Sound Advice for Signal Analysis and Synthesis with Maple 6

The incorporation of the NAG Linear Algebra routines in Maple 6 has taken the Maple technology beyond its earlier status as the world's leading symbolic computation system. While mathe-matical manipulations are key requirements in the early stages of the technical analysis process, the goal of most technical analyses is to generate numeric results. Driven by this need, Maple 6 not only incorporates the NAG routines but also provides some significant infrastructure changes to support purely numeric computations. The benefits of these important changes are always very difficult to put over in a product brochure, since they are not a self-contained feature but pervade the product "under the hood". The best way to highlight the advantages, therefore, is to demonstrate them with examples of intensely numeric applications.

The Maple Application Center now has a series of sound synthesis and analysis applications that incorporate the new Maple 6 symbolic and numeric computation, procedural programming, plotting, and documentation features to demonstrate their advantages. To add a further dimension, there is a new Maple package that exports and imports sound files (.wav) so you can listen to the results of the analysis. To access these applications go to www.maplesoft.com/apps/ categories/engineering/electrical_electronic/html/signal.htm The applications take you through the use of Maple 6 for some basic signal processing applications. They demonstrate the use of Maple for exploring the mathematics behind the analyses, including the principles of signal generation, spectral analysis and handling of large data sets, and then demonstrating the principles through the use of sound (.wav) files. Finally, the applications show how to create Maple 6 packages, and how to optimise the performance of Maple 6 for purely numeric applications.

From Fourier to Moog, the Mathematics of Synthesisers Electronic music has come a long way since Robert Moog invented the first music synthesiser in the 1960s. In the early days of synthesised music, a keyboard player had to understand what VCOs, LFOs, ADSRs, and Modulation meant in order to use their instrument effectively. This application will shed some light on the mysteries of sound generation, in the practical context of the music synthesiser. However, most of the principles discussed can be used in many areas of signal processing, vibration analysis, and RF applications. The application uses generated sounds, that can be played through a standard multimedia package or plug-in, to demonstrate the principles discussed.

Development of Attack and Decay Functions for Sound Synthesis

This describes the use of Maple for defining the functions required to generate the attack and decay envelopes for a synthesiser. Using a combination of MapleÍs algebraic solving and programming, the application creates attenuation functions that give natural attack and curves in a single Attack, Sustain, Decay (ASD) function.

Creating Sound Sample Files

This application outlines the development process for saving and reading large files of sound sample data. Ultimately, it covers the creation of a Maple 6 package for reading and writing standard .wav sound files that can be generated and played by using most standard multimedia tools. However, there are techniques in this worksheet that can be applied to any large data set „ from the creation of ASCII text files through to signed integer binary files.

Spectral Analysis

In Example 1, arbitrary waveforms are created by using Fourier Series of sine waves in Maple. This application shows how Maple can be used to perform the reverse analysis. That is, import a signal and then extract the component sine waves. The tool for doing this is called the Fast Fourier Transform (FFT). This is an intensely numeric process and illustrates Maple's handling of numerics at the hardware level through the use of the evalhf() function and hfarrays. The hfarray is a special array type that enables you to handle numeric data at the hardware level, thus dramatically speeding up purely numeric operations.

Analysis and Synthesis of Sound Samples

This worksheet brings together all the elements discussed in the previous examples in a way that allows us to explore the nature of a real sound from a musical instrument, and extract the necessary data from it in order to synthesise this sound.

It shows you how to import a sound sample as a large data set into the Maple 6 worksheet environment, analyse the frequency spectrum of the sound, extract the dominant frequencies, and their amplitudes, from the spectrum data, and construct a synthesised sound, based on these components.

It concludes with a look at different ways of visualising the frequency spectrum through animations, waterfall plots, and spectrographs.

Maple 6 Packages: SigGen and WAV

Many of the routines developed to illustrate the concepts discussed in these applications have been combined into two Maple 6 packages. SigGen contains the functions used to generate the sound signals in many of these applications. WAV contains two functions: ReadWAV and WriteWAV. These allow you to export a dataset as a standard sound file (.wav) and import a sound file as a dataset in Maple 6, respectively.

These examples clearly demonstrate how Maple 6 can be used for very intensive numeric computation with large amounts of data. In fact, the choice of application is almost secondary - the use of sound files makes for an interesting topic that allows the use of multimedia to experience the results of the analysis, instead of viewing graphs or numbers. Besides which, itÍs great fun!

The important point of these examples is to demonstrate the computational power of Maple 6, particularly in the analysis of sound/noise/vibration/radio signals. Indeed, with these basic building blocks, it would be possible to develop some powerful analysis tools by simply connecting your instrumentation, such as accelerometers and displacement sensors, to the standard sound card on your computer. For additional material www.maplesoft.com/Papers/Writing_Efficient_Maple_Code.htm Tips on writing efficient Maple code