Signals and Systems - Electrical Engineering

Problems 109

(a) Determine the periodT 0 ofx(t).

(b)Compute the powerPxofx(t).

(c)Verify that the powerPxis the sum of the powerP 1 ofx 1 (t)=cos( 2 πt)and the powerP 2 ofx 2 (t)=

2 cos(πt).

(d)In the above case you are able to show that there is superposition of the powers because the fre-

quencies are harmonically related. Suppose thaty(t)=cos(t)+cos(πt)where the frequencies are not

harmonically related. Find out whethery(t)is periodic or not. Indicate how you would find the power

Pyofy(t). WouldPy=P 1 +P 2 whereP 1 is the power ofcos(t)andP 2 is the power ofcos(πt)? Explain

what is the difference with respect to the case of harmonic frequencies.

1.4. Periodicity of sum of sinusoids—MATLAB
Consider the periodic signalsx 1 (t)=4 cos(πt)andx 2 (t)=−sin( 3 πt+π/ 2 ).
(a) Find the periods ofx 1 (t)andx 2 (t).
(b)Is the sumx(t)=x 1 (t)+x 2 (t)periodic? If so, what is its period?
(c)In general, two periodic signalsx 1 (t)andx 2 (t)having periodsT 1 andT 2 such that their ratioT 1 /T 2 =
M/Kis a rational number (i.e.,MandKare positive integers), then the sumx(t)=x 1 (t)+x 2 (t)is
periodic. Suppose the rationality condition is satisfied andM= 3 andK= 12. Determine the period of
x(t).
(d)Determine whetherx(t)=x 1 (t)+x 2 (t)is periodic when
n x 1 (t)=4 cos( 2 πt)andx 2 (t)=−sin( 3 πt+π/ 2 )
n x 1 (t)=4 cos( 2 t)andx 2 (t)=−sin( 3 πt+π/ 2 )
Use symbolic MATLAB to plotx(t)in the above two cases and confirm your analytic results about the
periodicity or lack of periodicity ofx(t).

1.5. Time shifting
Consider a finite-support signal

x(t)=t 0 ≤t≤ 1

and zero elsewhere.

(a) Carefully plotx(t+ 1 ).

(b)Carefully plotx(−t+ 1 ).

(c)Add the above two signals to get a new signaly(t). To verify your results, represent each of the above

signals analytically and show that the resulting signal is correct.

(d)How doesy(t)compare to the signal3(t)=( 1 −|t|)(u(t+ 1 )−u(t− 1 )? Plot them. Compute the

integrals ofy(t)and3(t)for all values oftand compare them. Explain.

1.6. Even and odd hyperbolic functions—MATLAB
According to Euler’s identity the sine and the cosine are defined in terms of complex exponentials. You
would then ask what if instead of complex exponentials you were to use real exponentials. Well, using
Euler’s identity we obtain the hyperbolic functions defined in−∞<t<∞:

cosh( 0 t)=

e^0 t+e−^0 t

2

sinh( 0 t)=

e^0 t−e−^0 t

2

(a) Let 0 = 1 rad/sec. Use the definition of the real exponentials to plotcosh(t)andsinh(t).

(b)Iscosh(t)even or odd?