1. The impulse response of a discrete time system is h[n] = 4"U[n]. Write down the difference equation for this system and calculate the zero-state step response of the system to the unit x[n] = 3U[n]. 2. Consider the Z transform of system H(Z) = Z*(Z-0.5)¹ and: (a) Compute the first 3 terms of the impulse response and write down the difference equation. (b) Sketch the Poles and Zeroes of the System. (c) Sketch the Magnitude Frequency Response of the System. 3. A Linear System has equation y[n] = 0.125y[n-2] -0.25y[n-1]+x[n] +0.25x[n-1] with initial condition y[-1] = 2 and y[-2]=-1. Calculate the Zero input response to this system up to the forth term. 4. A Linear System has difference equation y[n]-0.5y[n-1] x[n]+0.25x[n-1]. If the input is x[n] = 0.5" calculate the zero-state response using Z Transform and Inverse Z Transform method. Hint: The Binomial Theorem is given in the appendix to assist with this problem. 5. A Linear System has difference equation y[n]-0.5y[n-1] = x[n]+0.25x[n-1]. If the input is x[n] (-0.25)" calculate the total system response if the initial condition y[-1] 2. Hint: Find the System Zero to assist in calculating the Zero State Response. 6. For an RLC series circuit with parameters R = 2.52, C=1/6F and LH, use the Laplace transform to solve the initial value problem i(0) = 4, and i'(0) = % if i(t) is the current flow in the circuit. Note, the differential equation to the circuit is: Li"+Ri'+i/C V' with V the driving voltage and i the circuit current. 7. Consider the Transfer Function H(S)- = S² (1 + 25)(1+5)(3+5) Compute the impulse response to this circuit. 8. Consider the transfer function H(S) = S (1+25) Compute the frequency response, phase response and sketch the bode plot. 9. Compute the convolution between h(t) = 2e2t U(t) and x(t)=3eU(t). You can use any method you desire provided your working out is clear. 10. If the impulse response h(t) of a system is defined as 0.1e(-0.1) U(t), compute the Zero State Response of the System to an input x(t) = Arect(2t/Ts). Ts = 0.2s, A = 2V. You can use any method you desire provided your working out is clear. 11. Calculate the Fourier Transform for the product problem x(t)h(t) with the signal x(t) being a cosine with unit amplitude and frequency of 2Hz and the impulse response h(t) Arect(2t/Ts). Ts = 10s, A = 1V.