What are the Tools used in analog signal processing ?

A system's behavior can be mathematically modeled and is represented in the time domain as h(t) and in the frequency domain as H(s), where s is a complex number in the form of s=a+ib, or s=a+jb in electrical engineering terms (electrical engineers use j because current is represented by the variable i). Input signals are usually called x(t) or X(s) and output signals are usually called y(t) or Y(s).
Convolution
Convolution is the basic concept in signal processing that states an input signal can be combined with the system's function to find the output signal. The symbol for convolution is *.
That is the convolution integral and is used to find the convolution of a signal and a system; typically a = -∞ and b = +∞.
Fourier transform
The Fourier transform is a function that transforms a signal or system in the time domain into the frequency domain, but it only works for certain ones. The constraint on which systems or signals can be transformed by the Fourier Transform is that:
This is the Fourier transform integral:
Most of the time the Fourier transform integral isn't used to determine the transform. Usually a table of transform pairs is used to find the Fourier transform of a signal or system. The inverse Fourier transform is used to go from frequency domain to time domain:
Each signal or system that can be transformed has a unique Fourier transform; there is only one time signal and one frequency signal that goes together.
Laplace transform
The Laplace transform is a generalized Fourier transform. It allows a transform of any system or signal because it is a transform into the complex plane instead of just the jω line like the Fourier transform. The major difference is that the Laplace transform has a region of convergence for which the transform is valid. This implies that a signal in frequency may have more than one signal in time; the correct time signal for the transform is determined by the region of convergence. If the region of convergence includes the jω axis, jω can be substituted into the Laplace transform for s and it's the same as the Fourier transform. The Laplace transform is:
and the inverse Laplace transform is:
Bode plots
Bode plots are plots of magnitude vs. frequency and phase vs. frequency for a system. The magnitude axis is in Decibel (dB). The phase axis is in either degrees or radians. The frequency axes are in a logarithmic scale. These are useful because for sinusoidal inputs, the output is the input multiplied by the value of the magnitude plot at the frequency and shifted by the value of the phase plot at the frequency.