# Discrete Transform

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```					Discrete Fourier Transform

Prepared by Rejean Lau
M.Eng
Fourier 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
Fourier Analysis
Types of Signals
Type                   Description               Fourier
Transform
Aperiodic-Continuous   Includes decaying         Fourier Transform
exponentials and
Gaussian.
Periodic-Continuous    Includes: sine waves,     Fourier Series
square waves, and
any waveform that
repeats itself
Aperiodic-Discrete     Signal is defined only    Discrete Time
at discrete points, and   Fourier Transform
does not repeat.
Periodic-Discrete      Discrete signal that      Discrete Fourier
repeats.                  Transform
Types of Signals
Discrete 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.
Transforms
• 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.
Discrete 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.
Discrete 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)
DFT Notation
•    Lower case letters represent time domain signals
ie. x[ ], y[ ], z[ ]
•    Time domain runs from x[0] to x[N-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.
1)   ReX[ ]    Real part of X[ ] -amplitude of cos wave
- runs from ReX[0] to ReX[N/2]
2)   ImX[ ] Imaginary part of X[ ] -amplitude of sin wave
- runs from ImX[0] to ImX[N/2]
Discrete Fourier Transform
DFT 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
DFT Basis Functions
Synthesis Equation, Inverse DFT
• Given ReX[ ] and ImX[ ] (the frequency
components), determine x[ ] (the original time
signal).            Basis functions
Synthesis
equation
to
determine
x[i]
ReX[k], ImX[]
needs to be
scaled before
inserting in
synthesis
equation
Inverse DFT
Spectral 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 ReX[0] and ReX[N/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)
Two ways the synthesis
equation can be
programmed

Method (1)
Each of the scaled sinusoids
are generated one at a time
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

```
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