Convergence of the Fourier Series Used in Example 2.3 (and why analysis of the harmonics is important in communication systems)

Let’s see how the stairstep signal of Example 2.3 converges as we add more harmonics to its Fourier Series summation

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Summation of dc term plus first five harmonics

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Summation of dc term plus first ten harmonics

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Summation of dc term plus first fifteen harmonics

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Summation of dc term plus first twenty-five harmonics

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Summation of dc term plus all harmonics

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Let’s pass the stairstep function from Example 2.3 through channels with various bandwidths and evaluate the distortion in the received signal.

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The importance of analyzing harmonics in communication systems

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Channel with1 Hz bandwidth

Channel with 1 Hz bandwidth passes the first five harmonics

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Channel with2 Hz bandwidth

Channel with 2 Hz bandwidth passes the first ten harmonics

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Channel with3 Hz bandwidth

Channel with 3 Hz bandwidth passes the first fifteen harmonics

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Channel with5 Hz bandwidth

Channel with 5 Hz bandwidth passes the first twenty-five harmonics

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Channel with infinite bandwidth passes all harmonics

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Channel withinfinite bandwidth

For the stairstep function, we now see how channel bandwidth and accuracy of the received signal are related.