Answer these following Questions:
Express the following numbers in Cartesian (rectangular) form.
- (a) e^(jπ/4)
- (b) e^(-jπ/4)
2. Express the following numbers in polar form. Describe the quadrant of the complex plane, in which the complex number is located.
- (a) 1 + j1
- (b) 1 - j1
- (c) -1 + j1
- (d) -1 - j1
3. (a) A continuous-time sine wave has a frequency of 60 Hz, an amplitude of 117 V, and an initial phase of π/4 radians. Describe this signal in a mathematical form using the Sin function.
(b) Determine the period of the signal. Be sure to mention the units of the period.
(c) Describe this signal in a complex exponential form (see pages 16-17 of the text).
4. A sinusoidal signal described by 50 Cos (20πt + π/4) passes through a linear time invariant (LTI) system that applies a gain of 1.5 and a phase lag of π/2 radians to the signal. Write the mathematical expression that describes the signal that will come out of the LTI system.
5. A sinusoidal signal described by 20 Cos (2πt + π/4) passes through a linear time invariant (LTI) system that applies a gain of 2 and a time delay of 0.125 seconds to the signal. Write the mathematical expression that describes the signal that will come out of the LTI system.
6. Apply the principle of superposition to determine whether the following systems are linear. Sketch what the plot of the function looks like.
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(a)Y = 5x + 1
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(b)Y = 5x
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(c)Y = 5X2
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7. A continuous time system, described by y(t) = 5 Cos (2*π*20*t + π/2), is sampled at a rate 320 Hz.
(a) Write the mathematical expression that describes the discrete time (sampled) signal.
(b) Calculate the value of the discrete signal for n = 0 and n = 4.
(c) Does the sampling rate meet the Nyquist condition? Justify your answer.
8. Sketch the odd and even part of the following discrete signal.
9. Express the signal given in Problem 8 as the sum of the following.
(a) Delayed and undelayed (if any) impulses
(b) Delayed and undelayed (if any) step functions.