A Review of Complex Numbers la Pollak ECE 30 Signals and Sstems Section, Fall 00 Purdue Universit A complex number is represented in the form z = x + j, where x and are real numbers satisfing the usual rules of addition and multiplication, and the smbol j, called the imaginar unit, has the propert j =. The numbers x and are called the real and imaginar part of z, respectivel, and are denoted b x = R(z), = (z). We sa that z is real if = 0, while it is purel imaginar if x = 0. Example 0.. The complex number z = 3 + j has real part 3 and imaginar part, while the real number 5 can be viewed as the complex number z = 5+0j whose real part is 5 and imaginar part is 0. Geometricall, complex numbers can be represented as vectors in the plane (Fig. ). We will call the x-plane, when viewed in this manner, the complex plane, with the x-axis designated as the real axis, and the -axis as the imaginar axis. We designate the complex number zero as the origin. Thus, x + j = 0 means x = = 0. n addition, since two points in the plane are the same if and onl if both their x- and -coordinates agree, we can define equalit of two complex numbers as follows: x + j = x + j means x = x and =. z R θ x R Figure. A complex number z can be represented in Cartesian coordinates (x, ) or polar coordinates (R, θ).
Thus, we see that a single statement about the equalit of two complex quantities actuall contains two real equations. Definition 0. (Complex Arithmetic). Let z = x + j and z = x + j. Then we define: (a) z ± z = (x ± x ) + j( ± ); (b) z z = (x x ) + j(x + x ); (c) for z 0, w = z z is the complex number for which z = z w. Note that, instead of the Cartesian coordinates x and, we could use polar coordinates to represent points in the plane. The polar coordinates are radial distance R and angle θ, as illustrated in Fig.. The relationship between the two sets of coordinates is: x = R cos θ, = R sinθ, R = x + = z, ( θ = arctan. x) Note that R is called the modulus, or the absolute value of z, and it alternativel denoted z. Thus, the polar representation is: z = z cos θ + j z sin θ = z (cos θ + j sin θ). Definition 0. (Complex Exponential Function). The complex exponential function, denoted b e z, or exp(z), is defined b e z = e x+j = e x (cos + j sin ). n particular, if x = 0, we have Euler s equation: e j = cos + j sin. Comparing this with the terms in the polar representation of a complex variable, we see that an complex variable can be written as: z = z e jθ.
3 z = z z z = z z θ = θ + θ z θ θ x R Figure. Multiplication of two complex numbers z = z e jθ and z = z e jθ (z is not shown). The result is z = z e jθ with z = z z and θ = θ + θ. Properties of Complex Exponentials. cos θ = (ejθ + e jθ ), sin θ = j (ejθ e jθ ), e jθ =, e z e z = e z +z, e z = e z, e z+πjn = e x (cos( + πn) + j sin( + πn)) = e x (cos + j sin ) = e z, for an integer n. DT complex exponential functions whose frequencies differ b π are thus identical: e j(ω+π)n = e jωn+πjn = e jωn. We have seen examples of this phenomenon before, when we discussed DT sinusoids. t follows from the multiplication rule that z z = z e jθ z e jθ = z z e j(θ +θ ). Therefore, in order to multipl two complex numbers, add the angles; multipl the absolute values. Multiplication of two complex numbers is illustrated in Fig.. Definition 0.3 (Complex Conjugate). f z = x + j, then the complex conjugate of z is z = x j (sometimes also denoted z).
4 z = j z = x + j x R z = + j, z = θ = π/4 R /z = / j/ z = x j z = j (a) (b) Figure 3. (a) Complex number z and its complex conjugate z. (b) llustrations to Example 0.. This definition is illustrated in Fig. 3(a). Note that, if z = z e jθ, then z = z e jθ. Here are some other useful identities involving complex conjugates: R(z) = (z + z ), (z) = j (z z ), z = zz, (z ) = z, z = z z is real, (z + z ) = z + z, (z z ) = z z. Example 0.. Let us compute the various quantities defined above for z = + j.. z = j.. z = + =. Alternativel, z = zz = ( + j)( j) = + j j j =. 3. R(z) = (z) =. 4. Polar representation: z = (cos π 4 + j sin π 4 ) = e j π 4.
5 5. To compute z, square the absolute value and double the angle: z = (cos π + j sin π ) = j = ej π. The same answer is obtained from the Cartesian representation: ( + j)( + j) = + j + j = + j = j. 6. To compute /z, multipl both the numerator and the denominator b z : z = z zz = j ( + j)( j) = j Alternativel, use the polar representation: = j. z = e j π 4 = e j π 4 = ( (cos π ) ( + j sin π )) ( = 4 4 j ) = j. We can check to make sure that (/z) z = : These computations are illustrated in Fig. 3(b). ( j )(+j) = j + j j =.