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Ft and FFT
- 1. F T and F F T
- 2. Signal !
- 3. Signal Waves Data sampling time and Frequency domain representation The Nyquist Frequency The law of decomposition and superposition
- 4. Wave • is a vibration in space and time that continues in a repetitive pattern. Waves transfer energy from one place to another • In the time domain the amplitude of signals is plotted versus time ω. = 2 · π · ƒ. where ω. = angular frequency, in s–1 f. = signal frequency, in Hz x
- 5. Wave - Phase Is used to describe specific location within given cycle of periodic wave
- 6. periodic and non periodic wave periodic A signal which repeats itself after a specific interval of time is called periodic signal A signal that repeats its pattern over a period is called periodic signal They are deterministic signals Their value can be determined at any point of time Example: sine cosine square non periodic A signal which does not repeat itself after a specific interval of time is called aperiodic signal. A signal that does not repeats its pattern over a period is called aperiodic signal or non periodic. They are random signals Their value cannot be determined with certainty at any given point of time Example: sound signals from radio , all types of noise signals
- 7. periodic and non periodic wave
- 8. Data sampling continuous has a signal value for all times discrete has a signal value at certain times
- 9. The law of decomposition and superposition
- 10. The law of decomposition and superposition
- 11. The law of decomposition and superposition
- 12. Time and Frequency domain representation The bridge between the time domain and the frequency domain is defined by Fourier The Fourier Transform is simply a mathematical process that allows us to take a function of time (a seismic trace) and express it as a function of frequency (amplitude and phase spectra). Any repetitive waveform can be represented in the frequency domain by a pair of spectra The pair consists of an amplitude and a phase spectra
- 13. Time and Frequency domain representation signals may be examined in the time domain and in the frequency domain with the aid of a spectrum analyzer
- 14. The Nyquist Frequency The Nyquist frequency is equal to one-half of the sampling frequency. The Nyquist frequency is the highest frequency that can be measured in a signal.
- 15. Fourier
- 16. Fourier series in mathematics, a Fourier series is a way to represent a periodic function as the sum of simple sine, cosines waves + c .
- 17. Fourier series
- 18. Fourier series
- 19. Fourier series
- 20. close your eyes if you don’t like integrals !!
- 45. Discrete Fourier Transform Time ! frequency 1
- 46. Why we don’t use Discrete Fourier Transform !?
- 51. Fast Fourier Transform (FFT) The Fast Fourier Transform (FFT) is a very efficient algorithm for performing a discrete Fourier transform In 1969, the 2048 point analysis of a seismic trace took 13 ½ hours. Using the FFT, the same task on the same machine took 2.4 seconds!
- 52. Fast Fourier Transform (FFT)
- 53. Fast Fourier Transform (FFT)
- 54. Flow Diagram for a N=8 DFT
- 55. Further advantage of the FFT algorithm
- 56. FFT Derivation Summary The FFT derivation relies on redundancy in the calculation of the basic DFT A recursive algorithm is derived that repeatedly rearranges the problem into two simpler problems of half the size Hence the basic algorithm operates on signals of length a power of 2 It has M = log2 N stages, each using N / 2 butterflies At the bottom of the tree we have the classic FFT `butterfly’ structure
- 57. FFT Derivation Summary The radix-2 N-point FFT requires 10( N / 2 )log2 N real operations compared to about 8N2 real operations for the DFT. This is a huge speed-up in typical applications, where N is 128 – 4096
- 58. Applications of the FFT There FFT is surely the most widely used signal processing algorithm of all. It is the basic building block for a large percentage of algorithms in current usage Specific examples include: Spectrum analysis – used for analyzing and detecting signals Coding – audio and speech signals are often coded in the frequency domain using FFT variants (MP3, …)
- 59. Practical Spectral Analysis Say, we have a symphony recording over 40 minutes long – about 2500 seconds Compact-disk recordings are sampled at 44.1 kHz and are in stereo how we compute it !?
- 60. solution DTF! unrealistic – speed and storage requirements are too high FFT and filters useless – we will get a very much noise like wide-band spectrum covering the range from 20 Hz to 20 kHz at high resolution including all notes of all instruments with their harmonics
- 61. questions
- 62. thank you