05/24/2007

Time - Frequency representations of music

Introduction

The basic way to handle and store digitized music is in the time domain. This means that the amplitude of sound waves is sampled periodically. The result is a sequence of numbers, a function of discretized time. From them it is easy to rebuild an analog signal that can be sent to a loudspeaker for listening. Music CDs work this way. The amplitude values can also shown as a graph, and in this way it is easy to see how sound volume varies with time. It is the usual view for audio edition in music software.

This figure shows a temporal representation of this music excerpt:

However, when we listen to music, we distinguish between sounds with different pitch, and can tell which is higher. Musicians have been using notations that show the frequency of the notes at different times for centuries. It seems natural to ask for a way of representing digitized sound that has these properties, and that can also be modified and converted back to audible sound.

There are many applications that require time - frequency representations. They include multi band compression and other mastering tools, note identification and source separation, some kinds of equalizers and some musical effects.

Usual Time - Frequency representations

The standard way to do time - frequency analysis and processing is the Short Time Fourier Transform (STFT), with a fixed window size between 1024 and 8192, for the usual sampling frequencies of 44.1kHz and 48 kHz. However, the STFT has disadvantages. The following figure shows the spectrogram (magnitude of the STFT) for the same sound, with a window size of 4096. Only frequencies from 0Hz to 4kHz are shown. The frequency resolution is 21.6Hz of and the time resolution is 93mS. It has 185 bands.

The frequency resolution is not enough for discriminating semitones at lower frequencies. And the time frequency is not enough for discriminating fast notes. Can we do better?

What's really needed

A musical instrument with a low register, such as the double bass or the tuba can not play fast notes. And if we try playing a fast melody on lower notes on a synthesizer, it sound more and more like percussion and less and less like a melody. Lower notes need longer time to stabilize on a definite frequency. And the ear needs more time to make sense of them. On the other hand, high pitched notes can be very short and they need less frequency resolution. The higher the note, the wider its bandwidth.

What is needed is a way to represent sounds that uses high frequency resolution for low pitches and high time resolution for high pitches. The uncertainty principle is a theorem that bounds the simultaneous time and frequency localization of any signal. This means there is a trade off between them, and to get better localization in one domain we must give up some localization in the other. Continuous wavelet transforms do this and are successfully used for note identification. However, they give redundant representations and for this they are not good for getting back an audible sound after manipulation. Discrete wavelet transforms can be used for sound reconstruction, but they can not handle semitone bands.

My story

I heard about Wavelets for the first time around 1992 or 1993. They said they made it possible to decompose any signal in "time-frequency atoms". Each atom would span over a time interval and a frequency band, and would be indivisible. I understood they were the fundamental time-frequency elements any sound was made of. The idea of an audio editor that allows modification of each block, allowing a perfect control of the sound seemed unavoidable.

I had this idea in the back of my mind for several years. Later, closer to the end of my studies at the University, I knew that Dr. Ana Ruedin teaches an elective course on wavelets, called "New techniques for data compression". I decided to take it, willing to learn about the techniques to implement them. There I learned the truth: wavelets are great for image processing, but were not successful at audio. They are not "the elementary blocks sound is made of". There were many wavelet families, with different properties, but none was as fundamental and revolutionary as I imagined.

But I kept believing something could be done. A digital signal of n samples is a vector in Rn. Linear Algebra says there are infinite bases for such a space. And Heisenberg uncertainty principle states that a time frequency atom spans over a time interval and a frequency interval that can not be both arbitrary small. There would be many ways to decompose a signal in different sets of atoms. It is a matter of choosing the atoms to be close to the uncertainty principle, and spanning over appropriate areas of the time-frequency plane.

Fortunately, Dr. Ruedin believed it was worth supporting my effort, and accepted being my thesis adviser. The result was:

My transform

This picture shows the graph of my transform on the same music fragment. It is an invertible transformation consisting of an orthogonal base. The elements of the base are well located both in frequency and in time. The representation has one band per semitone in the equal temperament musical scale. Therefore higher bands have greater bandwidth. The frequency scale in the graph is logarithmic, so they have the same height. The bands are critically sampled, meaning that lower bands are longer in time than higher ones.

I superimposed two staffs at the usual clefs for piano music. On the left is a representation of the black and white notes on a piano. Together, these help interpreting the notes in the music. This transform has 4 octaves, or 48 semitone bands. They span from 110Hz to 1662Hz. You can clearly see the played notes. It is not hard to extract written music notation or MIDI data from it. The best part, is that given that the transformation is invertible; it is possible to modify the the data in the new domain, and rebuild high quality audio.

You can read more at Research.

 
 
   2006-2012 Juan M. Vuletich