The speed of sound

At the Parque de las Ciencias in Granada, Spain there is this long tube, open at the end nearest you and closed at the far end. We can calculate the length of this tube just from the audio recording, because we know the speed of sound. Here’s the waveform of part of the recording, showing […]

Wave propagation on the surface of water

At the Alhambra (Granada, Spain) I saw this nice example of waves from a point source propagating in all directions at a fixed speed.

Autocorrelation for estimating F0

Most methods for estimating F0 start from autocorrelation. The idea is pretty simple: we are just looking for a repeating pattern in the waveform, which corresponds to the periodic vocal fold activity. For some waveforms, it might be possible to do that directly in the time domain, but in general that doesn’t work very well. […]

The Gaussian probability density function: understanding the equation

The equation for the Gaussian probability density function looks a little scary at first, but this video should help you understand what each of the terms is doing, and how they fit together. After watching the video download the spreadsheet which shows the calculations and plots from this video (tip: the Apple version includes images […]

Token passing

Token passing is a really nice way to understand (and even to implement) Viterbi search for Hidden Markov Models. Here we see token passing in action, and you can look at the spreadsheet to see the calculations. To keep things simple, we are ignoring transition probabilities in this example. It would be simple to add them […]

TD-PSOLA …the hard way

Time-Domain Pitch Synchronous Overlap and Add (TD-PSOLA) can modify the fundamental frequency and duration of speech signals, without affecting the segment identity – that is, without changing the formants. Normally, it’s an automatic algorithm, but here we do it the hard way – by hand! If you want to follow-along, you will need Audacity and these materials (a […]

Entropy: understanding the equation

The equation for entropy is very often presented in textbooks without much explanation, other than to say it has the desired properties. Here, I attempt an informal derivation of the equation starting from uniform probability distributions. A good way to think about information is in terms of sending messages. In the video, we send messages […]

A simple synthetic vowel

Using Praat, we synthesise a simple vowel-like sound, starting with a pulse train, which we pass through a filter with resonant peaks.

Pipeline architecture for TTS

Most text-to-speech systems split the problem into two main stages. The first stage is called the front end and contains many separate processes which gradually build up a linguistic specification from the input text. The second stage typically uses language-independent techniques (although they still require a language-specific speech corpus) to generate a waveform. Here we see those two […]


In sampling and quantisation we saw that sampling a signal at a fixed rate means that there is an upper limit on the frequencies that can be represented. This limit is called the Nyquist frequency. Before sampling a signal, we must remove all energy above the Nyquist frequency, and here we will see what would […]

Classification and regression trees (CART)

A quick introduction to a very simple but widely-applicable model that can perform classification (predicting a discrete label) or regression (predicting a continuous value). The tree is learned from labelled data, using supervised learning. Before watching this video, you might want to check that you understand what Entropy is.


When we say that a signal is non-stationary we mean that its properties, such as the spectrum, change over time. To analyse signals like this, we need to first assume that these properties do not change over some short period of time, called the frame. We can then analyse individual frames of the signal, one at a […]