Math

Do a long hand division of [pmath] 1/{1-x} [/pmath] 

x greater than or equal to 1 does not result in convergence of this sum.  However this algorithm can still be used to do some interesting things.  Let us use a complex value of  [pmath] x = .707+.707i [/pmath] 

Each power of x yields a result one step around this unit circle. Thus this series is the Z transform of the associated sequence.  [1,0] , [0.707,0.707] , [0,1] ……. This sequence is [pmath] Sin( (n-1)*pi/{4}) [/pmath] 

Thus the z transform of this sequence is:   [pmath] 1/{1-(0.707+0.707i)*x}    [/pmath] 

If you want to get express in terms of n instead of n-1 you can multiply by 1/x.  Since x is the place holder it is easy to see if you want to slide a series one unit to the left by dividing by x. 

[pmath] Sin( n*pi/{4}) [/pmath]   : note this series starts at 45 degrees phase!

More information:

• Generating Functions
• Z Transform

The series:  [pmath] x^n  [/pmath]    [pmath]doubleleftright[/pmath] [pmath]1+ x + x^2 + x^3 + ….  = 1/{1-x}    n=0,1,2… [/pmath]   this converges for x < 1 : Both of these expressions are the Z transform of the [pmath] x^n  [/pmath] exponential decay sequence.  The first expression is easier to deal with because it is smaller and easier to work with.  The following diagram uses a decay sequence with [pmath] x = 0.75  [/pmath]

The filter takes what ever input it is fed and every interval multiplies it by 0.75.   To see the filters time response you can thus feed in a single ping.  This results in the filter output tracing out a special response.  This  is called the impulse response. It is the same as the filter plot above.

The exponential decay is maximum entropy.  That is to say this is how concentrated things soak out into the rest of the world as they become more dilute.