To compute the impulse response h(t) of a lowpass filter with frequency response function H(f) = 1, we can use the inverse Fourier transform. Specifically, we can use the fact that the impulse response of a filter is the inverse Fourier transform of its frequency response:
h(t) = F^-1{H(f)}
Since H(f) = 1 for all frequencies, we can simply take the inverse Fourier transform of the constant function 1 to obtain the impulse response:
h(t) = F^-1{1}
Using the Fourier transform property that F{1} = delta(f), where delta(f) is the Dirac delta function, we can write:
h(t) = F^-1{1} = F^-1{F{delta(f)}}
Applying the inverse Fourier transform of the Dirac delta function, which is a pulse of unit area centered at t = 0, we obtain:
h(t) = F^-1{F{delta(f)}} = 1/2pi * integral from -infinity to infinity of delta(f) * e^(j2pift) df
The integral evaluates to 1/2pi, since the Dirac delta function integrates to 1 over its support. Therefore, the impulse response of the lowpass filter is:
h(t) = 1/2pi * integral from -infinity to infinity of e^(j2pift) df
The integral evaluates to 2pi * delta(t), where delta(t) is the Dirac delta function. Therefore, the impulse response of the lowpass filter is:
h(t) = 2pi * delta(t)
This result indicates that the impulse response of a lowpass filter with frequency response function H(f) = 1 is a Dirac delta function scaled by a factor of 2pi. This impulse response represents the ideal response of a lowpass filter, which attenuates all high-frequency components of a signal while allowing low-frequency components to pass through.