The unit step sequence

u(n) =    1,       n≥ 0

     0,      n < 0

u(argument) =    1, if argument ≥ 0

 0, if argument < 0

a) The discrete delta function can be defined as the 1st difference of the unit step function:

δ(n) = u(n) - u(n-1)

b) The sum from -∞ to n of the δ function gives the unit-step as follows:

Results (a) and (b) are similar to the continuous-time derivative and integral respectively.

c) By going through the graph of u(n), which is shown below, we can write:

d) For any of the arbitrary sequence x(n), we have

x(n) δ(n-k) = x(k) δ(n-k)

which means that, the multiplication will pick out only one value x(k).

 If we find out the infinite sum of the above equation we get the sifting property:

e) We can express x(n) as follows:

x(n) = ...+ x(-1) δ(n+1) + x(0) δ(n) + x(1) δ(n-1) + x(2) δ(n-2) + ...

This true for all n by setting in turn

..., n = -2, n = -1, n = 0, n = 1, n = 2, etc. ... The above formula can be written compactly as follows

This is a weighted-sum of the unit sample functions which were delayed.

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