58 C H A P T E R 0: From the Ground Up!
(c) Give an expression of the above sum for all possible values ofα.
(d) Suppose now thatN=∞; under what conditions willSexist? If it does, what wouldSbe equal to?
Explain.
(e) Suppose the derivative ofSwith respect toαisS 1 =
dS
dα=∑∞n= 0nαnObtain an expression to findS 1.
0.10. Exponentials—MATLAB
The exponentialx(t)=eatfort≥ 0 and zero otherwise is a very common analog signal. Likewise,y[n]=αn
for integersn≥ 0 and zero otherwise is a very common discrete-time signal. Let us see how they are
related. Do the following using MATLAB:
(a) Leta=−0.5; plotx(t).
(b) Leta=− 1 ; plot the corresponding signalx(t). Does this signal go to zero faster than the exponential
fora=−0.5?
(c) Suppose we sample the signalx(t)usingTs= 1 ; what would bex(nTs)and how can it be related to
y(n)(i.e., what is the value ofαthat would make the two equal)?
(d) Suppose that a currentx(t)=e−0.5tfort≥ 0 and zero otherwise is applied to a discharged capacitor
of capacitanceC=1 Fatt= 0. What would be the voltage in the capacitor att=1 sec?
(e) How would you obtain an approximate result to the above problem using a computer? Explain.
0.11. Algebra of complex numbers
Consider complex numbersz= 1 +j1,w=− 1 +j1,v=− 1 −j 1 , andu= 1 −j 1.
(a) In the complex plane, indicate the point(x,y)that corresponds tozand then show a vectorEzthat joins
the point(x,y)to the origin. What is the magnitude and the angle corresponding tozorEz?
(b) Do the same for the complex numbersw,v, andu. Plot the four complex numbers and find their sum
z+w+v+uanalytically and graphically.
(c) Find the ratiosz/w,w/v, andu/z. Determine the real and imaginary parts of each, as well as their
magnitudes and phases. Using the ratios findu/w.
(d) The phase of a complex number is only significant when the magnitude of the complex number is
significant. Considerzandy= 10 −^16 z; compare their magnitudes and phases. What would you say
about the phase ofy?
0.12. Algebra of complex numbers
Consider a function ofz= 1 +j 1 ,
w=ez
(a) Findlog(w).
(b)Find the real and the imaginary parts ofw.
(c)What isw+w∗, wherew∗is the complex conjugate ofw?
(d)Determine|w|,∠w.
(e)What is|log(w)|^2?
(f) Expresscos( 1 )in terms ofwusing Euler’s equation.
0.13. Euler’s identity and trigonometric identities
Use Euler’s identity to obtain an expression forej(α+β)=ejαejβ; obtain its real and imaginary components
and show the following identities:
n cos(α+β)=cos(α)cos(β)−sin(α)sin(β)
n sin(α+β)=sin(α)cos(β)+sin(β)cos(β)
Hint:Find real and imaginary parts ofejαejβand ofej(α+β).