Basically, there are time domain, s domain and frequency domain in signal analysis. Signal is propagating in time domain naturally, we take the sample, and analyze. We need to convert time domain to s domain or frequency domain (there are many domains, but those 2 are the most important for signal analysis) to find other perspectives. There is a parameter that is same for both domains, called s parameter.
S domain is the domain without loss of the information of originating signal. It’s the generalization of power series formula. Convert time domain to s domain with Laplace transform for continuous signal. We can inverse s domain to time domain without loss of information. The parameter s mathematically is =σ+ω. It’s transient and steady state analysis.
Application:
- Math tool (simplify integral and derivative, ODE problem, PDE problem, anything else. Great tool for circuit analysis)
- Analyze the stability of system (but that’s not enough, there are routh hourtwitzh criterion, nquist criterion, analyze bode plot, etc)
Frequency domain is the domain to see how often the signal oscillate. It doesn’t take into account the stability parameter of s domain. Convert time domain to frequency domain with fourier transform. When we inverse frequency domain to time domain, we assume initial condition and stability. Mathematically the parameter =ω. It is steady state analysis.
Application:
- Analyze frequency response of signal (Resonance frequency, bandwidth size for example)
- Microwave telco hardware design (signal generator, amplifier, filter, attenuator, combiner, etc)
- Analyze system’s impulse response and telco signal (but not enough, sometimes you need Hilbert transform, etc)
- Math tool for convolution operation and Parseval’s theorem