Unusual trigonometric function - sin(Nx)/sin(x)

[squalho]

Hello,

I have been an avid reader of the physics forum, and now I think I finally have something worth asking. The problem is engineering-related, but it all comes down to a trigonometric equation, so I'm posting in general math.

The problem is the following:
$$\frac{1}{N^2} \frac{\sin^2(Nx)}{\sin^2(x)} = \frac{\sqrt{2}}{2}$$
I need to solve for x. N is an integer >1. I'm stuck because I don't know how to use arcsin in this case, and in most cases I would end up trying to do arcsin of something larger than 1, that doesn't make sense.

Another related problem is to find the maxima and minima (global would be the best, but local would be better than nothing) of:
$$\frac{1}{N} \frac{\sin(Nx)}{\sin(x)}$$
Here the problem is that computing the derivative I get something that has sine and cosine with argument x and Nx, and there is no good (easy) way to simplify those sin(Nx)'s in order to get something that is a function of just x.

I can plot and find all this information with MATLAB, but I'd like to know if it's possible to find an analytical form. I have been banging my head on this for a while now, but I can't come to a solution. Any idea? Especially about the first one...

Thanks!

--
Squalho

 

[CRGreathouse]

I can approximate one of the solutions fairly closely as A/n + B/n^3 + C/n^5, with A about 1.0019063576966069771401037205331682344703361047262779849, B about 0.4664222801098, and C about 0.32474447. This should be close enough to be a good start for a numerical approximation, but I'm not sure how to use this to find an analytic solution. I don't see anything special about these numbers, though, and the OEIS and RIES don't help either.

 

[jasonRF]

Hello,

I have been an avid reader of the physics forum, and now I think I finally have something worth asking. The problem is engineering-related, but it all comes down to a trigonometric equation, so I'm posting in general math.

The problem is the following:
$$\frac{1}{N^2} \frac{\sin^2(Nx)}{\sin^2(x)} = \frac{\sqrt{2}}{2}$$
I need to solve for x. N is an integer >1. I'm stuck because I don't know how to use arcsin in this case, and in most cases I would end up trying to do arcsin of something larger than 1, that doesn't make sense.

You have an N element antenna array, or some similar type of problem, yes?

In any case, as far as I know ther is no closed form solution for this equation and you must either use approximations that are good enough for your requirements, or resort to numerics (again, that are good enough for your requirements).

Another related problem is to find the maxima and minima (global would be the best, but local would be better than nothing) of:
$$\frac{1}{N} \frac{\sin(Nx)}{\sin(x)}$$
Here the problem is that computing the derivative I get something that has sine and cosine with argument x and Nx, and there is no good (easy) way to simplify those sin(Nx)'s in order to get something that is a function of just x.

I can plot and find all this information with MATLAB, but I'd like to know if it's possible to find an analytical form. I have been banging my head on this for a while now, but I can't come to a solution. Any idea? Especially about the first one...

Thanks!

--
Squalho

Note
$$-1 \leq \frac{1}{N} \frac{\sin(Nx)}{\sin(x)} \leq 1$$
At x=0, this is unity and that is a global maxima. Of course it also reaches this maxima at multiples of $$2\pi$$.

Note that this function is zero where Nx is a multiple of $$\pi$$ (except at the maxima), so your "main beam" has a null-to-null width of $$2 \pi/N$$. This provied n upper bound for your 3 dB width that you are looking for. In fact, between x=0 and x=pi/N, you could use a low order approximation to the function to get an analytical expression for your first question that may be good enough.

cheers,

jason

 

[squalho]

You have an N element antenna array, or some similar type of problem, yes?

In any case, as far as I know ther is no closed form solution for this equation and you must either use approximations that are good enough for your requirements, or resort to numerics (again, that are good enough for your requirements).



Note
$$-1 \leq \frac{1}{N} \frac{\sin(Nx)}{\sin(x)} \leq 1$$
At x=0, this is unity and that is a global maxima. Of course it also reaches this maxima at multiples of $$2\pi$$.

Note that this function is zero where Nx is a multiple of $$\pi$$ (except at the maxima), so your "main beam" has a null-to-null width of $$2 \pi/N$$. This provied n upper bound for your 3 dB width that you are looking for. In fact, between x=0 and x=pi/N, you could use a low order approximation to the function to get an analytical expression for your first question that may be good enough.

cheers,

jason

Jason,

You are right, I have an antenna array type of problem. I also feared that there wasn't an analytical solution.

I'm basically interested in knowing where the main beams are. Without making any assumption I can have one or more, and oriented in any direction. What I'd like to have is a ballpark estimate. Something like: you're going to have 2 main beams along these directions, plus other 4 secondary lobes somewhere in between them. That basically depends on N (for the number of lobes) and x (for where they are directed).

The low order approximation idea is good, but it's too limited: the lobes could really be anywhere.

Thanks for your help!

--
squalho

 

[CRGreathouse]

The low order approximation idea is good, but it's too limited: the lobes could really be anywhere.

What do you mean?

 

[jasonRF]

Jason,

You are right, I have an antenna array type of problem. I also feared that there wasn't an analytical solution.

I'm basically interested in knowing where the main beams are. Without making any assumption I can have one or more, and oriented in any direction. What I'd like to have is a ballpark estimate. Something like: you're going to have 2 main beams along these directions, plus other 4 secondary lobes somewhere in between them. That basically depends on N (for the number of lobes) and x (for where they are directed).

The low order approximation idea is good, but it's too limited: the lobes could really be anywhere.

Thanks for your help!

--
squalho

You obviously have some definition of "main beam" and "secondary beam" that you haven't shared with us. Right now I have no idea what you want. I was assuming that your first question was basically asking how to find the 3dB beamwidth of you main lobe. Obviously I was wrong. Please give us more information!

jason

jason

 

[squalho]

What do you mean?

In order to use low order approximation I must suppose that the argument of the sine is low (Jason mentioned somewhere between 0 and pi/N). But the argument of the sine is not small, can be anything from 0 to 2pi

 

[squalho]

You obviously have some definition of "main beam" and "secondary beam" that you haven't shared with us. Right now I have no idea what you want. I was assuming that your first question was basically asking how to find the 3dB beamwidth of you main lobe. Obviously I was wrong. Please give us more information!

jason

jason

I'm sorry, when you mentioned 3dB beamwidth I thought you knew already everything :)

Long story story short: I have an array with N elements. If you try to plot that function (1/N sin(Nx)/sin(x)), for x going from 0 to 2pi in polar coordinates (or cartesian, for what matters), you see that there is one or more "main lobes" (i.e. global maxima, whose magnitude is 1) and a certain number of secondary lobes (i.e. local maxima, whose magnitude is lower than 1). The greater N, the larger the number of these secondary maxima and of course, the larger the number of zeros too.

Originally I hoped it would be possible to solve the function analytically, knowing the location (in terms of x) of every global and local maxima. That is not possible, so that would answer my original question already.

Then I hoped there would be a way to know at least how many of these maxima, and how many global and how many local. Anyways this probably does not make much sense for my purpose, because when you study an array you already know how many global maxima there are, because that is your design constraint. So I would find N from that knowledge rather than knowing N and then guessing the number and location of these maxima.

In conclusion, I think my doubt has been solved. Thanks!

--
squalho

 

[trambolin]

why not making the "sin(x)^2" function a "cos(2x)" function? Complexify it by exponentials (Euler's formula) and try the real solution from there. I did not check it but I think it is doable.