Homework #1

Signals and Systems – Columbia University – Fall 2007

Due 09/13/07

 

 

1.1 Express the following numbers in polar form:

            (a) 12+j4 (b) 3-j8 (c) –46+j100 (d) -2-j21

 

1.2 Represent the following numbers in the complex plane and express them in Cartesian form:

            (a) ejπ/2 (b) 5ej2π/3 (c) 11e-2πj/5 (d) 6.2ej3π (e) 5ej4π/9 (f) ejπ/6

 

1.3 Determine z1z2 and z1/z2 in both polar and Cartesian forms.

            z1=-3-j5

            z2=5-7j

 

1.4 For z1=ejπ/7 and z2= 5e-j3π/5, find:

            (a) z2-3 z1

 

            (b) 1/z2

 

            (c) z2/z12

 

            (d) z1-1/2

 

1.5 Consider F(w), a complex function of a real variable w:

F(w) = (2 – j5w ) / ( -3 + jw)

 

            (a) Express F(w) in Cartesian form and find its real and imaginary parts.

            (b) Express F(w) in polar form and find its magnitude | F(w)| and angle ∠F(w).

 

1.6 In the following cases, express f(t) as a single sinusoid:

            (a) f(t) = 6cos(wt) – (5)1/2 sin(wt)

            (b) f(t) = -18cos(wt) + 2sin(wt)


 

Please drop assignment by 5PM on due date, in mailbox H2 or H8, in the EE Student Lounge (13th Floor Mudd Building)