1. Complex numbers

2. Fourier transform
The Fourier transform of a continuous-time signal may be defined as
The discrete version of this is
Both have a j term so we need a basic understanding of complex numbers

3. Complex numbers
There are 2 parts to a complex number, a real part and an imaginary part
z=a±jb

4. Complex number application

5. But what about this one?

6. Complex arithmetic Add: (a+jb) + (c+jd) = (a+c) + j(b+d)
e.g. (4+j5) + (3-j2) = (7+j3)
Subtract: (a+jb) – (c+jd) = (a-c) + j(b-d)
e.g. (4+j7) – (2-j5) = (2+j12)
Multiply: (a+jb)(c+jd) = ac+jad+jbc+j2bd
e.g. (3+j4)(2+j5) = 6+j15+j8+j220 = -14+j23
Note that all results are complex

7. Using complex arithmetic, check that the previous quadratic has the solutions:

8. Complex conjugate Complex conjugate: flip the sign of j
e.g. the complex conjugate of (5+j8) is (5-j8)
Multiplication of a complex number with its conjugate produces a real number
e.g. (5+j8)(5-j8) = 25-j40+j40-j264 = 25+64=89
Now consider division of complex numbers
But what about

9. Complex division
Division: make the denominator real by multiplying top and bottom by the denominator’s complex conjugate

10. Complex numbers as vectors
Complex numbers can not be enumerated but they can be represented diagrammatically
A vector (line with magnitude and direction) of a number pointing at an angle of zero can be represented as a line on the +ve x axis
Multiply the vector by -1 and it points the other way i.e. a 180° shift
As -1 = j2 then j lies between them i.e. a 90° shift

11. Complex plane or argand diagramj
-c+jd
Imaginary
a+jb
j2= -1
Real
g-jh
-e-jf
j3=-j

12. Polar form of a complex number
The number may be represented by its vector magnitude and angle

13. Or using trig for X and Y

14. Polar manipulation
It is easy to multiply and divide complex numbers in this form even if using degrees

15. Remember the series form?

16. Summary It is easy to add and subtract in Cartesian form
It is easy to multiply and divide in polar form
The exponential form is useful when dealing with sine and cosine waveforms

17. One final note, using even-odd trig identities
(or looking at waveforms), because the Fourier transform uses e-θ: