1) Using z-transform tables (page 776 of text or equivalent), find the z-transform of
a) x(n) = 2 (0.1^n) u(n) - (-0.5^n-1)u(n-1)
b) x(n) = (1/3)^n-2 u(n-2)
c) x(n) = cos (pi/4 n) u(n) (a) and express it in positive powers of z. When computing the value of trigonometric functions, keep in mind that the arguments are always in radians and not in degrees.
d) x(n) = (0.5)^n sin(pi/4 n) u(n) and express it in positive powers of z
2) Find the inverse z-transform, x(n), of the following functions by bringing them into a form such that you can look up the inverse z-transform from the tables. This will require some algebraic and /or trigonometric manipulation/calculation. You will also need a table of z-transforms (page 776 of text or equivalent). When computing the value of trigonometric functions, keep in mind that the arguments are always in radians and not in degrees.
(a) x(z) = (2z-0.5)/(Z+0.5)
then use the z-transform tables. Having found the inverse z-transform, determine its numerical value at n = 2.
(b) Using partial fraction expansion , find the inverse z-transform of 3/(z-0.2)(z+0.4)
(c) Using the tables, find the inverse z-transform of z^2-0.7071z/z^2-1.41z+1
d) Using the tables, find inverse z-transform of 0.2165z/z^2-0.25z+0.0625
3) Find the first seven values (i.e., x(n) for n = 0 to 6) of the function given below.
Hint: Manually calculate the three parts separately for various values of n and add or subtract them point by point for various values of n.
x(n) = [2nu(n) ] - [(n-3) u(n-3)] - [3u(n-4)]
Write the numerical values here as a row vector.