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The Discrete Fourier Transform. Jean Baptiste Fourier (1768-1830)showed that any signal or waveform could be made up just by adding together a series of pure tones (sine waves) with appropriate amplitude and phase Fourier's theorem assumes we add sine waves of infinite duration.
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The Discrete Fourier Transform • Jean Baptiste Fourier (1768-1830)showed that any signal or waveform could be made up just by adding together a series of pure tones (sine waves) with appropriate amplitude and phase • Fourier's theorem assumes we add sine waves of infinite duration
Any signal can be made up by adding together the correct sine waves with appropriate amplitude and phase It shows how a square wave can be made up by adding together pure sine waves at the harmonics of the fundamental frequency.
The Fourier transform is an equation to calculate the frequency, amplitude and phase of each sine wave needed to make up any given signal -The Fourier Transform (FT) is a mathematical formula using integrals -The Discrete Fourier Transform (DFT) is a discrete numerical equivalent using sums instead of integrals -The Fast Fourier Transform (FFT) is just a computationally fast way to calculate the DFT
Every next wave is the integral multiple of the fundamental wave. What does that mean? It means: every next wave has an integral multiple frequency of the fundamental wave, ex. ω, 2ω, 3ω, 4ω, …, nω. Green, blue , yellow, and orange waves are integral multiple of each othe
Simple FFT • http://lcni.uoregon.edu/downloads/fft.ppt
DFT - In the time domain, x[ ] consists of N points running from 0 to N-1 . - In the frequency domain, the DFT produces two signals, the real part, written: ReX[ ] , and the imaginary part, written: Im X[ ] - Each of these frequency domain signals are N/2+1 points long, and run from 0 to N/2 - While N can be any positive integer, a power of two is usually chosen, i.e., 128, 256, 512, 1024, etc.
DFT synthesis equation: add up sine and cosine to make up the signal Re X [k ] and Im X [k ] are the Fourier series coefficients at kthharmonics Where x [i ] is the signal value at sample time i In this relation, x [i ] is the signal being synthesized, with the index, i, running from 0 to N-1. Re X [k ] and Im X [k ] hold the amplitudes of the cosine and sine waves, respectively, known as Fourier series coefficientswith k running from 0 to N/2. Any N point signal, x[i] , can be created by adding N/2+ 1 cosine waves and N/2+ 1 sine waves. The amplitudes of the cosine and sine waves are held in the arrays Im X [k] and Re X [k], respectively. Note: All DSP operations consist of multiplying corresponding values contained in two numeric series and then calculating the sum of the products. Sometimes, the final sum is divided by the total number of values included in the sum. This is often referred to as a sum-of-products or multiply-add operation. (This is the digital equivalent of integrating the product of two continuous functions between two limits.)
Some useful links • http://www.falstad.com/fourier/ • Fourier series java applet.Make sure to check the sound on • http://www.jhu.edu/~signals/ • Collection of demonstrations about digital signal processing • http://www.ni.com/swf/presentation/us/fft/FFT - tutorial from National Instruments • http://www.cf.ac.uk/psych/home2/CullingJ/frames_dict.html -- Dictionary of DSP terms • http://lcni.uoregon.edu/downloads/fft.ppt • FFT Presentation very simple
