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Fast Fourier Transform

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.

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Fast Fourier Transform

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  1. Fast Fourier Transform Done by: AmerAlQaderi Ahmad Abdul-Rahman Ismail Kishtah

  2. 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. • 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.

  3. Introduction, cont • 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.

  4. Introduction, cont • For a signal x(t) with a spectrum X(f), the followings hold: FForward Fourier Transform Inverse Fourier Transform

  5. 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.

  6. Design Specification: After introducing FFT, we are going to go deep little bit to explain the way we followed to design FFT.

  7. FFT Design -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.

  8. FFT Design

  9. FFT Design Cont This picture illustrates the waveform before and after transformation (notice the frequency!!)

  10. FFT Design Cont So what if our signal has some noise??

  11. Built-in Filteration System: Do we need an external filter to deal with noise?? Actually ,No the FFT does filtering itself.

  12. Applications of Fast Fourier Transform: • 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.

  13. Applications of Fast Fourier Transform (Cont.): • 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.

  14. Features of Fast Fourier Transform: • 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

  15. Limitation of Fast Fourier Transform: • 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.

  16. Conclusion: • We have introduced the Fast Fourier Transform. • We have designed a circuit to implement the Fast Fourier Transform using Labview software. • The design specification • Application • Features • Limitation

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