Comparing The Walsh Domain To The Fourier Domain With A Labview Based Communication Systems Toolkit

Abstract

Although the Fourier Transform is the traditional frequency domain analysis tool in communications systems, other transforms are pointed out in the context of orthogonal series representation of periodic signals. Last year, we became interested in the Walsh Transform and developed virtual instruments (VIs) to compute the Walsh transform, to generate the Walsh basis functions and modified LabVIEW’s natural ordered fast Walsh Transform (FWT) routine to provide Walsh ordered Walsh transforms and a recent publication1 reported on this expansion of the Communication Systems Toolkit into the Walsh domain. This paper will describe the utilization of these most recent tools in order to compare the Walsh domain to the Fourier domain. We will compare the basis functions in each transformation and demonstrate similarities and differences between FFT and FWT. We will then propose a new arrangement of the FWT sequency plots that will correspond to the magnitude spectrum plots obtained by the FFT. We will conclude by a summary of the student responses to exercises comparing these two transform methods.

I. Introduction This paper follows recent papers that describe a simulation toolkit for communication systems2, its reception by students at two different institutions3 and its utilization in undergraduate student research4. In those papers we stated that in the absence of hardware that would reinforce the theoretical presentation, computer simulations of the systems described in class are the next available tools to bring these concepts to life. Those papers also describe the particular class environment and the process in which the software development tool, namely LabVIEW, was chosen. Although MATLAB is the standard software tool employed in the areas of signals and systems, as evidenced by the proliferation of books devoted to MATLAB based exercises in those subjects, the choice of the software tool is justified in several previous publications5, 2, 6.

This paper will report on the results of a term project carried out in ELE 402, Introduction to Communications Engineering class. In ELE 402, Fourier series expansion is presented in the context of orthogonal series representation of signals and noise. We define orthogonal functions over an interval, discuss how an arbitrary waveform may be expanded in a series of these orthogonal functions and present the various forms of the Fourier series as a particular type of orthogonal series whose basis functions are sinusoids or complex exponential functions. We mention, in passing, that there are other sets of orthogonal functions that may be employed to expand our waveform functions. In the Fall ’05 offering of ELE 402, we mentioned Walsh transforms in this context and one student decided to incorporate Walsh Transforms into the toolkit to provide an alternative example to orthogonal series representation of signals. This paper will describe how Walsh transforms were incorporated into the Communication Systems Toolkit and how the toolkit was used to demonstrate Walsh transforms in the Fall ’06 offering of ELE 402. Section 2 will provide a background for Walsh Transforms and section 3 will describe