EE 453 PSU Discrete Time Systems Linear Shift Invariant Causal & BIBO Stable Ques

  

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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(?n/6)(b)x[n] =ej(3?n/4)(c)x[n] = sin(?n/5)/(?n)(d)x[n] =ej?n/?24.  Consider a finite impulse response (FIR) filter described by the difference equationy[n] =x[n]+x[n?10].(a)  Compute the magnitude|H(ej?)|and phase?H(ej?) response(b)  Determine the system output if the input isx[n] = cos(?10n)+ 3 sin(?3n+?10)5.  Given thecomplexfunctionf(z) =1+z1?z:(a)  Find the derivative dfdz(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(?3n).  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(ej?)(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=1?ky[n?k] =x[n] +?x[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 of?1(such as?1= 5?+ 2h[3])10.   (a)  Prove that in polar form, the Cauchy-Riemann equations(f(z) =u(x,y) +jv(x,y) =??u?x=?v?yand?u?y=??v?y) can be written?u?r=1r?v???v?r=?1r?u??You 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?2??r2+1r???r+1r2?2???2= 0You only need to prove that eitheruorvsatisfies Laplace’s equation, you doNOTneed to showit for both (the proof is nearly identical).

EE 453: Homework 1
Due in PDF on Canvas: 5 February 2021
Fundamental 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+1
x[n]
(b) y[n] = sin(x[n + 1])
(c) y[n] = max{x[n − 1], x[n], x[n + 1]}
Pn
(d) y[n] = k=−∞ x[k]
2. (a) Simplify (1 − j)4 into a single real term.
(b) Find all solutions to z 6 = 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 the
fundamental period.
(a) x[n] = ej(Ï€n/6)
(b) x[n] = ej(3Ï€n/4)
(c) x[n] = sin(Ï€n/5)/(Ï€n)
√
(d) x[n] = ejπn/
2
4. Consider a finite impulse response (FIR) filter described by the difference equation y[n] = x[n]+x[n−10].
(a) Compute the magnitude |H(ejω )| and phase ∠H(ejω ) response

Ï€
(b) Determine the system output if the input is x[n] = cos 10
n + 3 sin
5. Given the complex function f (z) =
(a) Find the derivative
df
dz
Ï€
3n
+
Ï€
10

1+z
1−z :
(HINT: The quotient rule works for this function)
(b) Determine where f (z) is NOT analytic (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
pling rate was Fs = 5kHz.
Ï€
3n

. Suppose the sam-
(a) Determine the set of all continuous time sinusoidal signals, which when sampled yield the observed
discrete-time signal.
(b) Repeat, but this time prior to sampling, an ideal low pass antialiasing filter is used with a cutoff at
fc = 25kHz.
Pn
7. Find a 3-term difference equation for the accumulator system y[n] = k=−∞ x[k]
8. Consider the backward difference system
y[n] = ∂x[n] = x[n] − x[n − 1]
1
(a) Find the frequency response H(ejω )
(b) If x[n] = f [n] ? g[n], does ∂x[n] = (∂f [n]) ? g[n] = f [n] ? (∂g[n])? Prove your answer.
(c) Find the inverse system hi [n], such that hi [n] ? (∂x[n]) = x[n] i.e. hi [n] ? (x[n] − x[n − 1]) = x[n].
9. A causal and stable LTI system has input x[n] and output y[n] and is represented by the difference
equation
10
X
y[n] +
αk y[n − k] = x[n] + βx[n − 1]
k=1
let the impulse response be represented by h[n]
(a) Is the value at h[0] = 0? Prove your answer.
(b) Derive a two term relation for the value of α1 (such as α1 = 5β + 2h[3])
10. (a) Prove that in polar form, the Cauchy-Riemann equations
∂v
∂u
∂v
(f (z) = u(x, y) + jv(x, y) =⇒ ∂u
∂x = ∂y and ∂y = − ∂y ) can be written
∂u
1 ∂v
=
∂r
r ∂θ
∂v
1 ∂u
=−
∂r
r ∂θ
You only need to solve for ONE of the equations. You do NOT need to solve for both (the proof
is nearly identical).
(b) Prove that the real and imaginary parts of an analytic function of a complex variable, when expressed
in polar form, satisfy Laplace’s equation in polar form
1 ∂2Ψ
∂ 2 Ψ 1 ∂Ψ
+
+
=0
∂r2
r ∂r
r2 ∂θ2
You only need to prove that either u or v satisfies Laplace’s equation, you do NOT need to show
it for both (the proof is nearly identical).
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