In: Electrical Engineering

For the following causal difference equation,

given that y[-1] = 2, y[-2] = 3, and x[n] = 3^{n}u[n],
solve using z-Transforms.

(Hint: convert to delay operator form, find the z-Transform, use
PFE to find the inverse z-Transform)

y[n + 2] – 3y[n + 1] + 2y[n] = x[n + 1]

For the following causal difference equation,
given that y[-1] = 0, y[-2] = 1, and x[n] = u[n], solve using
z-Transforms.
(Hint: convert to delay operator form, find the z-Transform, use
PFE to find the inverse z-Transform)
4y[n + 2] + 4y[n + 1] + 2y[n] = x[n + 1]

Given a difference equation x[n+2] + 5x[n+1]+6x[n]=n with start
values x[0]= 0 and x[1]=0

A 5th filter is described by the difference equation: 2y(n)=2
x(n)+7 x(n-1)+3 x(n-2)-8 x(n-3)+ x(n-4)-8 x(n-5)+7 y(n-1)-3
y(n-2)+5y(n-3)- y(n-4) Determine the frequency response. Plot the
magnitude and the phase response of this filter. Consider the plot
-π≤w≤π for 501 points. Describe the magnitude response (Low pass
filter, High Pass filter, etc.) Determine the system stability.
Determine the impulse response h(n). You may set the period to
-100≤n≤100 Determine the unit step response for -100≤n≤100 .
(Matlab)

Given the surface x= (3 (y-1)^2)+ (2 (z+ 3)^2)+ 7. Find an
equation of the tangent plane at the point(12,2,2) in four
ways:
a) By using the surface as given,x=h(y,z)
b) By writing the surface as z=f(x,y)(only keep the square root
whose sign corresponds to the point(12,2,2)).
c) By writing the surface as y=g(x,z)(only keep the square root
whose sign corresponds to the point(12,2,2)).

1. Determine forced and natural response for the following
equation, where y[-1] =1, y[-2]=1 and x[n] = delta[n].
y[n]+1/2 y[n-1] -1/4 y[n-2] = x[n]
2. find the impulse response of the system described below
y[n]=1/2 y[n-1] + 2x[n]

Given the utility function U ( X , Y ) = X 1 3 Y 2 3, find the
absolute value of the MRS when X=10 and Y=24.
Round your answer to 4 decimal places.

Determine where the system is 1) linear 2) causal 3) stable and
4) time-invariant. a)y(n)= nx(n) b) y(n)=x square(n)

Consider the following equation: (3 − x^2 )y'' − 3xy' − y = 0
Derive the general solution of the given differential equation
about x = 0. Your answer should include a general formula for the
coefficents.

For each of the following difference equations
(i) y[n + 1] + 1.5y[n] = x[n]
(ii) y[n + 1] + 0.8y[n] = x[n]
(iii) y[n + 1] - 0.8y[n] = x[n]
For the difference equations given in Problem 2.16:
(a) Find a closed-form solution for y[n] when x[ n] = 0 for all
n and y[0] 2.
(b) Find a closed-form solution for y[n] when x[n] = u[n] and y[
0] = 0.
(c) Find a closed-form solution for y[n]...

A discrete system is described by the difference equation y(n)=
2.5y(n-1)-y(n-2)+3x(n)+3x(n-2)
a. Using Z-transform, Determine all
possible impulse responses h(n) and indicate the casuality and
stability
properties.
b.For the casual filter, determine the output y(n) if the input
is x(n)=g(n)-2g(n-1) where g(n)=cos (pin/s)u(n).

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