Discrete-time systems

EE 453: Homework 1Due in PDF on Canvas: 5 February 2021Fundamental Problems (20 pts)1. Determine whether each of the discrete-time systems is (i) Linear (ii) Shift-Invariant (iii) Causal (iv)BIBO-Stable:(a)y[n] =n+1x[n](b)y[n] = sin(x[n+ 1])(c)y[n] = max{x[n−1]x[n]x[n+ 1]}(d)y[n] =∑nk=−∞x[k]2. (a) Simplify (1−j)4into a single real term.(b) Find all solutions toz6= 1. Plot your solutions in the complex plane.3. Determine whether or not each of the following discrete signals is periodic and if it is periodic state thefundamental period.(a)x[n] =ej(πn/6)(b)x[n] =ej(3πn/4)(c)x[n] = sin(πn/5)/(πn)(d)x[n] =ejπn/√24. Consider a finite impulse response (FIR) filter described by the difference equationy[n] =x[n]+x[n−10].(a) Compute the magnitude|H(ejω)|and phase∠H(ejω) response(b) Determine the system output if the input isx[n] = cos(π10n)+ 3 sin(π3n+π10)5. Given thecomplexfunctionf(z) =1+z1−z:(a) Find the derivativedfdz(HINT: The quotient rule works for this function)(b) Determine wheref(z) isNOTanalytic (i.e. where the derivative does not exist)Advanced Problems (40 pts)6. After sampling a continuous-time signal we obtain the discrete-time signal sin(π3n). Suppose the sam-pling rate wasFs= 5kHz.(a) Determine the set of all continuous time sinusoidal signals which when sampled yield the observeddiscrete-time signal.(b) Repeat but this time prior to sampling an ideal low pass antialiasing filter is used with a cutoff atfc= 25kHz.7. Find a 3-term difference equation for theaccumulator systemy[n] =∑nk=−∞x[k]8. Consider thebackward differencesystemy[n] =∂x[n] =x[n]−x[n−1]1 (a) Find the frequency responseH(ejω)(b) Ifx[n] =f[n]? g[n] does∂x[n] = (∂f[n])? g[n] =f[n]?(∂g[n])? Prove your answer.(c) Find the inverse systemhi[n] such thathi[n]?(∂x[n]) =x[n] i.e.hi[n]?(x[n]−x[n−1]) =x[n].9. A causal and stable LTI system has inputx[n] and outputy[n] and is represented by the differenceequationy[n] +10∑k=1αky[n−k] =x[n] +βx[n−1]let the impulse response be represented byh[n](a) Is the value ath[0] = 0? Prove your answer.(b) Derive a two term relation for the value ofα1(such asα1= 5β+ 2h[3])10. (a) Prove that in polar form the Cauchy-Riemann equations(f(z) =u(xy) +jv(xy) =⇒∂u∂x=∂v∂yand∂u∂y=−∂v∂y) can be written∂u∂r=1r∂v∂θ∂v∂r=−1r∂u∂θYou only need to solve forONEof the equations. You doNOTneed to solve for both (the proofis nearly identical).(b) Prove that the real and imaginary parts of an analytic function of a complex variable when expressedin polar form satisfy Laplace’s equation in polar form∂2Ψ∂r2+1r∂Ψ∂r+1r2∂2Ψ∂θ2= 0You only need to prove that eitheruorvsatisfies Laplace’s equation you doNOTneed to showit for both (the proof is nearly identical).

Requirements: as long as needed