What's the Fourier transform of these functions?

[snickersnee]

Homework Statement

How can I figure out the Fourier transform of the following:
I'd prefer to use tables if at all possible.
1. d(z)=d_{eff}sign[\cos[2\pi z]/\Lambda])
(note this is one function inside another one.)
2. d(z)=d_{eff}(1/2)(sign[\cos[2\pi z]/\Lambda]+1)

3. d(z)=d_{eff}\frac{1}{2}a\{u(z)-u(z-\frac{\Lambda}{2})\}+\frac{1}{2}b\{u(z-\frac{\Lambda}{2}))-u(z-\Lambda)\}

d_eff, a, b and Lambda (the period) are constants. u(z) is the step function. (I'm using it to model a square wave)

Homework Equations

See above

The Attempt at a Solution

I took this class a long time ago. There were some kind of rules about what to do if a constant is added, or multiplied by a constant, or if functions are nested, please refresh my memory. For example, if two functions are added in time domain, does that also mean they are added in frequency domain?
FT of step function is this: \sum_{n\ odd}\frac{4}{n\pi}e^{iwt}-e^{-iwt}
FT of signum function: 1/(pi*i*f)
I need the exponential form but I can convert.

 

[Mute]

Homework Statement

How can I figure out the Fourier transform of the following:
I'd prefer to use tables if at all possible.
1. d(z)=d_{eff}sign[\cos[2\pi z]/\Lambda])
(note this is one function inside another one.)
2. d(z)=d_{eff}(1/2)(sign[\cos[2\pi z]/\Lambda]+1)

Homework Equations

See above

The Attempt at a Solution

I took this class a long time ago. There were some kind of rules about what to do if a constant is added, or multiplied by a constant, or if functions are nested, please refresh my memory.
FT of signum function is this: \sum_{n\ odd}\frac{4}{n\pi}e^{iwt}-e^{-iwt}

Do you want the Fourier transform, or the Fourier series? Since your functions are periodic, I assume you really should want the series; I'm not sure the transform is well-defined (at least I was having trouble getting the transform to work out).

Let me focus on your (1). The second should be similar. You know that $\mbox{sgn}(x)$ just gives the sign of its argument, x, which means that you basically just have a piecewise constant integrand.

The easiest way to approach it, or at least the most straight-forward, would be to split the Fourier integral up in regions where $\cos(2\pi z/\Lambda)$ is positive and regions where it is negative.

If you were doing an actual Fourier transform, there are infinitely many regions where it is positive, and infinitely many where it is negative, so you will have a sum over each kind of integral. I wasn't able to get the sum to work out to something sensible, though there's a fair chance I was making a mistake.

If you are computing a Fourier series, then your integrals for the coefficients are not over infinite bounds, but only the period of your periodic function. You can still split up the integral into pieces, but there are only a few of them now, so it will be easier to manage.

For part (2), it looks like you basically just have an additional constant term. Do you know what the Fourier transform representation or the Fourier series representation of a constant is?

Does this help you get started, or should I explain my suggestion in more detail?

 

[snickersnee]

FT of a constant is a multiple of the delta function.
Is this possible by just using tables? I just need the result. For example, in a table we can find the FT of the signum and the FT of the cosine. Is there any way to combine them?

 

[Mute]

FT of a constant is a multiple of the delta function.
Is this possible by just using tables? I just need the result. For example, in a table we can find the FT of the signum and the FT of the cosine. Is there any way to combine them?

No, you can't find the Fourier series by just knowing the Fourier transform of the signum and the Fourier transform of the cosine.

What you can do, however, is think about what kind of function $\mbox{sgn}(\cos(2\pi x/\Lambda))$ represents, and see if you can find a Fourier series for that kind of function. Hint: plot $\mbox{sgn}(\cos(2\pi x/\Lambda))$; what kind of wave is that?

 

[Ray Vickson]

Homework Statement

How can I figure out the Fourier transform of the following:
I'd prefer to use tables if at all possible.
1. d(z)=d_{eff}sign[\cos[2\pi z]/\Lambda])
(note this is one function inside another one.)
2. d(z)=d_{eff}(1/2)(sign[\cos[2\pi z]/\Lambda]+1)

3. d(z)=d_{eff}\frac{1}{2}a\{u(z)-u(z-\frac{\Lambda}{2})\}+\frac{1}{2}b\{u(z-\frac{\Lambda}{2}))-u(z-\Lambda)\}

d_eff, a, b and Lambda (the period) are constants. u(z) is the step function. (I'm using it to model a square wave)

Homework Equations

See above

The Attempt at a Solution

I took this class a long time ago. There were some kind of rules about what to do if a constant is added, or multiplied by a constant, or if functions are nested, please refresh my memory. For example, if two functions are added in time domain, does that also mean they are added in frequency domain?
FT of step function is this: \sum_{n\ odd}\frac{4}{n\pi}e^{iwt}-e^{-iwt}
FT of signum function: 1/(pi*i*f)
I need the exponential form but I can convert.

In (1) you have written, essentially,
\frac{\cos(2 \pi z)}{\Lambda}.
Is that what you want, or did you really mean
\cos\left(\frac{2 \pi z}{\Lambda}\right) ?