## Wavelets II - Vanishing Moments and Spectral Factorization

October 11, 2016

In the previous blog post I described the workings of the Fast Wavelet Transform (FWT) and how wavelets and filters are related. As promised, in this article we will see how to construct useful filters. Concretely, we will find a way to calculate the Daubechies filters, named after Ingrid Daubechies, who invented them and also laid much of the mathematical foundations for wavelet analysis.

Besides the content of the last post, you should be familiar with basic complex algebra, the...

## Wavelets I - From Filter Banks to the Dilation Equation

This is the first in what I hope will be a series of posts about wavelets, particularly about the Fast Wavelet Transform (FWT). The FWT is extremely useful in practice and also very interesting from a theoretical point of view. Of course there are already plenty of resources, but I found them tending to be either simple implementation guides that do not touch on the many interesting and sometimes crucial connections. Or they are highly mathematical and definition-heavy, for a...

## Digital Envelope Detection: The Good, the Bad, and the Ugly

Recently I've been thinking about the process of envelope detection. Tutorial information on this topic is readily available but that information is spread out over a number of DSP textbooks and many Internet web sites. The purpose of this blog is to summarize various digital envelope detection methods in one place.

Here I focus on envelope detection as it is applied to an amplitude-fluctuating sinusoidal signal where the positive-amplitude fluctuations (the sinusoid's envelope)...

## Autocorrelation and the case of the missing fundamental

[UPDATED January 25, 2016:  One of the examples was broken, also the IPython notebook links now point to nbviewer, where you can hear the examples.]

For sounds with simple harmonic structure, the pitch we perceive is usually the fundamental frequency, even if it is not dominant.  For example, here's the spectrum of a half-second recording of a saxophone.

The first three peaks are at 464, 928, and 1392 Hz.  The pitch we perceive is the fundamental, 464 Hz, which is close to...

## Generating pink noise

January 20, 20161 comment

In one of his most famous columns for Scientific American, Martin Gardner wrote about pink noise and its relation to fractal music.  The article was based on a 1978 paper by Voss and Clarke, which presents, among other things, a simple algorithm for generating pink noise, also known as 1/f noise.

The fundamental idea of the algorithm is to add up several sequences of uniform random numbers that get updated at different rates. The first source gets updated at...

## Amplitude modulation and the sampling theorem

I am working on the 11th and probably final chapter of Think DSP, which follows material my colleague Siddhartan Govindasamy developed for a class at Olin College.  He introduces amplitude modulation as a clever way to sneak up on the Nyquist–Shannon sampling theorem.

Most of the code for the chapter is done: you can check it out in this IPython notebook.  I haven't written the text yet, but I'll outline it here, and paste in the key figures.

Convolution...

## Multilayer Perceptrons and Event Classification with data from CODEC using Scilab and Weka

November 25, 2015

For my first blog, I thought I would introduce the reader to Scilab [1] and Weka [2].  In order to illustrate how they work, I will put together a script in Scilab that will sample using the microphone and CODEC on your PC and save the waveform as a CSV file.  Then, we can take the CSV file and open it in Weka.  Once in Weka, we have a lot of paths to consider in order to classify it.  I use the term classify loosely since there are many things you can do with data sets...

## Python scipy.signal IIR Filtering: An Example

May 19, 2013
Introduction

In the last posts I reviewed how to use the Python scipy.signal package to design digital infinite impulse response (IIR) filters, specifically, using the iirdesign function (IIR design I and IIR design II ).  In this post I am going to conclude the IIR filter design review with an example.

Previous posts:

## Beat Notes: An Interesting Observation

Some weeks ago a friend of mine, a long time radio engineer as well as a piano player, called and asked me,

"When I travel in a DC-9 aircraft, and I sit back near the engines, I hear this fairly loud unpleasant whump whump whump whump sound. The frequency of that sound is, maybe, two cycles per second. I think that sound is a beat frequency because the DC-9's engines are turning at a slightly different number of revolutions per second. My question is, what sort of mechanism in the airplane...

July 5, 2011

For the first time, the oral presentations of the International Conference on Accoustics, Speech, and Signal Processing (ICASSP) were recorded and posted online for free. This conference is the best in signal processing and it's diverse as well.

It has a bit speech processing, communication signal processing, and some interesting stuff like bio-inspired signal processing, where Prof. Sayed modeled the behaviour of a group of predetors attacking a herd of preys using distributed least mean...

## Adaptive Beamforming is like Squeezing a Water Balloon

Adaptive beamforming was first developed in the 1960s for radar and sonar applications. The main idea is that signals can be captured using multiple sensors and the sensor outputs can be combined to enhance the signals propagating from specific directions and attenuate (null out) signals from other directions. It has grown immensely in recent years as processors have become faster and cheaper. Today, adaptive beamforming applications include smart speakers (like the Amazon Echo),...

## Autocorrelation and the case of the missing fundamental

[UPDATED January 25, 2016:  One of the examples was broken, also the IPython notebook links now point to nbviewer, where you can hear the examples.]

For sounds with simple harmonic structure, the pitch we perceive is usually the fundamental frequency, even if it is not dominant.  For example, here's the spectrum of a half-second recording of a saxophone.

The first three peaks are at 464, 928, and 1392 Hz.  The pitch we perceive is the fundamental, 464 Hz, which is close to...

## Exploring Human Hearing Range

Human Hearing Range

In this post, I'll look at an interesting aspect of Audacity – using it to explore the threshold of human hearing. In my book Digital Signal Processing: A Gentle Introduction with Audio Examples, I go into this topic and I include a side note on the amazing hearing range of our canine companions.

Creating a Test Audio File

Audacity allows for the generation of a variety of test signals. If you click the Generate->Tone menu, it looks something like...

## Fitting Filters to Measured Amplitude Response Data Using invfreqz in Matlab

This blog post has been moved to the code snippet section and can now be found HERE.  Please update your bookmark.  Thanks!

## The Phase Vocoder Transform

February 12, 2019
1 Introduction

I would like to look at the phase vocoder in a fairly abstract'' way today. The purpose of this is to discuss a method for measuring the quality of various phase vocoder algorithms, and building off a proposed measure used in [2]. There will be a bit of time spent in the domain of continuous mathematics, thus defining a phase vocoder function or map rather than an algorithm. We will be using geometric visualizations when possible while pointing out certain group theory...

## In Search of The Fourth Wave

Last year I participated in the first DSP Related online conference, where I presented a short talk called "In Search of The Fourth Wave". It's based on a small mystery I encountered when I was working on Think DSP.  As you might know:

A sawtooth wave contains harmonics at integer multiples of the fundamental frequency, and their amplitudes drop off in proportion to 1/f.  A square wave contains only odd multiples of the fundamental, but they also drop off...

## Through the tube...

Hello all,

something completely different...

there was some recent discussion on the forum about modeling guitar amplifiers.I have been wondering for quite a while, whether the methods that I use to model radio frequency power amplifiers might also work for audio applications.

It's been a rainy day, so I found the time and energy for some experiments. Just for fun.

The device-under-test is a preamplifier with a single 12AX7 tube:

My good ol' Kurzweil (not in the picture) serves as "signal...

## Digging into an Audio Signal and the DSP Process Pipeline

In this post, I'll look at the benefits of using multiple perspectives when handling signals.A Pre-existing Audio File

Let's say we have an audio file of interest. Let's load it into Audacity and zoom in a little (using View → Zoom → Zoom In, multiple times). The figure illustrates the audio signal: just a basic single-tone signal.

By continuing to zoom into the signal, we eventually get to the point of seeing individual samples as illustrated below. Notice that I've marked one...