Signals and Systems - Electrical Engineering

108 C H A P T E R 1: Continuous-Time Signals

Problems............................................................................................

1.1. Signal energy and RC circuit—MATLAB

The signalx(t)=e−|t|is defined for all values oft.

(a) Plot the signalx(t)and determine if this signal is finite energy. That is, compute the integral

∫∞

−∞

|x(t)|^2 dt

and determine if it is finite.

(b)If you determine thatx(t)is absolutely integrable, or that the integral

∫∞

−∞

|x(t)|dt

is finite, could you say thatx(t)has finite energy? Explain why or why not.Hint:Plot|x(t)|and|x(t)|^2

as functions of time.

(c)From your results above, is it true the energy of the signal

y(t)=e−tcos( 2 πt)u(t)

is less than half the energy ofx(t)? Explain. To verify your result, use symbolic MATLAB to ploty(t)

and to compute its energy.

(d)To discharge a capacitor of 1 mF charged with a voltage of 1 volt we connect it, at timet= 0 , with a

resistor ofR. When we measure the voltage in the resistor we find it to bevR(t)=e−tu(t). Determine

the resistanceR. If the capacitor has a capacitance of 1 μF, what would beR? In general, how areR

andCrelated?

1.2. Power in RL circuits

Consider a circuit consisting of a sinusoidal sourcevs(t)=cos(t)u(t)volts connected in series to a resistor

Rand an inductorLand assume they have been connected for a very long time.

(a) LetR= 0 andL=1 H. Compute the instantaneous and the average powers delivered to the inductor.

(b)LetR= 1 andL=1 H. Compute the instantaneous and the average powers delivered to the resistor

and the inductor.

(c)LetR= 1 andL=0 H. Compute the instantaneous and the average powers delivered to the resistor.

Hint:In the above parts of the problem use phasors or the trigonometric formula

cos(α)cos(β)=0.5[cos(α−β)+cos(α+β)]

(d)The average power used by the resistor is what you pay to the electric company, but there is also a

reactive power for which you do not. The complex power supplied to the circuit is defined as

P=

1

2

VsI∗

whereVsandIare the phasors corresponding to the source and the current in the circuit, andI∗is the

complex conjugate ofI. Consider the values of the resistor and the inductor given above, and compute

the complex power and relate it to the average power computed in each case.

1.3. Power in periodic and nonperiodic sum of sinusoids

Consider the periodic signalx(t)=cos( 2  0 t)+2 cos( 0 t),−∞<t<∞, and 0 =π. The frequencies of

the two sinusoids are said to be harmonically related (one is a multiple of the other).