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Sine Wave Generation Using an Unstable IIR Filter

An unstable infinite impulse response (IIR) filter can be used as a sine wave generator. This little-known trick is useful when generating a pure tone with a microcontroller platform such as the Arduino or a computer such as the Raspberry Pi. Since the IIR filter is unstable, an input is not required to produce an output. Given appropriate initial conditions, an unstable recursive equation will produce an oscillating output for an indefinite length of time.

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The filter coefficients are obtained by taking the Laplace transform of and then converting the results to discrete time expressions in the domain. The coefficients of the domain denominator can be used in the recursion equation. The coefficients that produce an oscillation are not unique, since each of the methods for producing a discrete domain from a continuous domain (i.e. bilinear transform, zero-order hold, first-order hold, etc.) result in different coefficients.
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