Although complex notation is not complex to understand (see Chapter 8 in Understanding Digital Signal Processing by Richard Lyons for an excellent tutorial), one of the themes of this text is to avoid complex notation altogether.
A complex number is defined as an ordered pair of real numbers in -plane. In that respect, complex numbers can be considered as vectors with initial point on the origin . Addition of complex numbers is then similar to the addition of vectors in -plane from this perspective.
However, multiplication is well defined for complex numbers while it is not defined for vectors — the dot product of two vectors is a scalar, not a vector, while the cross product of two vectors in a plane is a vector that is outside of that plane. The product of complex numbers, on the other hand, is a complex number — an extremely useful property.
For our purpose, using the complex numbers but still staying clear of the complex notation means that we focus on a 2-dimensional plane with or real-axis named as (which stands for inphase) and or imaginary-axis named as (which stands for quadrature), as shown in this Figure. In the post about frequency, we will learn why the and components are called inphase and quadrature, respectively.
From a signal processing perspective, and are just two real signals that appear on two separate wires which can be manipulated according to the underlying algorithms. Through this approach, we can deal with complex numbers as a pair of real numbers only.
Magnitude and Phase
In polar representation of complex numbers, the magnitude of in an -plane is defined as
Defining the phase is a little trickier. It is tempting to define it as . However, a problem with this is that
There is no way to differentiate whether lies in quadrant I or III. The same holds for lying in quadrant II or IV. On the other hand, the above Figure tells us that can lie in any quadrant and its phase should be in the range because
- Quadrant I: should be in
- Quadrant II: should be in
- Quadrant III: should be in
- Quadrant IV: should be in
Similarly, from Figure below,
- Case 1: When and , the phase of should be , not
- Case 2: When and , the phase of should be
- Case 3: When and , the phase of should be
Taking into account all four quadrants, is defined in terms of as
From here onwards, we will call this adjustment as four-quadrant inverse tangent.
Addition and Multiplication Operations
The addition and multiplication rules for complex numbers are explained below.
Following rules apply to two complex numbers and in an -plane, which is basically a simpler way to write complex additions and multiplications.
Note that one complex addition results in two real additions, one each for and , as shown in Figure below.
We will call the above equation the multiplication rule. Note that one complex multiplication results in 4 real multiplications and 2 real additions. This is illustrated in Figure below.
Addition rule above makes perfect sense: both components are added together as well as both components. The multiplication rule seems a little strange though: is a difference between products of two aligned-axes terms (i.e., – ), while is a sum of products of two cross-axes terms (i.e., + ).
To see why multiplication of two complex numbers is perfectly logical, consider that and , while and . Plugging in these values in Eq (2),
which leads to
Using the identities and ,
which is equivalent to Eq (1). Hence, multiplication of two complex numbers is about multiplying their magnitudes and adding their phases.
Rotating a complex number in -plane by a phase seems very simple in complex notation but a bit complicated in terms. Rotation implies keeping the magnitude constant and adding to the angle of that complex number, as shown in Figure below.
To start, let us multiply a complex number by a complex number with magnitude and angle . Using and ,
Plugging in Eq (2),
Using and ,
which keeps the magnitude unchanged and adds the angle to the existing angle.
Similarly, for a complex number and an angle (i.e., clockwise rotation)
The phase rotation rule above is important because we are not using complex notation in this text. That implies having on our disposal a quick way to recognize an equation if it rotates a complex number by an angle. The above two equations help fulfill that purpose.
The conjugate of a complex number is defined as
A significance of conjugate of a complex number arises from the fact that a complex number multiplied by its complex conjugate cancels the phase and produces its magnitude squared. Using the above relations in multiplication rule of Eq (2),
which generates the following result.