Magnitude of abs(H(exp(jwT)) = 2

[Butters]

abs(H(exp(jwT))=abs((exp(jwT)+2)/(exp(jwT)+0.5))=2

How come the answer is 2. I just find it impossible to express it in terms of Imaginary and Real parts so that I could find the absolute value.

 

[meBigGuy]

the mag of the exponentials are 1 by definition, so (1+2)/1.5 = 2

As for expressing it as real and imaginary, do you understand this? http://en.wikipedia.org/wiki/Euler's_identity

 

[Butters]

Thanks but I didn't get what you mean by Euler's identity. I know that one can express exponential in terms of real (coswT) and imaginary (jsinwT) parts. Then the magnitude should be squared root of the sum of those parts squared.
Anyway, so whenever I have to find absolute value of exponentials I just substitute 1. Is it the same for exp(-2jwT) and exp(-jwT)?

 

[meBigGuy]

You need to understand such that you can prove the magnitude is 1. It is actually very simple. sin^2 + cos^2 = 1 by definition.

On the complex plane the comnplex exponential represents the magnitude and angle of a vector. If you draw that and the right triangle it defines, you see the cos part (horizontal axis) and the sin part (vert axis). It is that simple. If the exponential has an amplitude (like Aexp(-jwt)) then you modify the vector and adjust accordingly. (now it is Asin + Acos)

http://en.wikipedia.org/wiki/File:Euler's_formula.svg

 

[Babadag]

Abs(x+jy)=sqrt(x^2+y^2) see: http://www.clarku.edu/~djoyce/complex/abs.html
Let's say exp(jwt)=z=x+jy x=cos(wt) y=sin(wt) x^2+y^2=1
Abs(H(z))=Abs(x+2+jy)/Abs(x+0.5+jy)=sqrt(((x+2)^2+y^2)/((x+0.5)^2+y^2)))
x^2+4x+4+y^2=4x+5
x^2+x+0.25+y^2=x+1.25
sqrt((4x+5)/(x+1.25))=2 for all x!

 

[sophiecentaur]

Thanks but I didn't get what you mean by Euler's identity. I know that one can express exponential in terms of real (coswT) and imaginary (jsinwT) parts. Then the magnitude should be squared root of the sum of those parts squared.
Anyway, so whenever I have to find absolute value of exponentials I just substitute 1. Is it the same for exp(-2jwT) and exp(-jwT)?

The idea on a forum like this is that you come across a term like "Euler's Identity" and you look it up. If the maths is beyond you the, perhaps you need to get more familiar with it. I learned Euler's Identity at A level so it isn't that hard to understand - even less difficult to find it somewhere on the web. Maths is not a field where you can dip into it at random places and expect to 'get it'. You need to start at your present level and follow it through.
I am not being grumpy about this; I am simply being realistic.

 

[Babadag]

You are right, Sophiecentaur, it is not a big deal and I am doubt if Leonard Euler himself dealt with it, indeed.Of course, it is not Gamma or Bessel function but was as a joke for a smile.