The two different kinds of signal representation

signal representation?

Usually, we can see there are two kinds of signal representation in relative electronic books.
The first representation is Vm*cos(wt).
The other representation is Vm*exp(wt).

Does anyone can explain it in details?
I was confused in this issue for a long time, but few people can give me an answer.

Re: signal representation?

the second is exp(jwt) which is equal to coswt+j sinwt

so cos(wt)=re(exp(jwt))

representing the signal by exp(jwt) is for phasor anlysis: we use phaor analysis to simple the analysis of circuits. so if you refre to phasor analysis of circuits you'll get the reason in more details.

signal representation?

exp(jwt) is a complex signal. there is only one side spectrum while cos(wt) is a real signal, and it has both side spectrum. Usually, the baseband signal is a complex signal (PSK, QAM, etc) so we write exp(jwt). it a real word, to way to generate a complex signal is to use quadrature modulation (or IQ modulator).

Re: signal representation?

exp[j?t]=cos ?t + jsin ?t
exp[-j?t]=cos ?t - jsin ?t
Adding gives
exp[j?t]+exp[-j?t]=2cos ?t
cos ?t=[exp[j?t]+exp[-j?t]]/2

Re: signal representation?

In real world, there is only real part of exp(jwt) exists.
We use exp(jwt) as representation of signal is only convenient for mathematic operation.
Meanwhile, in fourier series formula, the input signal have to use real part only.
If there is only real part of complex representation exists, it is meaningless in analyzing phase variant.
Am I right?

Re: signal representation?

Imaginary is real enough. Try measuring current thru a reactance. Try running a 1000 horsepower electrical motor without doing something about the load factor (reactive parts of the power), and see your energy bill for the month.

Re: signal representation?

When you say signal REPRESENTATION ,remember its just a representation.
In practical would it be exactly like the representation is a question of design.

So our aim is to simplify the representation and get as close as we can get to practical waveforms.

First of all lets take first case Cosine wave representation. These are real waves. Have a look at the waves created when a stone is droped into a lake. Try to represent it, think of cosine waves and there you are with the best solution.
What am i trying to say is that there are a number of actual cases that follow the cosine rule. hence we find that representation both simple and practical.

But there was some problem with the cosine representations. Like there were still a number of signals which can't be easily defined by the cosine terms. Say for example charging and discharging cycle of a capacitor.
It increases very fast (or exponetially ) . And hence came this exp reprepresentation to deal with still a number of signals.
Finally these signals are intereralted and hence mathematical models were used to interrelate these two representation.

And that's the reason why today we use both these representations.
Which one you use is apllication and user specific.
I hope i made it clear, at least somewhat !

Re: signal representation?

In system viewpoint, there are R,L and C components in the circuit.
So, the equation of system can be described as complex formate, such as R+jXL, R-jXc.
But, in Signal viewpoint, if complex representation exists in real world.
How to make both signal sources as cos(wt)+j0 and cos(wt)+jsin(wt)?

Re: signal representation?

cos(wt)=sin(wt+pi/2)
Basically they are the same thing.
Just depend on the starting point of wave you observed.

You can use signal generator to make a cos wave and then
you get cos(wt)+j0.

You can shift this cos wave by pi/2 and you get sin(wt).
Adding both of signals you can get cos(wt)+jsin(wt)
Please ignore that "j".
cos(wt)+jsin(wt)=cos(wt)+sin(wt) in real world.