Calculate difference equation.b) Calculate the system impulse response

Calculate the z-tranform, draw the pole-zero plot and determine the stability for the followingdiscrete-time signalsi) x(n) ? 1 n ? 2= 0 elsewhereii) x(n) ? 1 4 ? n ? 8= 0 elsewhereiii)4x(n) sin n ?? n ? 0= 0 n ? 0iv)4x(n) sin n?? 0 ? n ? N ?1, N ? 8= 0 elsewherev)4x(n) 4n cos n ?? n ? 0= 0 n ? 0Question 2A 4 th order FIR system is defined asy(n) ? 0.0328 x(n) ? 0.24x(n ?1) ? 0.

455x(n ? 2) ? 0.24x(n ? 3) ? 0.0328 x(n ? 4)i) Calculate the transfer function H(z) for the difference equation.

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ii) Draw the pole-zero plot and determine if the system is stable.iii) Draw the signal flow graph for the IIR.iv) Given that ( ) ( ) ( ) 1 2 H z ? H z H z where H1(z) and H2(z) are 2 nd order transferfunctions. Find H1(z) and H2(z).

Question 3Calculate the inverse of the following z-transforms.a)11 241 14121 1( )?? ??? ??zz zX zb)1 218141 3( )? ??? ??z zX z zc)413cos( )22? ??z zX z z ?d)413cos3cos21( )2 ? ???z zzX z ??Question 4A 2 th order IIR system is defined asy(n)-0.866y(n ?1) ? 0.

25y(n ? 2) ? x(n) ? 2x(n ?1) ? x(n ? 2)a) Calculate the transfer function H(z) for the difference equation.b) Calculate the system impulse response h(n) directly using the partial fractionexpansion method and the shift property of the z-transform.c) Calculate and draw the frequency response |H(ej2?f)| based on the poles and zerospositions.Question 5A continuous-time linear time-invariant system is defined by the following transfer function? ?2 ? ?224 100 2 1000( ) 2 1000? ??? ??s sH sa) Calculate the poles and zeros for H(s).b) Based on the Nyquist sampling theorem, what is the suitable sampling frequency?c) If the sampling frequency is chosen at 4000 Hz, calculate the transfer function H(z)for the equivalent discrete-time system using the poles of H(s) and the impulseinvariant method.

d) Repeat part c), but use the bilinear transformation method.e) Compare the frequency response of the discrete-time system |H(ej2?f)| obtained in partb) and c). You may use MATLAB to solve this problem