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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^(-π|f|)
4. e^(-2πf²)

Correct answer: e^(-2πf²)

Solution

The output of the matched filter h(t) when g(t) is the input results in the convolution of their Fourier transforms. Since the Fourier transform of g(t) is a Gaussian function, the convolution with itself leads to a scaling of the exponent, resulting in e^(-2πf²).

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