1. Time and Frequency Representation
The most common representation of signals and waveforms is in the time domain
Most signal analysis techniques only work in the frequency domain
This can be a difficult concept when first introduced to it
The frequency domain is just another way of representing a signal
Fist consider a simple sinusoid
The time-amplitude axes on which the sinusoid is shown define
the time plane.
If an extra axis is added to represent frequency then the sinusoid would illustrated as ……

2. Time and Frequency Representation
The frequency-amplitude axes define the frequency plane in the same way as the time-amplitude axes defines the time plane
The frequency-plane is orthogonal to the time-plane and intersect with it a line on the amplitude axis.
The actual sinusoid can be considered to be existing some distance along the frequency domain

3. Fourier Series
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Any periodic function f(t), with period T, may be represented by an infinite series of the form:
where the coefficients are calculated from:

4. Fourier Series
Provides a means of expanding a function into its major sine / cosine or complex exponential components
These individual terms represent various frequency components which make up the original waveform
Example: Square wave

5. Complex Fourier Series
Using Eulers formula to derive the complex expressions for , and substituting these into the Fourier series it can be shown that the complex form of the Fourier series is:
where

6. Discrete Fourier Transform (DFT)
The Fourier transform provides the means of transforming a signal in the time domain into one defined in the frequency domain.
The DFT is given by:
DFTExpanded.m
DFT.m
Example: Find the DFT of the sequence {1, 0, 0, 1}
Solution……..

7. Discrete Fourier Transform (DFT)
Example: Find the DFT of the sequence {1, 0, 0, 1} Solution: { 2, 1+j, 0, 1-j }

8. Computational Complexity of the DFT
Consider an 8-point DFT
Letting
Each term consists of a multiplication of an exponential term by another term which is either real or complex.
Each of the product terms are added together.
There are also eight harmonic components (k = 0, … ,7)
Therefore for an 8-point DFT there are 82 = 64 multiplications and 8 x 7 additions .
For an N-point DFT - N2 multiplications and N(N-1) additions

9. Computational Complexity of the DFT
For an N-point DFT - N2 multiplications and N(N-1) additions
Therefore for a 1024-point DFT (N=1024)
Multiplications: N =
Additions: N(N-1) =
Clearly some means of reducing these numbers is desirable

10. Computational Complexity of the DFT
where
X(k)
x(0)
x(1)
x(2)
x(3)
x(4)
x(5)
x(6)
x(7) 1
π/4
π/2
3π/4 π
5π/4
3π/2
7π/4 2 2π
5π/2 3π
7π/2 3
9π/4
15π/4
9π/2
21π/4 4 4π 5π 6π 7π 5
25π/4
15π/2
35π/4 6 9π
21π/2 7
49π/4

11. FFT Algorithmic Development
Computational Savings