# Convolution, Occurence and Special Cases.

Cuthbert Nyack

There are a few special cases where the response of a circuit can be calculated without
convolutions. However these represent ideal cases which can be approximated but not
achieved. Some of these cases are represented by transfer functions of the following
type.:-
Without the phase factor in green when n = 0, the system is an ideal amplifier with
constant gain A at all frequencies from 0
to infinity. For this case a unit impulse input produces an impulse of magnitude A at
the output and an input f(t) results in an output Af(t). If the phase factor in green is
included in the T/F then a unit impulse input produces an impulse of magnitude A
delayed by an amount T at the output. Similarly an input f(t) produces an output Af(t - T).
When n = 1, the system is an integrator and an impulse input produces a step output and
a function f(t) produces INT(f(t)dt) at the output. For n = -1, T/F is that of a
differentiator, an impulse input produces a differentiated impulse output and input f(t)
produces df(t)/dt at the output. An integrator is described by the simple differential
equation dy(t)/dt = Ax(t) and the differentiator by y(t) = Adx(t)/dt. In practice real
amplifiers, integrators and differentiators can only approximate these limiting cases.

When measuring waveforms with an oscilloscope, the oscilloscope display is a convolution
of the input signal with the impulse response of the oscilloscope. When properly used this
should not matter, but if inappropriate probes are used or the signal has significant
components above the upper 3dB frequency of the oscilloscope, then it does matter.

In Electronics besides system response convolution also occurs in *convolution*
codes where the output bit sequence is the convolution of the input bit sequence with the
generator functions.

## Other areas where Convolution occurs

Convolution occurs in many areas of science besides electronics, a sampling is indicated
below:-

### Differential Equations

A first order differential equation with an arbitrary function on the RHS is shown below.
This DE can be solved by integrating factors to give:-
This solution is readily recognised as a convolution of the solution of the equation with
an impulse on the RHS(Impulse Response) with the arbitrary RHS funtion. Similar results are
obtained for 2nd order differential equations.

### Physics

Convolution occurs in many areas of physics where the dynamics can be described by
differential equations eg:-

Mechanics:- Motion of a body with friction force av and with an applied force F(t) results
in a differential equation mdv/dt + av = F(t) and a convolution solution.

Atomic Physics:- When quantum systems are subjected to radiation of sufficient frequency
then some will go into a higher energy state. The number in the higher state at any given
time is also determined by a convolution.

In Control Engineering, Temperature, velocity, level, pressure control systems are all described by
differential equations and the solution requires convolution.

### Green's function

The "Green's" function which arises from partial differential equations is shown
below and occurs in many different areas of physics - diffusion, heat, waves, fields etc.
Green's function can be loosely described as a generalisation of the impulse response to
include space as well as time, the
function above relates the response at point r at time t to an impulse at location rho and
time tau. When position is not included the function becomes identical to the
impulse response. This function plays the same role for partial differential equations that
the impulse response plays for ordinary differential equations with time as the independent
variable.
The complete value of f(r,t) can be obtained by integrating the above equation which in effect
is a convolution over space and time.

### Polynomial multiplication

When 2 polynomials p_{1} and p_{2} are multiplied together:-
The coefficients of the product polynomial p_{1}p_{2} are obtained by a *discrete* convolution.

### Biological Dynamics

If a molecule is added to a biological system at rate A(t), the system tries to remove it at
a rate bc(t) which is dependent on the concentration. The equation for the concentration is then
given by dc(t)/dt + bc(t) = A(t) with a convolution solution. How do you calculate how much
"stuff" is in your stomach at a given time.

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COPYRIGHT © 1999 Cuthbert A. Nyack.