Quadrature Signals: Compex, But Not Complicated note

Yesterday, I read a paper about Complex in digital communication system. I thought that was a good paper to comprehend the complex meaning in engineering. So I copied some points from this paper and made it briefer.

Introduction :

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

Quadrature signal processing is uesd in many fields of science and engineering, and quadrature signals are necessary to describe the processing and implementation that takes place in modern digaital communication systems. In this tutorial we will review the fundamentals of complex numbers and get comfortable with how they are used to represent quadrature signals. Next we examine the notion of negative frequency as it relates to quadrature signal algebraic notation, and learn to speak the language of quadrature processing. In addition, we'll use 3-dimentsional time and frequency-domain plots to give some physical meaning to quadrature signals. This tutorial concludes with a brief look at how a quadrature signal can be generated by means of quadrature-sampling.

Why Care About Quadrature Signals ?

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

- digital communications systems,

- radar systems,

- time difference of arrival processing in radio direction finding schemes,

- coherent pulse measurement systems,

- antenna beamforming applications,

- single sideband modulators,

- etc.

A quadrature signal is a 2-dimentsional signal whose value at some instant in time can be specified by a single complex number having two parts; what we call real part and the imaginary part.

The Development and Notation of Complex Numbers

We define a real number to be those numbers we use in every day life, like a voltage, a temperature on the Fahrenheit scale, or the balance of your checking account. These 1-dimensional numbers can be either positive or negative as depicted in Figure 1(a). A complex number, c, is shown in Figure 1(b) where it's also represented as a point.

 

 

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

Compex number c is represented in a number of different ways in the literature, 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, sometimes called the modulus of c, is

 

The phase angle Φ, or argument, is the arctangent of the radio  or

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

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

 

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

Here's the point to remember. If you have a single complex number, represented by a point on the complex plane, multiplying that number by j or by e^(jπ/2) will result in a new complex number that's rotated 90 counterclockwise(CCW) on the complex plane. Don't forget this, as it will be useful as you begin reading the literature of quadrature processing systems!

 

Represnting Real Signals using Complex Phasors

 

 

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

Thinking about these phasors, it's clear now why the cosine wave 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 complex exponentials 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 nature and 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 about zero Hz.

Conclusions:

This ends our little quadrature signals tutorial. We learned that using the complex plane to visualize the mathematical descriptions of complex numbers enabled us to see how quadrature and real signals are related. We say how three-dimentional frequency-domain depictions help us understand how quadrature signals are generated, translated in frequency, combined, and separated. Finally we reviewed an example of quadrature-sampling and two schemes for inverting the spectrum of quadrature sequence.