fast fourier transform
DESCRIPTION
Fast Fourier Transform. Done by: Amer AlQaderi Ahmad Abdul- Rahman Ismail Kishtah. Introduction. The Fast Fourier Transform (FFT) is a set of mathematical formulas used to convert a time function to a function in the frequency domain (Fourier analysis) and back. - PowerPoint PPT PresentationTRANSCRIPT
The Fast Fourier Transform (FFT) is a set of mathematical formulas used to convert a time function to a function in the frequency domain
(Fourier analysis) and back. The Fast Fourier Transform is used extensively
in Signal processing to design filters and remove
coherent noise. Many Filtering operations are performed in the
frequency domain. The Fast Fourier Transform has applications in
image analysis.
Fast Fourier transform is same as the Fourier transform but it is much faster than it, so it achieves the conversion in very short period of time.
Even if a function is not periodic, it can be described as a linear combination of an infinite number of orthogonal functions (In case of Fourier Transform, sinusoids). i.e. spectrum consists of a continuum of
frequencies.
For a signal x(t) with a spectrum X(f), the followings hold:
dtetxfX ftj 2 )()(
dfefXtx ftj 2 )()(
FForward Fourier Transform
Inverse Fourier Transform
Notice that: The narrower a function is in one domain, the wider its transform in the
other domain.
A function is narrower in time domain.
The same function is wider in frequency domain.
A function is wider in time domain.
The same function is narrower in frequency
domain.
After introducing FFT, we are going to go deep little bit to explain the way we followed to design FFT .
-Actually, FFT can be designed by many software or even by some programming languages, but we decided to
use Lab View Software for flexibility.
Let us take a look at the whole design block.
Filtering: - representing the function as the sum of
sine functions. - By eliminating undesirable high- and/or
low-frequency components. - By taking an inverse Fourier transform to
get us back into the time domain.
Image Compression : - By eliminating the coefficients of sine
functions that contribute relatively little to the image.
- we can further reduce the size of the image, at little cost.
Fast performance (Real and Complex, Forward and Inverse)
Easy to use with excellent documentation Includes examples with compiling
instructions Allows any array size up to the practical
limits of the PCs memory
The signal must be band limited, and the sampling rate must be sufficiently high to avoid aliasing.
Components lying between discrete frequency lines are subject to error in magnitude .
The magnitude level may be different from that of the continuous-time transform due to the variation in definitions.