %s' % (69, P(69, Fs, N)))
In [ ]:
# Compute the sets for p=57, p=45, and p=33
In [ ]:
# try out lower values of p
Lab Experiment 3
Use the file Scale_Cmajor_Piano.wav to define a signal $x$.
In [ ]:
# your code here...

Audio(x, rate=Fs)
Compute the STFT and the spectrogram as above using a Hann window of size $N=4096$ and a hop size $H=2048$. In the following you also need the information contained in the frequency vector F.
In [ ]:
# your code here...
Compute the log-frequency spectrogram ${\mathcal Y}_\mathrm{LF}$.
In [ ]:
# your code here...
Visualize the log-frequency spectrogram with the axes given in seconds and MIDI pitch, respectively.
In [ ]:
# your code here...
Use log-compression to enhance the visualization.
In [ ]:
# your code here...
Play around with different parameter settings for $N$ and $H$. Also, try out some other audio files.
In [ ]:
# your code here...

The human perception of pitch is periodic in the sense that two pitches are perceived as similar in color playing a similar harmonic role) if they differ by one or several octaves (where, in our scale, an octave is defined as the distance of $12$ pitches). For example, the pitches $p=60$ and $p=72$ are one octave apart, and the pitches $p=57$ and $p=81$ are two octaves apart. A pitch can be separated into two components, which are referred to as the height and chroma. The tone height refers to the octave number and the chroma to the respective pitch spelling attribute. In Western music notation, the $12$ pitch attributes are given by the set $\{\mathrm{C},\mathrm{C}^{\sharp},\mathrm{D},\ldots,\mathrm{B}\}$. Enumerating the chroma values, we identify this set with $[0:11]$ where $c=0$ refers to chroma $\mathrm{C}$, $c=1$ to $\mathrm{C}^{\sharp}$, and so on. A pitch class is defined as the set of all pitches that share the same chroma. For example, the pitch class that corresponds to the chroma $c=0$ ($\mathrm{C}$) consists of the set $\{0,12,24,36,48,60,72,84,96,108,120\}$ (which are the musical notes $\{\ldots,\,\mathrm{C}\mathrm{0},\mathrm{C}\mathrm{1},\mathrm{C}\mathrm{2},\mathrm{C}\mathrm{3}\ldots\}$).

The main idea of chroma features is to aggregate all spectral information that relates to a given pitch class into a single coefficient. Given a pitch-based log-frequency spectrogram ${\mathcal Y}_\mathrm{LF}:{\mathbb Z}\times[0:127]\to {\mathbb R}_{\geq 0}$, a chroma representation or chromagram ${\mathbb Z}\times[0:11]\to {\mathbb R}_{\geq 0}$ can be derived by summing up all pitch coefficients that belong to the same chroma:

\begin{equation} {\mathcal C}(m,c) := \sum_{\{p \in [0:127]\,|\,p\,\mathrm{mod}\,12 = c\}}{{\mathcal Y}_\mathrm{LF}(m,p)} \end{equation}

for $c\in[0:11]$.

Lab Experiment 4
Derive the chroma representation ${\mathcal C}$ from the log-frequency spectrogram as computed in the last exercise.
In [ ]:
# your code here...
Visualize the chroma representation with the axes given in seconds and chroma indices, respectively. Try to explain what you see in the chroma visualization.
In [ ]:
# your code here...
Also play around with different parameter settings for $N$ and $H$ and try out some other audio files.
In [ ]:
# your code here...

Reference


Meinard Müller
Fundamentals of Music Processing – Audio, Analysis, Algorithms, Applications
ISBN: 978-3-319-21944-8
483 p., Springer, 2015
www.music-processing.de
Accompanying Website

Acknowledgment


The International Audio Laboratories Erlangen are a joint institution of the Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU) and Fraunhofer Institute for Integrated Circuits IIS.