digital signal analysis assignment z transforms

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Determine the z-transform of each of the following. Include with your answer the region of convergence in the z-plane and a sketch of the pole-zero patterns. Express all sums in closed form.
a can be complex.

a. a^|n|, 0<=|a|<1

b. x(n)= 1, 0<=n<=N-1
       = 0, N<=n
       = 0, n<0

c. x(n)= n,     0<=n<=N
       = 2N-n,  N+1<=n<2N
       = 0,     n<0 or n>2N


Determine the inverse z-transform of:

a. X(z)= 1/(1+1/2*(z)^(-1)), |z|>1/2
b. X(z)= 1/(1+1/2*(z)^(-1)), |z|>1/2


Calculate the Z transforms of
a. x(n)= [1, 3/4, 1/2, 1/4, 0]
b. y(n)= [1, -3/4, 1/2, -1/4, 0]
c. Convolve x and y using the product of the z-transforms and then take the inverse Z-transform by inspection


Spiking  deconvolution using the z-transform. Given the time series:

a. x= [1, 1/2, -1/16, -1/32] for time t=[0, 1, 2, 3]
b. Take its z-transform.
c. Find the inverse filter for the z-transform of x by polynomial division for different orders of z^(-n). Let the maximum n be 1, 2, 3, 4 and 5. (In other words you truncate the deconvolution filter to 2, 3, 4, 5 and 6 points). This gives you 5 different filters w_n, to apply to the time series x to try to turn it into a delta function. A spiked time series is the delta function, which has the energy E=1.

i. Take the inverse z-transform of each filter by inspection
ii. Convolve each filter w_n with x in the time domain and plot the deconvolved time series.
y_n= w_n*x
iii. Calculate the energy in each deconvolved time series
E_n= sum|(k=0 to n) (y_n^2(k))
iv. Plot the quantitiy T=E_n-1, where E_n is the energy i the deconvolved time series, and R is energy difference between desired (E=1) and calculated energies in the spiked time series. Comment on the effect of increasing the filter length.

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