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EE 453: Homework 1Due in PDF on Canvas: 5 February 2021Fundamental Problems (20 pts)1. Ã‚Â 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. Ã‚Â Ã‚Â (a) Ã‚Â Simplify (1?j)4into a single real term.(b) Ã‚Â Find all solutions toz6= 1. Ã‚Â Plot your solutions in the complex plane.3. Ã‚Â 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. Ã‚Â Consider a finite impulse response (FIR) filter described by the difference equationy[n] =x[n]+x[n?10].(a) Ã‚Â Compute the magnitude|H(ej?)|and phase?H(ej?) response(b) Ã‚Â Determine the system output if the input isx[n] = cos(?10n)+ 3 sin(?3n+?10)5. Ã‚Â Given thecomplexfunctionf(z) =1+z1?z:(a) Ã‚Â Find the derivative dfdz(HINT: The quotient rule works for this function)(b) Ã‚Â Determine wheref(z) isNOTanalytic (i.e. Ã‚Â where the derivative does not exist)Advanced Problems (40 pts)6. Ã‚Â After sampling a continuous-time signal, we obtain the discrete-time signal sin(?3n). Ã‚Â Suppose the sam-pling rate wasFs= 5kHz.(a) Ã‚Â Determine the set of all continuous time sinusoidal signals, which when sampled yield the observeddiscrete-time signal.(b) Ã‚Â Repeat, but this time prior to sampling, an ideal low pass antialiasing filter is used with a cutoff atfc= 25kHz.7. Ã‚Â Find a 3-term difference equation for theaccumulator systemy[n] =?nk=??x[k]8. Ã‚Â Consider thebackward differencesystemy[n] =?x[n] =x[n]?x[n?1]1

(a) Ã‚Â Find the frequency responseH(ej?)(b) Ã‚Â Ifx[n] =f[n]? g[n], does?x[n] = (?f[n])? g[n] =f[n]?(?g[n])? Ã‚Â Prove your answer.(c) Ã‚Â Find the inverse systemhi[n], such thathi[n]?(?x[n]) =x[n] i.e.hi[n]?(x[n]?x[n?1]) =x[n].9. Ã‚Â A Ã‚Â causal Ã‚Â and Ã‚Â stable Ã‚Â LTI Ã‚Â system Ã‚Â has Ã‚Â inputx[n] Ã‚Â and Ã‚Â outputy[n] Ã‚Â and Ã‚Â is Ã‚Â represented Ã‚Â by Ã‚Â the Ã‚Â differenceequationy[n] +10?k=1?ky[n?k] =x[n] +?x[n?1]let the impulse response be represented byh[n](a) Ã‚Â Is the value ath[0] = 0? Ã‚Â Prove your answer.(b) Ã‚Â Derive a two term relation for the value of?1(such as?1= 5?+ 2h[3])10. Ã‚Â Ã‚Â (a) Ã‚Â 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. Ã‚Â You doNOTneed to solve for both (the proofis nearly identical).(b) Ã‚Â Prove that the real and imaginary parts of an analytic function of a complex variable, when expressedin polar form, satisfy LaplaceÃ¢â‚¬â„¢s equation in polar form?2??r2+1r???r+1r2?2???2= 0You only need to prove that eitheruorvsatisfies LaplaceÃ¢â‚¬â„¢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 Ã¢Ë†â€™ 1], x[n], x[n + 1]}

Pn

(d) y[n] = k=Ã¢Ë†â€™Ã¢Ë†Å¾ x[k]

2. (a) Simplify (1 Ã¢Ë†â€™ 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(Ãâ‚¬n/6)

(b) x[n] = ej(3Ãâ‚¬n/4)

(c) x[n] = sin(Ãâ‚¬n/5)/(Ãâ‚¬n)

Ã¢Ë†Å¡

(d) x[n] = ejÃâ‚¬n/

2

4. Consider a finite impulse response (FIR) filter described by the difference equation y[n] = x[n]+x[nÃ¢Ë†â€™10].

(a) Compute the magnitude |H(ejÃâ€° )| and phase Ã¢Ë†Â H(ejÃâ€° ) response

Ãâ‚¬

(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

Ãâ‚¬

3n

+

Ãâ‚¬

10

1+z

1Ã¢Ë†â€™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.

Ãâ‚¬

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=Ã¢Ë†â€™Ã¢Ë†Å¾ x[k]

8. Consider the backward difference system

y[n] = Ã¢Ë†â€šx[n] = x[n] Ã¢Ë†â€™ x[n Ã¢Ë†â€™ 1]

1

(a) Find the frequency response H(ejÃâ€° )

(b) If x[n] = f [n] ? g[n], does Ã¢Ë†â€šx[n] = (Ã¢Ë†â€šf [n]) ? g[n] = f [n] ? (Ã¢Ë†â€šg[n])? Prove your answer.

(c) Find the inverse system hi [n], such that hi [n] ? (Ã¢Ë†â€šx[n]) = x[n] i.e. hi [n] ? (x[n] Ã¢Ë†â€™ x[n Ã¢Ë†â€™ 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] +

ÃŽÂ±k y[n Ã¢Ë†â€™ k] = x[n] + ÃŽÂ²x[n Ã¢Ë†â€™ 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 ÃŽÂ±1 (such as ÃŽÂ±1 = 5ÃŽÂ² + 2h[3])

10. (a) Prove that in polar form, the Cauchy-Riemann equations

Ã¢Ë†â€šv

Ã¢Ë†â€šu

Ã¢Ë†â€šv

(f (z) = u(x, y) + jv(x, y) =Ã¢â€¡â€™ Ã¢Ë†â€šu

Ã¢Ë†â€šx = Ã¢Ë†â€šy and Ã¢Ë†â€šy = Ã¢Ë†â€™ Ã¢Ë†â€šy ) can be written

Ã¢Ë†â€šu

1 Ã¢Ë†â€šv

=

Ã¢Ë†â€šr

r Ã¢Ë†â€šÃŽÂ¸

Ã¢Ë†â€šv

1 Ã¢Ë†â€šu

=Ã¢Ë†â€™

Ã¢Ë†â€šr

r Ã¢Ë†â€šÃŽÂ¸

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 LaplaceÃ¢â‚¬â„¢s equation in polar form

1 Ã¢Ë†â€š2ÃŽÂ¨

Ã¢Ë†â€š 2 ÃŽÂ¨ 1 Ã¢Ë†â€šÃŽÂ¨

+

+

=0

Ã¢Ë†â€šr2

r Ã¢Ë†â€šr

r2 Ã¢Ë†â€šÃŽÂ¸2

You only need to prove that either u or v satisfies LaplaceÃ¢â‚¬â„¢s equation, you do NOT need to show

it for both (the proof is nearly identical).

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