Sinusoidal and complex equation signals

[dervast]

Homework Statement

I am reading the book Wiley Signal And Systems-Simon Haykin and at the page 34 i read the followings

Consider the complex exponential e^j*theta. Using Euler's identity we may expand this term as e^j*theta=cos(theta)+jsin(theta)
This result indicates that we may express the continuous time sinusoidal signal x(t)=Acos(wt+f) as the real part of the complex exponential signal Be^jwt (w is the angle like theta) where B itself a complex quantity defined by B=Ae^i*f (f another angle)
that is we may write
Acos(wt+f)=Re{Be^jwt}

I ask this cause i need firstly to understand this part befroe i can understand what
a)is an exponentially damped sinusoidal signal
and b)why these signals can't be ever periodics

Homework Equations

The Attempt at a Solution

I am trying to understand why the author assumes that x(t)=Acos(wt+f) is the real part of Be^jwt and why the B is in exponential form again B=Ae^i*f
I am asking cause i am have to understand this first so i can later on i can understand what an exponentially damped sinusoidal signal is
and why such signals can't ever be periodics

 

[denverdoc]

I'm confused by your post, what makes you believe they can't ever be periodic functions?

You might start here for some background about Euler's identity:
http://www.sosmath.com/complex/number/eulerformula/eulerformula.html
edit: I see you had separate post about Euler's, which has been answered.

 

[berkeman]

dervast, one helpful mental image for you to use as you are studying these complex functions, is to picture the 2-D Real-Imaginary plane.

Draw the Real axis left-to-right, and the Imaginary axis vertically. The two axes intersect at 0+j0. Now start a point on the real axis at A+j0, and rotate it counterclockwise with some velocity omega. The position vector to that point is the complex vector Ae^j*theta, where theta is the angle formed between the position vector and the + Real axis. Do you see now how the Real part is the Acos(wt) part, and the Imaginary part is the jAsin(wt) part?

Hope that helps.

 

[dervast]

Thx a lot for your answers. The book i am reading says that the exponentially damped sinusoidal signals of whatever kind can't be periodic? Anmd i am trying to thing why?
@berkeman the signal u have represented in the x-y axis is an exponentially damped sinusoidal signals?
Finally can u also explain me why what the book is correct?

Consider the complex exponential e^j*theta. Using Euler's identity we may expand this term as e^j*theta=cos(theta)+jsin(theta)
This result indicates that we may express the continuous time sinusoidal signal x(t)=Acos(wt+f) as the real part of the complex exponential signal Be^jwt (w is the angle like theta) where B itself a complex quantity defined by B=Ae^i*f (f another angle)
that is we may write
Acos(wt+f)=Re{Be^jwt}

Why we have to express the signal as 2 complex quantitities? (Be^jwt where B is the 2ndcomplex quantity)

 

[Integral]

Consider the definition of periodic: f(x) = f(x+a) (IIRC)

This cannot be true for a damped signal, since the amplitude continues to decrease with time. For large enough time the signal will vanish altogether.

 

[denverdoc]

Consider the definition of periodic: f(x) = f(x+a) (IIRC)

This cannot be true for a damped signal, since the amplitude continues to decrease with time. For large enough time the signal will vanish altogether.

sure, but a lightly damped system such as a near perfect pendulum looks very much periodic, and is often treated as such. Its a matter of degree I suppose.

A heavily damped system is neither a purely exponential process either.

This is where I think the confusion may lay, at least for me, that the eqn is then product of exponential decay and harmonic term, in order to satisfy say an eqn such as
ay"-by'+cy where b^2-4ac is negative. Am I recalling this right?

 

[berkeman]

The general form of the damped complex exponential is like this:

$$f(t) = A e^{s t} = Ae^{(\sigma + j \omega) t}$$

The $$e^{\sigma t}$$ term is the lossy term, and the $$e^{ j \omega t}$$ is the complex sinusoid term.

 

[Integral]

Denverdoc,
What you have posted is correct, put this is mathematics. We have to apply the definitions rigorously. The fact is, by strict mathematical definitions periodic motion does not exist in nature. Even the motion of the planets changes over time, to the extent that eventually they will be devoured by the expanding sun thus do not meet the requirements of the mathematical definition of periodic motion.

 

[denverdoc]

Right I see your point and appreciate taking the time to address my confusion. I want very badly to understand complex analysis as all engineering texts I have, make abundant use of these tools, but when I see paragragh such as the one quoted in the OP, my eyes begin to glaze over... The problem is i took 3 out of 4 qtrs of engr math sequence and missed the last quarter which was exclusively about complex variables. So what I know is in connection to ode and pde solutions, Fourier transforms and the like, but not the real nuts and bolts of their use. This ignorance is frustrating as I find I am not so good at abstract math and so cannot just pick up a text like the one cited and put it to good use.
Again thanks,
John