You are asking why use imaginary numbers in the exponentials.

This is simply because \({e^{it}}\) is oscillatory and \({e^{t}}\) is not

This can be seen from Euler's formulae connecting the complex exponential to the

\({e^{it}} = \cos t + \sin t\)

circular sine and cosine functions which are oscillatory

compared to the real exponential to the hyperbolic sine and cosine which are not oscillatory.

\({e^t} = \cosh t + \sinh t\)


This is why the exponential solution to the wave equation is complex.

Question: isn't the first equation missing the i that multiplies the sin function?
shouldn't that equation read \({e^{it}} = \cos t + i\sin t\) ?

 

"viewing frequencies" on what ? will math allow you to "view frequencies"? a spectrum analyser would be much better, then you can work out the transforms.

 

"viewing frequencies" on what ? will math allow you to "view frequencies"? a spectrum analyser would be much better, then you can work out the transforms.

You're right... regarding math, maybe a more proper expression would be "visualizing frequencies" ....

 

Question: isn't the first equation missing the i that multiplies the sin function?
shouldn't that equation read ?

Well spotted, thanks that was a mistake.

 

Only if you pay some attention to what I, and others have said to you.

We have assumed that the only things you haven't understood are embodied in the questions you have asked.

But I think it runs deeper than that.

Is it a language difficulty, is English your first language?

Edit: Please note this is not meant as a criticism, I'm simply trying to find a way to help.

Yeah my first language is not English.But I think that I only have doubts regarding these questions not anything deeper than this.

 

Thanks.That's really a good answer.But let me say I know all these 5 points that you have mentioned.Yes of course I know Euler's identity makes a circle of unit radius.But what still confuses me is these questions:

1. Why do we take a unit circle for analyzing signals?
2. How can a circle be manipulated to analyze a signal?(I haven't never seen a signal like a circle)
3. Why doesn't the real part holds information about frequency while the complex part does?(It might be as you said that "the complex part could also be used in polar representation".But how is frequency related to polar representation?)

Could you please help me in answering these questions/

Perhaps it will help if you consider the following:

1) Practically all signals can be described as being the sum of a (possibly infinite) set of sinusoidal signals.
2) Linear systems can be analysed as being the sum of responses to a set of signals (sinusoidal or otherwise).
3) Thus, if we can adequately describe how linear systems respond to sinusoidal signals, then we can describe how they respond to practically any signal.
4) Working directly with sinusoidal signals is not very convenient.
5) Working with exponentials is actually quite convenient, even if they involve complex quantities.
6) Working with differential equations is a bitch.
7) Working with systems of linear equations is straightforward and very well understood.
8) We can express sinusoids in terms of exponentials in a few ways, such as:

\(
y(t) = A \cos \( \omega t + \phi \) = Re\lbrace Ae^{j \omega t} e^{j \phi} \rbrace
\)

9) A transform, such as Laplace or Fourier, is merely a mathematical process that changes how information is represented which makes something easier to work with (and some things harder). With respect to this conversation, they turn differential equations into algebraic ones, which permit us to avoid dealing with differential equations and, instead, deal with linear systems of algebraic ones.
10) Because the information is represented in a different way, and because the goal of the transform is not to make the information easy to see, but rather the system of equations easy to solve, there may or may not be any simple or intelligible relationship between information in the time domain and its representation in the transformed domain. We can often see how some things map, but it is usually not a complete picture.

 

Perhaps it will help if you consider the following:

1) Practically all signals can be described as being the sum of a (possibly infinite) set of sinusoidal signals.
2) Linear systems can be analysed as being the sum of responses to a set of signals (sinusoidal or otherwise).
3) Thus, if we can adequately describe how linear systems respond to sinusoidal signals, then we can describe how they respond to practically any signal.
4) Working directly with sinusoidal signals is not very convenient.
5) Working with exponentials is actually quite convenient, even if they involve complex quantities.
6) Working with differential equations is a bitch.
7) Working with systems of linear equations is straightforward and very well understood.
8) We can express sinusoids in terms of exponentials in a few ways, such as:

\(
y(t) = A \cos \( \omega t + \phi \) = Re\lbrace Ae^{j \omega t} e^{j \phi} \rbrace
\)

9) A transform, such as Laplace or Fourier, is merely a mathematical process that changes how information is represented which makes something easier to work with (and some things harder). With respect to this conversation, they turn differential equations into algebraic ones, which permit us to avoid dealing with differential equations and, instead, deal with linear systems of algebraic ones.
10) Because the information is represented in a different way, and because the goal of the transform is not to make the information easy to see, but rather the system of equations easy to solve, there may or may not be any simple or intelligible relationship between information in the time domain and its representation in the transformed domain. We can often see how some things map, but it is usually not a complete picture.

Thanks.Let me first dig into the representation of a signal.I'm still confused with the representation after that I'll come back to your answer.Anyway this answer helped me to know that you should first understand the representation of signal to learn these transforms.

 

Hello again,

Just a couple more quick notes...

With just an imaginary part as the exponent we have as before:
e^((j*b)*t)=cos(b*t)+j*sin(b*t)

but with a real part as well we have:
e^((a+j*b)*t)=e^(a*t)*e^(j*b*t)=e^(a*t)*(cos(b*t)+j*sin(b*t))

Note that the b*t always appears in the sin and/or cos terms, while the a*t only appears in the exponential multiplier.
b*t is the "w*t" we normally see inside the cos and sin terms.
So the real part brings in the exponential multiplier while the imaginary part beings in the sinusoidal (frequency) terms.
'a' is usually negative but that's not mandatory if the system output increases without limit.

 

e^((a+j*b)*t)=e^(a*t)*e^(j*b*t)=e^(a*t)*(cos(b*t)+j*sin(b*t))

So the real part brings in the exponential multiplier while the imaginary part beings in the sinusoidal (frequency) terms.
'a' is usually negative but that's not mandatory if the system output increases without limit.

I didn't follow this. Surely the real part also contains a sinusoid?


\(f(t) = {e^{at}}\left( {\cos bt + j\sin bt} \right)\)

\( = \left( {{e^{at}}\cos bt} \right) + \left( {j{e^{at}}\sin bt} \right)\)

\( = {\rm Re\nolimits} \{ f(t)}\ + {\rm Im\nolimits} \{ f(t)\} \)

 

Hi,

Usually when someone writes "a+b*j" that means the 'a' is the real part and the 'b' is the imaginary part. So the 'b' in the left hand side exponential brings in the sinusoidal terms on the right, while the 'a' brings in the exponential multiplier. Remember we are converting the exponential with real and imaginary parts in the exponent to an equivalent form, not really analyzing the result of that equivalent form. When we do in fact analyze the resulting equivalent form we end up with a real sinusoidal part and imaginary sinusoidal part, as you noted, which i think is good to look at too

Also to add a little general info, for me the significance of the Laplace transform is that 's' is the complex frequency. But if we default to s just imaginary, then we can take it to be an imaginary sinusoidal frequency in order to calculate the sinusoidal response to circuits.
The physical significance of the transform i think can be looked at as a process that repeats over and over again, and starting with a certain initial value, we end up with some solution at some t. So in a way it answers the question of what we should start with to get that result at that t or even several values of t. We could look at this more i guess, as well as comparing Fourier to Laplace.