II. Echo generation and cancelation Given a signal x(n) and (0,1), one can simulate an echo by creating the following signal y(n)=x(n)+x(nd), where dN is the delay factor. 1. Explain in your own words why this is an echo filter. 2. Write the impulse response of the filter, h(n), that transforms x(n) to y(n). Write its Z-transorm and compute its Fourier transform. What kind of filter is it? 3. Write the inverse filter that transforms y(n) to x(n) in the z-plane. Compute the poles. Is it stable? Does it have a Fourier transform? If yes, compute it, what kind of filter is it? 4. Let us define the auto-correlation xx(k)=n=x(n)x(nk) compute the relationship between xx and yy. 5. Explain why yy has a strong local maximum at 0,d,d. 6. Design an algorithm for finding a,d from y(n). 7. Implement your algorithm in Python and test it on your own recorded voice for a=0.5,d=3. In applications filters might need to adapt their defining parameters based on real-time needs, such filters are called adaptive filters. Adaptive filtering is necessary in many cases, like adaptive antenna systems, digital communication receivers, adaptive noise canceling techniques, systems modeling ... In the following exercise we witl implement a simple adaptive filter via least mean square method. Least mean square for coefficient adiustment Suppose that we have a EIR-filter with adaptable coefficients {h(k)kN1} Denote by x(n) and y(n) the input and output signals of the filter, respectively. Suppose we have a desired output signal d(n) which we can compare to the output y(n). Define e(n)=y(n)d(n) and set =n=0Me(n)2 where M is the common length of y(n) and d(n). 1. Explain in your own words why can be considered as the distance or error between the required output d(n) and the resulting output y(n) 2. Prove the following equality =n=0Md(n)22k=0N1h(k)rdx(k)+k=0N1l=0N1h(k)h(l)rxx(kl) When a Python function is requested, you must both wite the code in your theoretical answer and aiso provide a Python file ready to run. Write detailed and well explained answers. 1. Ellier analyazis Consider the filter T yn+21Jn1=xnxn4 and denote by h=T(n) its impulse response. 1. Is the filter conditional? 2. Compute H(z)=Z[n](z). 3. Compute the ROC of H(z)= 4. Is the filter stable? 5. Is the filter FIR or IIR. 6. Explain why h^(u) the FT of hn exists. Compute h^(u). 7. Compute h^(u) at u=0,41,21. Is the filter high-pass, low-pass or neither of the two? 8. Find yn for n5 for the input xn={2n10n0n