How to do this unusual Fourier transform?

I am trying to compute a sine transform:

I'm not sure about the non-standard bound of the integral (why it's not infinity???).

Anyway, let's say f_t = rand(40,1). Then,

N = 1000;
omega = logspace(-2,2,N);
for ii = 1:N
    f_omega{ii} = omega(ii).*integral(@(t) f_t.*sin(omega(ii).*t),0,2*pi./omega(ii),'ArrayValued',true);
end

gives a vector of length 40 for each omega, which doesn't seem right to me. I should have just a vector of length N at the end. How do I do this integral? And should it matter what range in t I choose for f(t)?

[Star Strider]

The result looks strange —

syms omega t 
sympref('AbbreviateOutput',false);
f(t) = cos(5*pi*t)

f(t) = 

f(omega) = omega * int(f(t) * sin(omega*t), t, 0, 2*pi/omega);
f = simplify(f, 500)

f(omega) = 

figure
fplot(f,[0.01 1000])
grid
Ax = gca;
Ax.XScale = 'log';

.

[L'O.G.]

@Star Strider How would you recommend doing this integral then? Please see my comment below as well.

[Star Strider]

I could not get it to work numerically, even using the non-random sine function for ‘f_t’ so that I could be certain that it had a non-random result. I did the symbolic calculation and plot to see what the correct result would be (sort of like looking in the back of the book to see what the correct answer is).

Answer 1

[Walter Roberson]

Anyway, let's say f_t = rand(40,1).

Let's not say that.

f is a function of t. By saying that f_t = rand(40,1) you are saying that f is constant in t, and is a 40-dimensional (constant) point. When you integrate that, of course you are going to end up with a 40-dimensional result.

[L'O.G.]

@Walter Roberson Can you provide some tips then? f(t) will have a different value for each time point, which is why I thought to treat it as a vector.

[Walter Roberson]

Could you confirm that the f on the left side of your equation is the same f on the right side of your equation? You talk about a sine transform, and "transforms" are suitable for application to classes of functions, but if you have the same f on both sides of the equation you are going to be restricting yourself to a rather narrow set of functions in a way that would never be called a "transform".

@Star Strider chose f(t) = cos(5*pi*t) and since I do not see any reference to that in your discussion, I gather that he choose an arbitrary function to attempt to transform? That would be more in keeping with it being considered a "transform" -- but there is the possibility that particular function does not happen to be one of the ones that the transform is valid for.

[Star Strider]

I considered to be the result of that transform, although the usual representation would be (upper case) denoting the transformed function.

I chose arbitrarily in order to see what the result was (in both the numeric and symbolic calculations). Transforming a random vector (such as the one defining the original ‘f_t’) would simply produce noise, and that would not be sufficiently informative to determine whether the numeric function worked correctly.

[L'O.G.]

Thank you both. @Walter Roberson Excellent point about f(t). I was trying to simplify the notation, but I might've simplified it too much. The actual equation is:

where is the "unit vector along the element axis". is just a constant. only has nonzero values in x, so I think the element axis here should just be the x-axis. There's just one value of for a particular time t, so perhaps constructing it as a vector is not right as you indicated. Based on what you're saying, I would also not regard as the transform of . I am really in the dark about how to do this. Would appreciate your and @Star Strider's insight.

[Walter Roberson]

Using that notation, but with the same functionality as Star Strider already posted:

syms f(t) G(omega) varepsilon_0 e_1
Pi = sym(pi);
sympref('AbbreviateOutput',false);
f(t) = cos(5*Pi*t)

f(t) = 

G(omega) = omega/(varepsilon_0 * Pi) * int((f(t)*e_1) * sin(omega*t), t, 0, 2*Pi/omega);
G = simplify(G, 500)

G(omega) = 

G_representative = subs(G, {e_1, varepsilon_0}, {1,1})

G_representative(omega) = 

figure
fplot(G_representative,[0.01 1000])
grid
Ax = gca;
Ax.XScale = 'log';