Appendix ASliding Correlators, Delay-BasedDiscriminators, and Processing Gainwith GPS ApplicationsThis appendix is meant to introduce the reader to the principles of sliding correlatorsand how they are used to create a delay discriminator. Additionally, the spreadspectrum term “processing gain” is discussed. Although there are many other typesof correlators, here we will focus on the sliding type. Before we dive right in on thesliding correlator, we need some intro material on the correlation operation.One of the key operations that distinguish the GPS receiver from classic narrowband receivers is the use of a correlator. The correlation process in the GPS receiveris used to align the replica C/A code with the transmitted C/A code. Additionally,this results in the recovery of timing signals that are ultimately used in the receiverTOA measurement process. But what is correlation and how does it work?Fundamentally, correlation is a statistical process, that is, it is related to averagesand probabilities. We intuitively know that when we roll a pair of dice, the outcomefrom one throw to another is not correlated, that is, the previous throws have noeffect on the subsequent throws.When two events are in some way interrelated suchthat the outcome of one affects the other, we could say that the events have somesort of correlation.There is another interpretation of correlation and that is as a measure ofsimilarity. Particularly in electronics, it would be desirable to compare varioustime signals to one another and see if they have anything in common. By havingsuch a tool, it should be possible to determine quantitatively how much and wheretwo signals are correlated and where they are not correlated. It is this interpretationof correlation that is used in the GPS receiver correlation process.Finding the point in time where two signals are similar, or in GPS receiver wherethe receivers replica C/A code is lined up with the received C/A code from thesatellite, is the prime reason we need to understand the correlation process. We willsee that correlation allows to us determine, to a very high degree of accuracy, whenwe have C/A code alignment.Before we discuss correlation in more depth, we need to examine briefly aconcept called “Time Average Value” of a time varying voltage or current signal.D. Doberstein, Fundamentals of GPS Receivers: A Hardware Approach,DOI 10.1007/978-1-4614-0409-5, # Springer Science+Business Media, LLC 2012271Time AveragingWhat is the “time averaging value” function? Let us look at the waveforms inFig. A.1. Figure A.1a shows a sine wave that has a maximum value of 1 V and aminimum value of ?1 V. Its time average is zero volts. The reason is that thewaveform spends as much time positive as it does negative, and the magnitude ofabcd+1V0VTIMEAverage. Value= 0.5v+5V0VTIMEAverage. Value= 2.5v+1V-1V0VTIMEAverage. Value= 0v+1V-1V0VTIMEAverage. Value= 0vFig. A.1 Average value of time waveforms272Appendix Athese positiveand negativeexcursions isidentical. The waveform shown in Fig. A.1bis a square wave that goes from 0 to 1 V. The average value would be 0.5 V.Waveform A1c is a sine wave that goes from zero volts to 5 V maximum. Theaverage value of this waveform is 2.5 V. Waveform A1d is a small segment of arandombinarybit sequence, withvalues 1 or?1.Ifthe sequenceistruly “random”and of large length, the average value will be very close to zero. It should be apparentthat the time average function of a waveform is the DC value of the waveform. If onehad a perfect DC voltmeter (reads true independent of waveform type) and appliedthe waveforms of Fig. A.1 to it, it would read the average value. In lieu of the DCmeter, an analog power meter can be used. The total power minus the AC coupledpower is equivalent to measuring the average or DC value (if properly scaled).After studying various waveforms and the time average function, the readershould be able to draw a line on the waveform indicating the approximate averageor DC value. It is hoped that the reader will develop an intuitive feel for the timeaverage function and be able to approximate this value for most commonwaveforms.Correlation, the Mathematical StatementFor the readers who are familiar with convolution, correlation is a closely relatedoperation. Convolution and correlation both use shifting, multiplying, and integra-tion operations on time waveforms (typically). This discussion is restricted to timewaveforms only. The mathematical formula for correlation of two time signals iszt xt ? ytt dttcovers ? 1to 1;(A.1)where x(t) and y(t) are time waveforms for the purposes of this discussion.The variable t is the time shift applied to y(t) and the variable of integration.z(t) is the correlation waveform that we seek.As usual, writing such equations down does little to inform our intuition on whatis really happening! But the math is the exact model that we will attempt to executein the imperfect world of electronic circuits.We need to break down (A.1) int