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Quadrature Signals: Complex, But Not Complicated

by Richard Lyons

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 numbers and their strange

terminology of j-operator, complex, imaginary, real, and orthogonal. If you're a little unsure

of the physical meaning of complex numbers and the j =

-1

operator, don't feel bad because

you're in good company. Why even Karl Gauss, one the world's greatest mathematicians,

called the j-operator the "shadow of shadows". Here we'll shine some light on that shadow so

you'll never have to call the Quadrature Psychic Hotline for help.

Quadrature signal processing is used in many fields of science and engineering, and

quadrature signals are necessary to describe the processing and implementation that takes

place in modern digital communications systems. In this tutorial we'll review the

fundamentals of complex numbers and get comfortable with how they're 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 three-dimensional 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.

These applications fall in the general category known as quadrature processing, and they

provide additional processing power through the coherent measurement of the phase of

sinusoidal signals.

A quadrature signal is a two-dimensional signal whose value at some instant in time can be

specified by a single complex number having two parts; what we call the real part and the

imaginary part. (The words real and imaginary, although traditional, are unfortunate because

their of meanings in our every day speech. Communications engineers use the terms in-phase

and quadrature phase. More on that later.) Let's review the mathematical notation of these

complex numbers.

The Development and Notation of Complex Numbers

To establish our terminology, 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 one-dimensional numbers can be either positive or negative as depicted in

Figure 1(a). In that figure we show a one-dimensional axis and say that a single real number

can be represented by a point on that axis. Out of tradition, let's call this axis, the Real axis.

(b)

This point represents

the complex number

c = 2.5 + j2

Real

axis

2.5

Imaginary

axis (j)

+2.5 line

+j2 line

This point represents the

real number a = -2.2

01234-5 -4 -3 -1-2

(a)

Real

axis

Figure 1. An graphical interpretation of a real number and a complex number.

A complex number, c, is shown in Figure 1(b) where it's also represented as a point.

However, complex numbers are not restricted to lie on a one-dimensional line, but can reside

anywhere on a two-dimensional plane. That plane is called the complex plane (some

mathematicians like to call it an Argand diagram), and it enables us to represent complex

numbers having both real and imaginary parts. For example in Figure 1(b), the complex

number c = 2.5 + j2 is a point lying on the complex plane on neither the real nor the

imaginary axis. We locate point c by going +2.5 units along the real axis and up +2 units

along the imaginary axis. Think of those real and imaginary axes exactly as you think of the

East-West and North-South directions on a road map.

We'll use a geometric viewpoint to help us understand some of the arithmetic of complex

numbers. Taking a look at Figure 2, we can use the trigonometry of right triangles to define

several different ways of representing the complex number c.

c = a + jb

Real

axis

Imaginary

axis (j)

Figure 2 The phasor representation of complex number c = a + jb on the complex plane.

Our complex number c is represented in a number of different ways in the literature, such as:

jφ

= cos(φ) + jsin(φ)(7)

The suspicious reader should now be asking, "Why is it valid to represent a complex number

using that strange expression of the base of the natural logarithms, e, raised to an imaginary

power?" 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

in the top line of

Figure 3. That substitution is shown on the second line. Next we evaluate the higher orders of j

to arrive at the series in the third line in the figure. Those of you with elevated math skills like

Euler (or those who check some math reference book) will recognize that the alternating terms in

the third line are the series expansion definitions of the cosine and sine functions.

= 1 + z +

+ ...

jφ

= 1 + jφ +

(jφ)

+ ...

jφ

= 1 + jφ -

- j

+ j

+ ...

jφ

= cos(φ) + jsin(φ)

Figure 3 One derivation of Euler's equation using series expansions for e

, cos(φ), and sin(φ).

Figure 3 verifies Eq. (7) and our representation of a complex number using the Eq. (3) polar

form: Me

jφ

. If you substitute -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:

-jφ

= cos(φ) - jsin(φ)(8)

The polar form of Eqs. (7) and (8) benefits us because:

- It simplifies mathematical derivations and analysis,

-- turning trigonometric equations into the simple algebra of exponents, and

-- math operations on complex numbers follow exactly the same rules as real numbers.

- It makes adding signals merely the addition of complex numbers (vector addition),

- It's the most concise notation,

- It's indicative of how digital communications system are implemented, and described in

the literature.

We'll be using Eqs. (7) and (8) to see why and how quadrature signals are used in digital

communications applications. But first, let’s take a deep breath and enter the Twilight Zone

of that 'j' operator.

You've seen the definition j =

-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

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linfeng97521

2014-05-26

学习资源，不错的文章，就是贵了。

深刻理解正交信号英文原著容易理解力

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