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Let g(t) = e^(-πt²) and h(t) is a filter matched to g(t). If g(t) is applied as input to h(t), then the Fourier transform of the output is

1. e^(-πf²)
2. e^(-πf²/2)
3. e^(-|f|)
4. e^(-2πf²)

Correct answer: e^(-2πf²)

Solution

The correct option is right because the output of a matched filter is the convolution of the input's Fourier transform with the filter's Fourier transform. Since the Fourier transform of g(t) is e^(-πf²), and the matched filter has the same Fourier transform, the result of their convolution leads to e^(-2πf²).

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