1.13 Show the use of the inverse Z transform

1.13.1 example 1
1.13.2 example 2

These examples show how to use the inverse a Z transform.

1.13.1 example 1

Problem: Given $F\left (z\right ) =\frac {z}{z-1}$ ﬁnd $$x[n]=F^{-1}\left (z\right )$$ which is the inverse Ztransform.

Mathematica

 Clear["Global*"]; x[n_]:= InverseZTransform[z/(z-1),z,n] ListPlot[Table[{n,x[n]},{n,0,10}], Frame->True, FrameLabel->{{"x[n]",None}, {"n","Inverse Ztransform of z/(z-1)"}}, FormatType->StandardForm, RotateLabel->False, Filling->Axis, FillingStyle->Red, PlotRange->{Automatic,{0,1.3}}, PlotMarkers->{Automatic,12}] 

Matlab

 function e19() nSamples = 10; z = tf('z'); h = z/(z-1); [num,den] = tfdata(h,'v'); [delta,~] = impseq(0,0,nSamples); xn = filter(num,den,delta); stem(xn); title('Inverse Z transform of z/(z-1)'); xlabel('n'); ylabel('x[n]'); ylim([0 1.3]); set(gcf,'Position',[10,10,400,400]); end %function from Signal Processing by Proakis %corrected a little by me for newer matlab version function [x,n] = impseq(n0,n1,n2) % Generates x(n) = delta(n-n0); n1 <= n,n0 <= n2 % ---------------------------------------------- % [x,n] = impseq(n0,n1,n2) % if ((n0 < n1) || (n0 > n2) || (n1 > n2)) error('arguments must satisfy n1 <= n0 <= n2') end n = n1:n2; x = (n-n0) == 0; end 

1.13.2 example 2

Problem: Given $F\left ( z\right ) =\frac {5z}{\left ( z-1\right ) ^{2}}$ ﬁnd $$x[n]=F^{-1}\left ( z\right )$$

In Mathematica analytical expression of the inverse Z transform can be generated as well as shown below

Mathematica

 Clear["Global*"]; x[n_]:= InverseZTransform[(5 z)/(z-1)^2,z,n]; ListPlot[Table[{n,x[n]},{n,0,10}], Frame->True, FrameLabel->{{"x[n]",None}, {"n","Inverse Ztransform of (5 z)/(z-1)^2"}}, FormatType->StandardForm, RotateLabel->False, Filling->Axis, FillingStyle->Red, PlotRange->{Automatic,{0,60}}, PlotMarkers->{Automatic,12}, ImageSize->350] 

Matlab

 function e19_2() nSamples = 10; z = tf('z'); h = (5*z)/(z-1)^2; [num,den] = tfdata(h,'v'); [delta,~] = impseq(0,0,nSamples); xn = filter(num,den,delta); stem(xn); title('Inverse Z transform of 5z/(z-1)^2'); xlabel('n'); ylabel('x[n]'); ylim([0 60]); set(gcf,'Position',[10,10,400,400]); end