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# fft2 - Intuitive Guide to Principals of Communications...

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Tutorial 6 - Fourier Analysis Made Easy Part 2 Charan Langton, complextoreal.com Page 1 Intuitive Guide to Principals of Communications Tutorial 6 - Fourier Analysis Made Easy Part 2 Complex representation of Fourier series cos sin jwt e wt i wt (1) Bertrand Russell called this equation “the most beautiful, profound an d subtle expression in mathematics.”. Richard Feyman., the noble laureate said that it is “the most amazing equation in all of mathematics”. In electrical engineering, this enigmatic equation is equivalent in importance to F = ma. This perplexing looking equation was first developed by Euler (pronounced Oiler) in the early1800’s. A student of Johann Bernoulli, Euler was the foremost scientist of his day. Born in Switzerland, he spent his later years at the University of St. Petersburg in Russia. He perfected plane and solid geometry, created the first comprehensive approach to complex numbers and is the father of modern calculus. He was the first to introduce the concept of log x and e x as a function and it was his efforts that made the use of e, i and pi the common language of mathematics. He derived the equation e x + 1 = 0 and its more general form given above. Among his other contributions were the consistent use of the sin, cos functions and the use of symbols for summation. A father of 13, he was a prolific man in all aspects, in languages, medicine, botany, geography and all physical sciences. The secret to this equation lies in understanding that sinusoids are a special case of a general polynomial function of the form e jwt in Euler’s equatio n is a decidedly confusing concept. What exactly is the role of j in e jwt ? We know that it stands for 1 but what is it doing here? Can we visualize this function?

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Tutorial 6 - Fourier Analysis Made Easy Part 2 Charan Langton, complextoreal.com Page 2 Before we continue the discussion of Fourier Series and its complex re presentation, let’s first try to make sense of e jwt as it relates to signal processing. Take any real number, say 3, and plot it on a X-Y plot as in Fig 1a. Multiply this number by j, so it becomes 3j. Where do we plot it now? Herein lies our answer to what multiplication with j does. Figure 1a - Relationship of real Figure 1b Multiplication with j and imaginary numbers represents a phase shift The number stays exactly the same, 3j is the same as 3, except that multiplication with j shifts the phase of this number by +90 o . So instead of an X-axis number, it becomes a Y-axis number. Each subsequent multiplication rotates it further by 90 o in the X-Y plane as shown in Figure 1b. 3 become 3j, then -3 and then -3j and back to 3 doing a complete 360 degree turn. Division by j means the opposite. It shifts the phase by -90 o .
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fft2 - Intuitive Guide to Principals of Communications...

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