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Problem 1 A) Write down Euler’s identity b) Write x{n}=cos wn as a sum of complex exponentials

Electrical Engineering

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Problem 1

A) Write down Euler’s identity

b) Write x{n}=cos wn as a sum of complex exponentials.

 

Problem 2

?₁ =1-j    ?₃ -1-j      ?₅ =2ej^π/3

?₂ =e^jr/2    ?₁ = e^jπ/4

Give the answers in polar form:

a) ?₁^*  = -----------------

b) ?₂ ?₁^* = -----------------

c) 1/ ?₃ = ----------------------

d) ?₃ + ?₃ ^* = -----------------

e) 1/ ?₅ = ----------------------

f) ?₃/ ?₂ = ---------------------

g) ?₃ ?₃/ ?₅* = ----------------

 

Problem 3

Consider the following 3 complex numbers.

a) Write down each complex number in polar form : -------------------

b) Are they solutions to ?³ = 1 ? Explain

                                                                  Im(?)

                                               120          120

                                -1/2                                     1/2               Re(?)

                                                    120

                                                                    -√3/2

Problem 4

Convert x(t) to a sum of complex exponentials and sketch the magnitude and phase plots

X(t) = 2cos (2π100t + π/4) – 2 cos (2 π 200t+ π/2)

 

Problem 5

X(t) – (A/D)-------X(n)

       F_s = 200H?

X(t) = 5cos (2π50t)

a) Find x{n}

b) Find the following, including unit’s frequency:

Angular frequency:

Normalized frequency :

Normalize angular frequency:

                                                                                                    X(n)

Problem 6                                                                                  

X(t) ---------{A/D}------------x(n)                        -1                      1                  3                              

   F₃ = 100H?                                                                     -1                                                            5

Provide two possible functions x(t) that could have produced x{n}

 

 

Problem 7

X(t) = 2cos 2π100 t + 3cos 2π200 t + 4 cos 2π400 t

Is sampled at f_s= 350 H ?.

                                                                   

Find y^(t) and give the values of any alias frequencies in H?.

 

Problem 8

Please circle final answers

A⁽¹⁾ =     120-1

         A⁽²⁾ =     21-10

         B =     d-d1+jd

        

 

V⁽¹⁾ = -12

               V⁽²⁾ = -11

                      V⁽³⁾ = 11

                     V⁽⁴⁾ = -jj

 

 

 

Find

a) A⁽¹⁾ A⁽²⁾

b) A⁽²⁾ V⁽²⁾

c) 2V⁽³⁾ + V⁽⁴⁾

d) B^h

e) (V⁽¹⁾ , V⁽²⁾)

f) (V⁽⁴⁾ , V⁽³⁾)

 

Problem 9

a) Let   A= abcdefghi

       and P = 001010100

 

Find AP and PA

b) Find B = (e⁽²⁾)^T Ae⁽¹⁾

 

Problem 10

 

1000

A) Find the projection of  a = 2-22-2

              onto B =                                                     

 

 

B) Find the Projection of b onto a