Smoothing in frequency domain and impulse reponse of the filter in time-domain

1. What is the relation between time-domain and frequency domain?
2. What is the difference between time-domain and frequency domain analysis?
3. How does a filter modify the input signal in the time-domain?
4. Is time-domain an impulse response?

What is the relation between time-domain and frequency domain?

Parseval's theorem gives the relationship between the squared integral of a time function and that of its Fourier transform, namely, the energy in the time domain is equal to the energy in the frequency domain.

What is the difference between time-domain and frequency domain analysis?

As stated earlier, a time-domain graph displays the changes in a signal over a span of time, and frequency domain displays how much of the signal exists within a given frequency band concerning a range of frequencies.

How does a filter modify the input signal in the time-domain?

In the time domain, the filtering operation involves a convolution between the input and the impulse response of the finite impulse response (FIR) filter. In the frequency domain, the filtering operation involves the multiplication of the Fourier transform of the input and the Fourier transform of the impulse response.

Is time-domain an impulse response?

The impulse that is referred to in the term impulse response is generally a short-duration time-domain signal. For continuous-time systems, this is the Dirac delta function δ(t), while for discrete-time systems, the Kronecker delta function δ[n] is typically used.