Title: Discrete Fourier Transform
1Discrete Fourier Transform
- Prepared by Rejean Lau
- M.Eng
2Fourier Analysis
- Named after mathematician Jean Baptiste Joseph
Fourier (1768-1830) - Fourier claimed that any continuous periodic
signal could be represented as the sum of
properly chosen sinusoidal waves
3Fourier Analysis
4Types of Signals
5Types of Signals
6Discrete Fourier Transform
- The only type of Fourier Transform that can be
used in DSP is the DFT. - Why?
- Digital computes can only work with information
that is discrete and finite in length.
7Transforms
- Examples of other types of transforms Fourier,
Laplace, Z, Hilbert, Discrete Cosine etc. - What is a transform?
- A function which allows both the input and output
to have multiple values. - Eg. A signal composed of 100 samples. A transform
changes the 100 samples into another 100 samples.
8Discrete Fourier Transform
- The discrete Fourier transform changes an N point
input signal into two point output signals. - The input signal contains the signal being
decomposed, while the two output signals contain
the amplitudes of the component sine and cosine
waves - The input signal is said to be in time domain,
while the output signals are said to be in
frequency domain.
9Discrete Fourier Transform
- The inverse DFT performs the reverse of the DFT
- Transform a frequency domain signal to time
domain - The input length N is usually selected to be a
power of 2 . Ie. 128,256,512, 1024 - This is a requirement by the most efficient
algorithm which calculates the DFT, called the
FFT (fast fourier transform)
10DFT Notation
- Lower case letters represent time domain signals
- ie. x , y , z
- Time domain runs from x0 to xN-1
- Upper case letters represent the corresponding
frequency domain signals - ie. X , Y , Z
- Frequency signal X consists of two parts, each
an array of N/2 1 samples. - ReX Real part of X -amplitude of cos
wave - - runs from ReX0 to ReXN/2
- ImX Imaginary part of X -amplitude of sin
wave - - runs from ImX0 to ImXN/2
11Discrete Fourier Transform
12DFT Basis Functions
- The sine and cosine waves used in the DFT are
called DFT basis functions. - The index value of ReX and ImX represented
by k, is the amplitude of the corresponding basis
function.
k is also equal to the number of cycles that
occur over the N points of the signal
13DFT Basis Functions
14Synthesis Equation, Inverse DFT
- Given ReX and ImX (the frequency
components), determine x (the original time
signal).
Basis functions
Synthesis equation to determine xi
ReXk, ImX needs to be scaled before
inserting in synthesis equation
15Inverse DFT
16Spectral Density
- The scaling is because the frequency components
are expressed as spectral density - ReX / (2/N) ReX where 2/N is the spectral
density - Therefore ReX ReX (2/N)
- The difference in scaling of the first and last
frequency components ReX0 and ReXN/2 they
have half the spectral density 1/N - ReX / (1/N) ReX where 1/N is
the spectral density - Therefore ReX ReX (1/N)
-
17Two ways the synthesis equation can be
programmed Method (1) Each of the scaled
sinusoids are generated one at a time and added
to an accumulation array, which ends up becoming
time domain signal Method (2) Each sample in the
time domain signal is calculated one at a time,
as the sum of all the corresponding samples in
the cosine and sine waves