The impulse response of a high-pass filter

[elgen]

Consider the passive high-pass filter that consists of a resistor (R) and a capacitor (C). The frequency response is
[itex]H(\omega)=\frac{j\omega C R}{j\omega C R + 1}[/itex].

What would be the impulse response of this high pass filter? This question translates to how to evaluate the following Fourier integral:
[itex]h(t) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty} H(\omega ) e^{j\omega t} d\omega [/itex]

If this integral cannot be evaluated in the close form, would it be possible to seek the asymptotic expansion? as R becomes large or C becomes large?

Many thanks.


elgen

 

[lippyka]

ovG,The impulse response of the high pass filter can be evaluated by computing the Fourier integral h(t) = 1/√2π∫−∞∞H(ω)ejωt dω. This integral cannot be evaluated in a closed form, but an asymptotic expansion can be found when either R or C is large. In particular, when R is large, the Fourier integral can be approximated with h(t) ≈ 1/√2πCR ∫−∞∞ejωt dω, which yields the impulse response h(t) ≈ 1/(RC)δ(t), where δ(t) is the Dirac delta function. When C is large, the Fourier integral can be approximated with h(t) ≈ 1/√2πR ∫−∞∞ e−ω2CRt dω, which yields the impulse response h(t) ≈ (1/√2πR)exp(−t²/4CR).