Fast Fourier Transform

Done by:

Amer AlQaderi

Ahmad Abdul-Rahman

Ismail Kishtah

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.

This picture illustrates the waveform before and after transformation (notice the frequency!!)

So what if our signal has some noise??

Do we need an external filter to deal with noise??

Actually ,No the FFT does filtering itself.

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.

We have introduced the Fast Fourier Transform.

We have designed a circuit to implement the Fast Fourier Transform using Labview software.

The design specificationApplicationFeaturesLimitation