Editing Analog signal processing (section)

==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" instead of "i" 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. It is the integral of the product of two waveforms after one has reversed and shifted; the symbol for convolution is *.  
: <math>y(t) = (x * h )(t) = \int_{a}^{b} x(\tau) h(t - \tau)\, d\tau</math>
That is the convolution integral and is used to find the convolution of a signal and a system; typically a = -∞ and b = +∞.

Consider two waveforms f and g. By calculating the convolution, we determine how much a reversed function g must be shifted along the x-axis to become identical to function f. The convolution function essentially reverses and slides function g along the axis, and calculates the integral of their (f and the reversed and shifted g) product for each possible amount of sliding. When the functions match, the value of (f*g) is maximized. This occurs because when positive areas (peaks) or negative areas (troughs) are multiplied, they contribute to the integral.

===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 functions.  The constraint on which systems or signals can be transformed by the Fourier Transform is that:  
:<math>\int^\infty_{-\infty} |x(t)|\, dt < \infty</math>
This is the Fourier transform integral:
:<math>X(j\omega) = \int^\infty_{-\infty} x(t)e^{-j\omega t}\, dt</math>
Usually the Fourier transform integral isn't used to determine the transform; instead, 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:  
:<math>x(t)=\frac{1}{2\pi}\int^\infty_{-\infty} X(j\omega )e^{j\omega t}\, d\omega </math>
Each signal or system that can be transformed has a unique Fourier transform. There is only one time signal for any frequency signal, and vice versa.

===Laplace transform===
{{further|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:
:<math>X(s) = \int^\infty_{0^-} x(t)e^{-s t}\, dt</math>
and the inverse Laplace transform, if all the singularities of X(s) are in the left half of the complex plane, is:
:<math>x(t)=\frac{1}{2\pi}\int^\infty_{-\infty} X(s )e^{s t}\, d s </math>

===Bode plots===
[[Bode plot]]s 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.