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# fft1 - SIGNAL PROCESSING SIMULA TION NEWSLETTER Fourier...

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Easy Fourier Analysis Part 1 Complextoreal.com 1 SIGNAL PROCESSING & SIMULATION NEWSLETTER Fourier analysis made Easy Part 1 Jean Baptiste Joseph, Baron de Fourier, 1768 - 1830 While studying heat conduction in materials, Baron Fourier (a title given to him by Napoleon) developed his now famous Fourier series, approximately 120 years after Newton published the first book on calculus. It took Fourier another twenty years to develop the Fourier transform which made the theory applicable to a variety of disciplines such as signal processing where Fourier analysis is now an essential tool. Fourier did little to develop the concept further and most of that work was done by Euler, LaGrange, Laplace and others. Fourier analysis is now also used in thermal analysis, image processing, quantum mechanics and physics. Why do we need to do Fourier analysis ° In communications, we can state the problem at hand this way; we send an information-laced signal over a medium. The medium and the hardware corrupt this signal. The receiver has to figure out from the received signal which part of the corrupted received signal is the information signal and which part the extraneous noise and distortion. The transmitted signals have well defined spectral content, so if the receiver can do a spectral analysis of the received signal then it can extract the information. This is what Fourier analysis allows us to do. Fourier analysis can look at an unknown signal and do an equivalent of a chemical analysis, identifying the various frequencies and their relative ±quantities² in the signal. Fourier noticed that you can create some really complicated looking waves by just summing up simple sine and cosine waves. For example, the wave in Figure 1a is sum of the just three sine waves shown in Figures 1b, 1c and 1d of assorted frequencies and amplitudes.

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Easy Fourier Analysis Part 1 Complextoreal.com 2 (a) - A complicated looking wave (b) - Sine wave 1 (c)- Sine wave 2 (d) - Sine wave 3 Figure 1 - Sine waves Let³s look at signal 1a in three dimensions. With time progressing to the right we see the amplitude going up and down erratically, we are looking at the signal in Time domain. From this angle, we see the sum of the three sine waves as shown in Fig (1b,c,d). When we look at the same signal from the side along the z-axis, what we see are the three sine waves of different frequencies. We also see the amplitude but only as a line with its maximum excursion. This view of the signal from this point of view is called the Frequency Domain . Another name for it is the Signal Spectrum . Figure 3 - Looking at signals from two different points of view The concept of spectrum came about from the realization that any arbitrary wave is really a summation of many different frequencies. The spectrum of the composite wave f(t) of Fig (1) is composed of just three frequencies and can be drawn as in Fig (3.1).
Easy Fourier Analysis Part 1 Complextoreal.com 3 This is called a one-sided magnitude spectrum . One-sided not because anything has been left out of it, but because only positive frequencies are represented. (

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fft1 - SIGNAL PROCESSING SIMULA TION NEWSLETTER Fourier...

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