# What is the magnitude response and phase of this function?

Given the system/filter $$H(\omega)=\frac{1}{5-j\omega}$$, find $$h(t)$$, it's magnitude response and phase and identify what type of filter it is.

Now clearly given it's form, $$h(t)=e^{5t}u(-t)$$, but I'm confused on finding it's magnitude and phase as every example uses the form $$\frac{1}{a+j\omega}$$.

Another confusing function in regards to its magnitude, phase and type is $$H(\omega)=\pi\delta(\omega)+\frac{1}{j\omega}$$, again this is simply $$h(t)=u(t)$$ but has infinite magnitude at $$\omega=0$$ with $$\omega$$ approaching positive and negative infinity the manitude approaches 0 while it's phase is $$\frac{\pi}{2}$$, correct? Would this be a high-pass filter?

The response is non-causal so $$h(t)$$ looks correct as done.
For the frequency response, assuming a non-causal input, don’t use the step function u(t) as the input but use an impulse (as we seek the FT or the impulse response not the step response, and the FT of the impulse response is the freq response you seek ). Since the FT of an impulse is 1, thus reduces to simply finding the magnitude and phase of $$H(\omega)$$. When $$\omega$$ is 5 rad/sec the magnitude will be at -3 dB and the phase will be at -45 degrees consistent with the expected 3 dB cutoff for this non-causal first order low pass filter.
• Right it is non-causal and I agree with your h(t). The magnitude and phase response is just the magnitude and phase of $H(\omega)$ as you sweep $\omega$, assuming a time reversed input. Apr 13, 2022 at 21:12