**Exercise**

We consider the cascaded LTI system of the figure 1

Figure 1

And the impulsional response h2[n]=u[n]-u[n-2] where u[n] is a discrete Heavside signal.

The global impulsional response of the system is shwon on Figure 2.

Figure 2

1- Find the impulsional response h1[n].

2-Find the global response of the system if the input is x[n]=δ[n]-δ[n-1].

**Answers**

1- Answer to question 1

1- Answer to question 1

Since the blocks of Figure 1 are in cascade, theglobal system h[n]=h1*h2[n]*h2[n] * is the symbol of convolution

so

From Figure 2 we can obtain the Z transform of the global response

From the prperties of Z transform in convolution we know that the H[z]=H1[z].H2[z].H2[z]

so H1[z]=Y[z]/(H2[z].H2[z])

And h1[n] is the discrete time signal defined by

h[n]=1 if n=0

h[n]=3 if n=1

h[n]=3 if n=2

h[n]=2 if n=3

h[n]=1 if n=4

h[n]=0 else

**Answer to question 2**

No idea.

No idea.

Thank you.