Digital Signal Processing
Moslem Amiri, Václav Přenosil
Masaryk University
Understanding Digital Signal Processing, Third Edition, Richard Lyons
(0-13-261480-4) © Pearson Education, 2011.
The Arithmetic of Complex
Numbers
2
Graphical Representation of Real and Complex Numbers
 Real number
 Can be represented by a point on a onedimensional
axis, called real axis
3
Graphical Representation of Real and Complex Numbers
 Complex number
 Has two parts: a real part and an imaginary part
 Can be treated as a point on a complex plane
4
Arithmetic Representation of Complex Numbers
 A complex number C is represented in a
number of different ways
 Rectangular form
 Trigonometric form
 Exponential form
 Magnitude and angle form
jIRC +=
)]sin()[cos( φφ jMC +=
φj
MeC =
φ∠= MC
5
Arithmetic Representation of Complex Numbers
 Magnitude (modulus) of C
 Phase angle (argument) of C
 In exponential form
 Variable ø need not be constant
 A phasor of magnitude M that rotates in a
(counter)clockwise direction at a radian frequency
of (+ω) –ω radians per second
22
IRCM +==






= −
R
I1
tanφ
)2( njj
MeMeC πφφ +
==
tjtj
MeCMeC ωω −
== or
6
Arithmetic Operations of Complex Numbers
 Addition and subtraction
 Rectangular form is the easiest to use here
)()()(
)(
2121221121
2121221121
IIjRRjIRjIRCC
IIjRRjIRjIRCC
−+−=+−+=−
+++=+++=+
7
Arithmetic Operations of Complex Numbers
8
Arithmetic Operations of Complex Numbers
 Multiplication
 Can use rectangular form to multiply
 In exponential form, product takes simpler form
 Product of magnitudes of two complex numbers
 Scaling
)()())(( 12212121221121 IRIRjIIRRjIRjIRCC ++−=++=
)(
212121
2121 φφφφ +
== jjj
eMMeMeMCC
2121 CCCC =⋅
φφ jj
kMeMek
jkIkRjIRkkC
==
+=+=
)(
)(
9
Arithmetic Operations of Complex Numbers
 Conjugation
 Complex conjugate of a complex number is
obtained by changing sign of its imaginary part
 Conjugate of a product = product of conjugates
 Sum of conjugates = conjugate of the sum
φφ jj
MejIRCMejIRC −∗
=−=→=+=
∗∗−−
+−∗+∗∗
==
===
=
2121
)(
21
)(
2121
21
21
2121
)()(
CCeMeM
eMMeMMCCC
CCC
jj
jj
φφ
φφφφ
∗
∗∗
+++=+−+=
−+−=+++
)]([)(
)()()()(
21212121
22112211
IIjRRIIjRR
jIRjIRjIRjIR
10
Arithmetic Operations of Complex Numbers
 Conjugation
 Product of a complex number and its conjugate is
complex number’s magnitude squared
202
MeMMeMeCC jjj
=== −∗ φφ
11
Arithmetic Operations of Complex Numbers
 Division
2
2
2
2
21122121
22
22
22
11
22
11
2
1
21
2
1
2
1
)(
2
1
2
1
2
1
)()(
)(
21
2
1
IR
IRIRjIIRR
jIR
jIR
jIR
jIR
jIR
jIR
C
C
M
M
C
C
e
M
M
eM
eM
C
C j
j
j
+
−++
=
−
−
⋅
+
+
=
+
+
=
−∠=
== −
φφ
φφ
φ
φ
12
Arithmetic Operations of Complex Numbers
 Inverse of a complex number
222
11
111
M
C
IR
jIR
jIR
jIR
jIRC
e
MMeC
j
j
∗
−
=
+
−
=
−
−
⋅
+
=
== φ
φ
13
Arithmetic Operations of Complex Numbers
 Complex numbers raised to a power
φφφ jkkkjkkj
eMeMCMeC ==→= )(
14
Arithmetic Operations of Complex Numbers
 Roots of a complex number
 Next, we assign values 0, 1, 2, 3, . . ., k–1 to n to
get the k roots of C
knjkk njk
njj
eMMeC
MeMeC
/)360()360(
)360(


++
+
==
==
φφ
φφ
15
Arithmetic Operations of Complex Numbers
 Natural logarithms of a complex number
where 0 ≤ ø < 2π
φφφ
φ
jMeMMeC
MeC
jj
j
+=+==
=
ln)ln(ln)ln(ln
16
Arithmetic Operations of Complex Numbers
 Logarithm to base 10 of a complex number
)43429.0(log
)(loglog)(loglog)(loglog
10
101010101010
φ
φφφ
φ
⋅+≈
⋅+=+==
=
jM
ejMeMMeC
MeC
jj
j
17
Arithmetic Operations of Complex Numbers
 Log to base 10 of a complex number using
natural logarithms
)(ln43429.0)(ln43429.0
))(ln(log
10ln
ln
log
)10ln(
)ln(
)(log
1010
10
φ
φ
jMC
Ce
C
C
MeC
x
x
j
+⋅=⋅≈
==
=
=
18
Some Practical Implications of Using Complex Numbers
 Representing numbers in a computer
 Rectangular form has advantage over polar form
 Example: represent complex numbers using a
four-bit sign-magnitude binary number format
 Integral numbers ranging from –7 to +7
 Range of complex numbers covers a square on
complex plane (Fig. A-4(a)) using rectangular form
 If we use four-bit numbers to represent magnitude in
polar form, those numbers reside on or within a circle
whose radius is 7 (Fig. A-4(b))
 Four shaded corners in Fig. A-4(b) represent locations
of valid complex values using rectangular form but are
out of bounds if we use polar form
 Acceptable result in rectangular could overflow in polar
19
Some Practical Implications of Using Complex Numbers