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This module demonstrates the effect that a time-invariant filter can have on periodic waveforms.

The Fourier series representation of a periodic signal makes it easy to determine how a linear, time-invariant filter reshapessuch signals in general . The fundamental property of a linear system is that its input-output relationobeys superposition: L a 1 s 1 t a 2 s 2 t a 1 L s 1 t a 2 L s 2 t . Because the Fourier series represents a periodic signal as alinear combination of complex exponentials, we can exploit the superposition property. Furthermore, we found for linearcircuits that their output to a complex exponential input is just the frequency response evaluated at the signal's frequencytimes the complex exponential. Said mathematically, if x t 2 k t T , then the output y t H k T 2 k t T because f k T . Thus, if x t is periodic thereby having a Fourier series, a linear circuit's output to this signal will be the superposition of the output toeach component.

y t k c k H k T 2 k t T
Thus, the output has a Fourier series, which means that it too is periodic. Its Fourier coefficients equal c k H k T . To obtain the spectrum of the output, we simply multiply the input spectrum by the frequency response . The circuit modifies the magnitude and phase of each Fouriercoefficient. Note especially that while the Fourier coefficients do not depend on the signal's period, the circuit'stransfer function does depend on frequency, which means that the circuit's output willdiffer as the period varies.

Filtering a periodic signal

Periodic pulse signal
Top plots show the pulse signal's spectrum for various cutoff frequencies. Bottom plots show the filter's outputsignals.
A periodic pulse signal, such as shown on the left part ( Δ T 0.2 ), serves as the input to an R C lowpass filter. The input's period was 1 ms (millisecond). The filter's cutoff frequency was set to the various valuesindicated in the top row, which display the output signal's spectrum and the filter's transfer function. The bottom rowshows the output signal derived from the Fourier series coefficients shown in the top row.

The periodic pulse signal shown on the left above serves as the input to a R C -circuit that has the transfer function (calculated elsewhere )

H f 1 1 2 f R C
[link] shows the output changes as we vary the filter's cutoff frequency. Note how thesignal's spectrum extends well above its fundamental frequency. Having a cutoff frequency ten times higher thanthe fundamental does perceptibly change the output waveform, rounding the leading and trailing edges. As the cutofffrequency decreases (center, then left), the rounding becomes more prominent, with the leftmost waveform showing a smallripple.

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What is the average value of each output waveform? The correct answer may surprise you.

Because the filter's gain at zero frequency equals one, the average output values equal the respective average inputvalues.

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This example also illustrates the impact a lowpass filter canhave on a waveform. The simple R C filter used here has a rather gradual frequency response, which means that higher harmonics are smoothly suppressed. Later, wewill describe filters that have much more rapidly varying frequency responses, allowing a much more dramatic selection ofthe input's Fourier coefficients.

More importantly, we have calculated the output of a circuit toa periodic input without writing, much less solving, the differential equation governing the circuit'sbehavior. Furthermore, we made these calculations entirely in the frequency domain. Using Fourier series, we can calculatehow any linear circuit will respond to a periodic input.

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Source:  OpenStax, Fundamentals of electrical engineering i. OpenStax CNX. Aug 06, 2008 Download for free at http://legacy.cnx.org/content/col10040/1.9
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