Google Search Adds Graphing Calculator

Google will now show a graph of a function entered in the search box.  Here is an example for the sinc function.

The official google search blog gives more details on the new graphing feature.

If you want more than a quick graph, I recommend Wolfram Alpha.  Here is the output for the same function.

Here is my original post on Wolfram Alpha.

TI Releases Free DSP Software Development Tool

TI is providing DSP developers C6EZFlo, a free graphical tool for use with C6000 DSPs.  You can find more information at the TI C6Flo Wiki, including and introduction, DSP block descriptions, and instructions to extend C6Flo by creating  your own blocks using C (and a little bit of Javascript).

National Instruments/Tektronix Embedded Seminars

Measurement and Analysis Techniques for Embedded System Designers will be presented by National Instruments and Tektronix in 10 U.S. cities and Toronto starting March 2010. The half day session will include talks on automated control of measurements, advanced analysis tools, and examples of enhancing embedded system testing. They will showcase Tektronix test equiptment and National Instruments LabVIEW and LabVIEW SignalExpress software. Both analog and digital techniques will be presented.

Wolfram Alpha

The makers of Mathematica have created a website to "make it possible to compute whatever can be computed about anything." They need some help at the moment and will give you a link to participate if you find something the site is missing like digital filter, frequency spectrum, Hamming window, or impulse function.

Wolfram Alpha is currently more powerful than using Google's built in calculator but I find the interface to be cumbersome. If you perform more than one calculation the history function can interfere with your text input and you lose the focus from the text entry box after each result.

It does look useful for learning about mathematical functions, try e^pi*x for example, you can see a quick graph and they give a lot of useful formulae like series representations and integrals.

Scilab for Embedded

 Scilab is an open-source mathematical package similar to Matlab. These applications are commonly used in embedded development to prototype DSP or control algorithms on a PC before porting them to the target hardware. There are several embedded projects using scilab.
"hArtes aims to lay the foundation for a new holistic (end-to-end) approach for complex real-time embedded system design, with the latest algorithm exploration tools and reconfigurable hardware technologies."
The Evidence Scilab/Scicos code generator uses a graphical interface to program a Microchip dsPIC DSC micro-controller board.

Check Your Math

How well do you trust the basic mathematical operations of your processor? Subtle math errors in code can allow access secure data. Even if you are not implementing a security algorithm, potential problems from basic mathematical errors can cause results that would seem impossible. There are methods to help prevent these problems:

• Use asserts before any critical operations.
• Test important (security) results for consistency.
• Use a watchdog timer.
• Check processor errata sheets (both for development and production versions).
• Use mature, well tested math libraries.
• Write unit tests for the mathematical sections of the code.

This all adds another layer of complexity to embedded system design, but that cannot be avoided with the increasing complexity and security of modern systems. So make sure to check your math.

Adding Math to List of Security Threats - NY Times Math Error Could Compromise Cryptographic Systems - Information Week

Benford's Law

While updating my page of DSP links, I noticed a new chapter to The Scientist and Engineer's Guide to Digital Signal Processing about the solution to Benford's law. This phenomenon was first noticed in 1881 by Simon Newcomb, but the rediscovery by Frank Benford in 1938 gives it its name. Both noticed something odd about books containing tables of logarithms, pages with data with 1 as the first digit were worn more than pages with other digits. Benford found this type of pattern in a wide variety of data sets such as numbers in magazine articles and baseball statistics. It seemed that the data should be random and the first number would have an equal probability to be any digit between 1 and 9, or 11.1% for each. In fact, the pattern of leading digits fits a logarithmic distribution, with 1 having a 30.1% probability.

This phenomenon was observed but not understood until recently, and that gives it the connection with DSP. A proof can be constructed using signal analysis techniques, in particular convolution, the Fourier transform and homomorphic processing. See Steven Smith's new chapter for details, but you might not need mathematics to understand the mystery. According to Smith, the key is to realize that we are transforming the data when we extract the first digit. Since it so easy to see the first digit, we do not realize that we are performing any operation on the data. This operation is taking the anti-logarithm of the data, and that gives the logarithmic pattern to the leading digits. Wikipedia has a few other explanations as well, but I think there is a counterexample that is much easier to understand:

If we assume that a random event has a random leading digit, then we are saying that a number with a leading 1 has an 11% probability. That would be true if we had a single die with 9 sides. Any of the sides would correspond to a different leading digit, but what about a die with 20 sides? There are 11 digits that would start with 1 out of the 20 for a 55% probability of a 1. For an average 7 sided die, it would be 14%. In fact, I think you will find it would never be less than 11% for any number of sides, but we find many cases where it is greater than 11%. So for data that can have any value between 1 and N, the probability of a 1 must be greater than 11%.

This is not a complete proof, and even if true, the counter-example does not address events like the roll of two or more die, or fractional data, but it is enough to remove my initial disbelief in Bedford's law. If books with tables of numbers are worn the most on pages with data that has a leading 1, it could just be that the type of data being analyzed follows this 1 to N pattern. I would imagine that many types of data sets have this property.

For more information about Bedford's law I recommend the pages on Mathworld and Wikipedia. It was also featured in the episodeThe Running Man, of the TV show NUMB3RS.