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Which one of the following is a Dirichlet condition?

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  1. \(\int\limits_{{t_1}}^\infty {\left| {x\left( t \right)} \right| < \infty } \)
  2. Signal x(t) must have a finite number of maxima and minima in the expansion interval
  3. x(t) can have an infinite number of finite discontinuities in the expansion interval

  4. x2(t) must be absolutely summable

Answer (Detailed Solution Below)

Option 2 : Signal x(t) must have a finite number of maxima and minima in the expansion interval
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Detailed Solution

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Dirichlet Conditions in Fourier Transformation are as follows:

  • f(x) must absolutely integrable over a period, i.e., \(\rm \mathop \smallint \limits_{ - \infty }^\infty \left| {f\left( t \right)} \right|dt < \infty\)
  • f(x) must have a finite number of extrema in any given interval, i.e. there must be a finite number of maxima and minima in the interval.
  • f(x) must have a finite number of discontinues in any given interval, however, the discontinuity cannot be infinite.
  • f(x) must be bounded.
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