difference equation: filter design

Hi all:

I have a very basic question about difference equations. I seem to be going in circles trying to find the answer online.

1. if someone asks you to obtain the difference equation of a low pass filter with cut of freq say 60Hz, how would one go about doing it? Once you get the difference equation you can write the software, but I just can't remember how to get the difference equation.

2. The coefficients for the difference equations are scaling factors right?
I mean, if y = a1x + a2x(n-1), the coeff's are scaling factors for signals x and x(n-1), correct?

Someone please help me!

To pass from differential equation (continuous domain) to difference equation (discrete domain) you can approximate the derivative of a function with its difference quotient. That is:

dy/dt ≈ {y[i*T] - y[(i-1)*T]}/T i*T and (i+1)*T are the time instants the function is evaluated

if we consider, for instance, a simple RC lowpass filter having input "x" and output "y", using Laplace we have: transfer function = y/x = 1/(S*R*C+1)
this means a differential equation R*C*dy/dt + y = x
using, now, the difference quotient:

R*C*{y[i*T] - y[(i-1)*T]}/T + y(i*T) = x(i*T)

we can omit "T" as argument of the funcion, then:

R*C*[y(i) - y(i-1)]/T + y(i) = x(i)

rearranging:

(R*C/T+1)*y(i) - R*C/T*y(i-1) = x(i)

thus:

y(i) = 1/(R*C/T+1)*x(i) + (R*C/T)/(R*C/T+1)*y(i-1)

T is the so called timestep, the smaller the more accurate (and slower) the simulation.