Quadrature Signals: Compex, But …

Yesterday, I read a paper about Complex in digital communicationsystem. I thought that was a good paper to comprehend the complexmeaning in engineering. So I copied some points from this paper andmade it briefer.

Introduction :

Quadrature signals are based on the notion of complex numbersand perhaps no other topic causes more heartache for newcomers toDSP than these number and their strange terminology of j-operator,complex, imaginary, real, and orthogonal. Why even Karl Gauss, onethe world's greatest mahematicians, called the j-operator the"shadow of shadows". Here we'll shine some light on that shadow soyou will never have to call the Quadrature Signal Psychic Hotlinefor help.

Quadrature signal processing is uesd in many fields of scienceand engineering, and quadrature signals are necessary to describethe processing and implementation that takes place in moderndigaital communication systems. In this tutorial we will review thefundamentals of complex numbers and get comfortable with how theyare used to represent quadrature signals. Next we examine thenotion of negative frequency as it relates to quadrature signalalgebraic notation, and learn to speak the language of quadratureprocessing. In addition, we'll use 3-dimentsional time andfrequency-domain plots to give some physical meaning to quadraturesignals. This tutorial concludes with a brief look at how aquadrature signal can be generated by means ofquadrature-sampling.

 

Why Care About Quadrature Signals ?

Quadrature signal formats, also called complex signals, are usedin many digital signal processing applications such as :

- digital communications systems,

- radar systems,

- time difference of arrival processing in radio directionfinding schemes,

- coherent pulse measurement systems,

- antenna beamforming applications,

- single sideband modulators,

- etc.

 

A quadrature signal is a 2-dimentsional signal whose value atsome instant in time can be specified by a single complex numberhaving two parts; what we call real part and the imaginarypart.

 

The Development and Notation of Complex Numbers

We define a real number to be those numbers we use in every daylife, like a voltage, a temperature on the Fahrenheit scale, or thebalance of your checking account. These 1-dimensional numbers canbe either positive or negative as depicted in Figure 1(a). Acomplex number, c, is shown in Figure 1(b) where it's alsorepresented as a point.

 

We'll use a geometric viewpoint to help us understand some ofthe arithmetic of complex numbers. Take a look at Figure 2, we canuse the trigonometry of right triangles to define complex numberc.

Compex number c is represented in a number of different ways in theliterature, such as :

Eqs.(3)and(4) remind us c is a complex number and the variables a,b, M and Φare all real numbers. The magnitude of c, sometimescalled the modulus of c, is

The phase angle Φ, orargument, is the arctangent of theradio  or

We can validate Eq.(7) as did the world's greatest expert oninfinite series, Herr Leonard Euler, by plugging jΦ in for z in theseries expansion definition of e^z in the top line of Figure 3.Those of you with elevated math skills linke Euler will recognizethat alternating terms in the third line are the series expansiondefinitions of the cosine and sine functions.

So if you substitue -jΦ for z in the top line of Figure 3, you'dend up with a slightly different, and very useful, form of Euler'sidentity :

You've seen the definition j = sqrt(-1)before. Stated in words, wesay that j represents a number when multiplied by itself results ina negative one. Well, this definition causes difficulty forbeginner because we all know that any number multiplied by itselfalways results in a positive number.

Here's the point to remember. If you have a single complexnumber, represented by a point on the complex plane, multiplyingthat number by j or by e^(jπ/2) will result in a new complex numberthat's rotated 90 counterclockwise(CCW) on the complex plane. Don'tforget this, as it will be useful as you begin reading theliterature of quadrature processing systems!

Represnting Real Signals using ComplexPhasors

Let's now call our two e^(j2πfot)and e^(-j2πfot) complex expressions quadraturesignals.

Thinking about these phasors, it's clear now why the cosinewave can be equated to the sum of two compex exponentials by

Representing Quadrature Signals In the Frequency Domain

Figure 8 tells us the rules for representing complexexponentials in the frequency domain.

If you understand the notation and operations in Figure 10,pat yourself on the back because you know a great deal about natureand mathematics of quadrature signals.

Bandpass Quadrature Signals In the Frequency Domain

A Quadrature-Sampling Example

The complex spectrum at the bottom Figure 18 shows what we wanted;a digitized version of the compelx bandpass signal centered aboutzero Hz.

Conclusions:

This ends our little quadrature signals tutorial. We learnedthat using the complex plane to visualize the mathematicaldescriptions of complex numbers enabled us to see how quadratureand real signals are related. We say how three-dimentionalfrequency-domain depictions help us understand how quadraturesignals are generated, translated in frequency, combined, andseparated. Finally we reviewed an example of quadrature-samplingand two schemes for inverting the spectrum of quadraturesequence.