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Fields and Complex Numbers

Module by: Stephen Kruzick. E-mail the author

Based on: Complex Numbers by Don Johnson

Summary: An introduction to fields and complex numbers.

Fields

In order to propely discuss the concept of vector spaces in linear algebra, it is necessary to develop the notion of a set of “scalars” by which we allow a vector to be multiplied. A framework within which our concept of real numbers would fit is desireable. Thus, we would like a set with two associative, commutative operations (like standard addition and multiplication) and a notion of their inverse operations (like subtraction and division). The mathematical algebraic construct that addresses this idea is the field. A field (S,+,*)(S,+,*) is a set SS together with two binary operations ++ and ** such that the following properties are satisfied.

  1. Closure of S under ++: For every x,ySx,yS, x+ySx+yS.
  2. Associativity of S under ++: For every x,y,zSx,y,zS, (x+y)+z=x+(y+z)(x+y)+z=x+(y+z).
  3. Existence of ++ identity element: There is a e+Se+S such that for every xSxS, e++x=x+e+=xe++x=x+e+=x.
  4. Existence of ++ inverse elements: For every xSxS there is a ySyS such that x+y=y+x=e+x+y=y+x=e+.
  5. Commutativity of S under ++: For every x,ySx,yS, x+y=y+xx+y=y+x.
  6. Closure of S under **: For every x,ySx,yS, x*ySx*yS.
  7. Associativity of S under **: For every x,y,zSx,y,zS, (x*y)*z=x*(y*z)(x*y)*z=x*(y*z).
  8. Existence of ** identity element: There is a e*Se*S such that for every xSxS, e*+x=x+e*=xe*+x=x+e*=x.
  9. Existence of ** inverse elements: For every xSxS with xe+xe+ there is a ySyS such that x*y=y*x=e*x*y=y*x=e*.
  10. Commutativity of S under **: For every x,ySx,yS, x*y=y*xx*y=y*x.
  11. Distributivity of ** over ++: For every x,y,zSx,y,zS, x*(y+z)=xy+xzx*(y+z)=xy+xz.

While this definition is quite general, the two fields used most often in signal processing, at least within the scope of this course, are the real numbers and the complex numbers, each with their typical addition and multiplication operations.

The Complex Field

The reader is undoubtedly already sufficiently familiar with the real numbers with the typical addition and multiplication operations. However, the field of complex numbers with the typical addition and multiplication operations may be unfamiliar to some. For that reason and its importance to signal processing, it merits a brief explanation here.

Definitions

The notion of the square root of -1 -1 originated with the quadratic formula: the solution of certain quadratic equations mathematically exists only if the so-called imaginary quantity -1 -1 could be defined. Euler first used ii for the imaginary unit but that notation did not take hold until roughly Ampère's time. Ampère used the symbol ii to denote current (intensité de current). It wasn't until the twentieth century that the importance of complex numbers to circuit theory became evident. By then, using ii for current was entrenched and electrical engineers now choose jj for writing complex numbers.

An imaginary number has the form jb=b2 j b b 2 . A complex number, zz, consists of the ordered pair (aa,bb), aa is the real component and bb is the imaginary component (the jj is suppressed because the imaginary component of the pair is always in the second position). The imaginary number jb jb equals (00,bb). Note that aa and bb are real-valued numbers.

Figure 1 shows that we can locate a complex number in what we call the complex plane. Here, aa, the real part, is the xx-coordinate and bb, the imaginary part, is the yy-coordinate.

Figure 1: A complex number is an ordered pair (aa,bb) that can be regarded as coordinates in the plane. Complex numbers can also be expressed in polar coordinates as rθ rθ .
The Complex Plane
The Complex Plane (complex.png)
From analytic geometry, we know that locations in the plane can be expressed as the sum of vectors, with the vectors corresponding to the xx and yy directions. Consequently, a complex number zz can be expressed as the (vector) sum z=a+jb z a j b where jj indicates the yy-coordinate. This representation is known as the Cartesian form of zz. An imaginary number can't be numerically added to a real number; rather, this notation for a complex number represents vector addition, but it provides a convenient notation when we perform arithmetic manipulations.

The real part of the complex number z=a+jb z a j b , written as Rez z , equals aa. We consider the real part as a function that works by selecting that component of a complex number not multiplied by jj. The imaginary part of zz, Imz z , equals bb: that part of a complex number that is multiplied by jj. Again, both the real and imaginary parts of a complex number are real-valued.

The complex conjugate of zz, written as z* z , has the same real part as zz but an imaginary part of the opposite sign.

z=Rez+jImz z*=RezjImz z z j z z z j z
(1)

Using Cartesian notation, the following properties easily follow.

  • If we add two complex numbers, the real part of the result equals the sum of the real parts and the imaginary part equals the sum of the imaginary parts. This property follows from the laws of vector addition. a1+jb1+a2+jb2=a1+a2+j(b1+b2) a1 j b1 a2 j b2 a1 a2 j b1 b2 In this way, the real and imaginary parts remain separate.
  • The product of jj and a real number is an imaginary number: ja j a . The product of jj and an imaginary number is a real number: j(jb)=b j j b b because j2=1 j 2 1 . Consequently, multiplying a complex number by jj rotates the number's position by 9090 degrees.

Exercise 1

Use the definition of addition to show that the real and imaginary parts can be expressed as a sum/difference of a complex number and its conjugate. Rez=z+z*2 z z z 2 and Imz=zz*2j z z z 2 j .

Solution

z+z*=a+jb+ajb=2a=2Rez z z a j b a j b 2 a 2 z . Similarly, zz*=a+jb(ajb)=2jb=2jImz z z a j b a j b 2 j b 2 j z

Complex numbers can also be expressed in an alternate form, polar form, which we will find quite useful. Polar form arises arises from the geometric interpretation of complex numbers. The Cartesian form of a complex number can be re-written as a+jb=a2+b2(aa2+b2+jba2+b2) a j b a 2 b 2 a a 2 b 2 j b a 2 b 2 By forming a right triangle having sides aa and bb, we see that the real and imaginary parts correspond to the cosine and sine of the triangle's base angle. We thus obtain the polar form for complex numbers. z=a+jb=rθ r=|z|=a2+b2 a=rcosθ b=rsinθ θ=arctanba z a jb r θ r z a 2 b 2 a r θ b r θ θ b a The quantity rr is known as the magnitude of the complex number zz, and is frequently written as |z| z . The quantity θθ is the complex number's angle. In using the arc-tangent formula to find the angle, we must take into account the quadrant in which the complex number lies.

Exercise 2

Convert 32j 3 2 j to polar form.

Solution

To convert 32j 3 2 j to polar form, we first locate the number in the complex plane in the fourth quadrant. The distance from the origin to the complex number is the magnitude rr, which equals 13=32+22 13 3 2 2 2 . The angle equals arctan23 2 3 or -0.588-0.588 radians (33.733.7 degrees). The final answer is 13(33.7) 13 33.7 degrees.

Euler's Formula

Surprisingly, the polar form of a complex number zz can be expressed mathematically as

z=rejθ z r j θ
(2)
To show this result, we use Euler's relations that express exponentials with imaginary arguments in terms of trigonometric functions.
ejθ=cosθ+jsinθ j θ θ j θ
(3)
cosθ=ejθ+e(jθ)2 θ j θ jθ 2
(4)
sinθ=ejθe(jθ)2j θ j θ jθ 2 j The first of these is easily derived from the Taylor's series for the exponential. ex=1+x1!+x22!+x33!+ x 1 x 1 x 2 2 x 3 3 Substituting jθ j θ for xx, we find that ejθ=1+jθ1!θ22!jθ33!+ j θ 1 j θ 1 θ 2 2 j θ 3 3 because j2=-1 j 2 -1 , j3=j j 3 j , and j4=1 j 4 1 . Grouping separately the real-valued terms and the imaginary-valued ones, ejθ=1θ22!++j(θ1!θ33!+) j θ 1 θ 2 2 j θ 1 θ 3 3 The real-valued terms correspond to the Taylor's series for cosθ θ , the imaginary ones to sinθ θ , and Euler's first relation results. The remaining relations are easily derived from the first. Because of (Reference), we see that multiplying the exponential in Equation 3 by a real constant corresponds to setting the radius of the complex number by the constant.

Calculating with Complex Numbers

Adding and subtracting complex numbers expressed in Cartesian form is quite easy: You add (subtract) the real parts and imaginary parts separately.

z 1 ± z 2 =( a 1 ± a 2 )+j( b 1 ± b 2 ) ± z 1 z 2 ± a 1 a 2 j ± b 1 b 2
(5)
To multiply two complex numbers in Cartesian form is not quite as easy, but follows directly from following the usual rules of arithmetic.
z 1 z 2 =( a 1 +j b 1 )( a 2 +j b 2 )= a 1 a 2 b 1 b 2 +j( a 1 b 2 + a 2 b 1 ) z 1 z 2 a 1 j b 1 a 2 j b 2 a 1 a 2 b 1 b 2 j a 1 b 2 a 2 b 1
(6)
Note that we are, in a sense, multiplying two vectors to obtain another vector. Complex arithmetic provides a unique way of defining vector multiplication.

Exercise 3

What is the product of a complex number and its conjugate?

Solution

zz*=(a+jb)(ajb)=a2+b2 z z a j b a j b a 2 b 2 . Thus, zz*=r2=|z|2 z z r2 z 2 .

Division requires mathematical manipulation. We convert the division problem into a multiplication problem by multiplying both the numerator and denominator by the conjugate of the denominator.

z1z2=a1+jb1a2+jb2=a1+jb1a2+jb2a2jb2a2jb2=( a 1 +j b 1 )( a 2 j b 2 ) a 2 2+ b 2 2= a 1 a 2 + b 1 b 2 +j( a 2 b 1 a 1 b 2 ) a 2 2+ b 2 2 z1 z2 a1 j b1 a2 j b2 a1 j b1 a2 j b2 a2 j b2 a2 j b2 a 1 j b 1 a 2 j b 2 a 2 2 b 2 2 a 1 a 2 b 1 b 2 j a 2 b 1 a 1 b 2 a 2 2 b 2 2
(7)
Because the final result is so complicated, it's best to remember how to perform division—multiplying numerator and denominator by the complex conjugate of the denominator—than trying to remember the final result.

The properties of the exponential make calculating the product and ratio of two complex numbers much simpler when the numbers are expressed in polar form.

z 1 z 2 = r 1 ej θ 1 r 2 ej θ 2 = r 1 r 2 ej( θ 1 + θ 2 ) z 1 z 2 r 1 j θ 1 r 2 j θ 2 r 1 r 2 j θ 1 θ 2
(8)
z 1 z 2 = r 1 ej θ 1 r 2 ej θ 2 = r 1 r 2 ej( θ 1 θ 2 ) z 1 z 2 r 1 j θ 1 r 2 j θ 2 r 1 r 2 j θ 1 θ 2 To multiply, the radius equals the product of the radii and the angle the sum of the angles. To divide, the radius equals the ratio of the radii and the angle the difference of the angles. When the original complex numbers are in Cartesian form, it's usually worth translating into polar form, then performing the multiplication or division (especially in the case of the latter). Addition and subtraction of polar forms amounts to converting to Cartesian form, performing the arithmetic operation, and converting back to polar form.

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