How do I find the magnitude of this function?

[interxavier]

Homework Statement

I'm asked to find the magnitude of the following complex function:

$$R(j \omega) = 1 + \exp{(-j \omega)} + \exp{(-j2 \omega)} + \exp{(-j3 \omega)} + \exp{(-j4 \omega)}$$

Homework Equations

None

The Attempt at a Solution

I did the following but got stuck immediately as I'm not sure on how to proceed:

$$|R(j \omega)| = |1 + \exp{(-j \omega)} + \exp{(-j2 \omega)} + \exp{(-j3 \omega)} + \exp{(-j4 \omega)}|$$

 

[NascentOxygen]

I'm not sure what you are doing. Maybe finding the peak value of a signal and its harmonics?

I think the idea would be to use the identity from maths: e^jW=cosW + j sinW

 

[interxavier]

I'm not sure what you are doing. Maybe finding the peak value of a signal and its harmonics?

I think the idea would be to use the identity from maths: e^jW=cosW + j sinW

I'm trying to find the magnitude. If I use euler's identity, then I would get multiple values of sin's and cosine's:
$$R(j \omega) = 1 + \cos{(\omega)} - j\sin{(\omega)} + \cos{(2\omega)} -j\sin{(2\omega)} ...$$

How do you find the magnitude of that?

 

[Hootenanny]

I'm trying to find the magnitude. If I use euler's identity, then I would get multiple values of sin's and cosine's:
$$R(j \omega) = 1 + \cos{(\omega)} - j\sin{(\omega)} + \cos{(2\omega)} -j\sin{(2\omega)} ...$$

How do you find the magnitude of that?

Initially, I would stick with exponential notation and recall that $|z|^2=zz^*$. Once you have computed $|z|^2=zz^*$ and simplified, you should find that you are left with a rather nice expression, which you should be able to easily convert into a real, non-negative, trigonometric expression.

 

[interxavier]

Initially, I would stick with exponential notation and recall that $|z|^2=zz^*$. Once you have computed $|z|^2=zz^*$ and simplified, you should find that you are left with a rather nice expression, which you should be able to easily convert into a real, non-negative, trigonometric expression.

Does that mean I take the conjugates like this:

$$R^{*} = 1 - \exp{(-j \omega)} - \exp{(-j2 \omega)} - \exp{(-j3 \omega)} - \exp{(-j4 \omega)}$$

or

$$R^{*} = 1 + \exp{(j \omega)} + \exp{(j2 \omega)} + \exp{(j3 \omega)} + \exp{(j4 \omega)}$$

 

[Hootenanny]

Does that mean I take the conjugates like this:

$$R^{*} = 1 - \exp{(-j \omega)} - \exp{(-j2 \omega)} - \exp{(-j3 \omega)} - \exp{(-j4 \omega)}$$

No. Like this:

$$R^{*} = 1 + \exp{(j \omega)} + \exp{(j2 \omega)} + \exp{(j3 \omega)} + \exp{(j4 \omega)}$$

Remember that if $z = Re^{i\theta}$, then $z^* = (e^{i\theta})^*R^*$, but since R is real: $z^* = R(e^{i\theta})^* = Re^{-i\theta}$ (check this using Euler's formula if you are unsure).