## Bernard Widrow in Signal Processing

*Foreward: Widrow is highly respected for his accomplishments, and for his attention to detail in learning the mechanisms of error propagation. Insight Inc Founder’s methods for designing adaptive filters are different from Widrow’s methods because stock market price prediction dynamics have special requirements.*

In the arena of signal processing and modeling, Bernard Widrow has developed predictive algorithms for various cases of time-varying, nonstationary signals. He has discovered many sources of error which can affect prediction.

## An Interview with Widrow

Oral-History: Bernard Widrow – Engineering and Technology History Wiki

In the interview, Widrow describes **“quantization noise,”** a nonlinear problem in sampled data. Widrow studied this problem within the frequency domain (i.e., using Fourier transforms and LaPlace transforms).

In statistics, these two transforms are also known as **“characteristic functions.”** Widrow used probability density functions and the characteristic functions to explore the nonlinear error effects in great detail, step by step.

Interestingly, he found that he could use the sampling theorem to understand the quantization problem. In essence, the quantization noise is uniformly distributed and uncorrelated with the signal being quantized. In other words, a linear theory could be used to understand and explain something that’s nonlinear.

Thus, in the mathematical equations, Widrow replaced the quantizer with a source of additive noise. It is simple and elegant. That step makes everything linear, so one can examine how the noise propagates through the system. Widrow then, extends this fundamental idea to prediction applications, such as neural networks and adaptive filters. (Not quite price prediction)

## More Resources

- Widrow’s most recent Abstract reveals his great influence.
- Widrow and Kollár describe general quantization errors in the following work: Quantization Noise: Roundoff Error in Digital Computation, Signal Processing, …
- A list of papers and books by Widrow and collaborators.