Fourier Transforms: Evaluating Periodic Triangle Pulses

[mcfetridges]

I am having a little trouble. I am asked to evaluate the Fourier transform of a periodic train of triangle shaped pulses. Then I have to evaluate the power spectral density. Now it is very easy to find the Fourier transform of one of the triangles, but what do I do when it is periodic? Do I have evaluate the signal with the Fourier series and using the relationships between the magnitude of Dn to find the mag. of F(w) and then get the spectral density?

 

[LeBrad]

A train of triangles could also be written as a single triangle convolved with an impulse train. Do you know what convolution in the time domain becomes in the frequency domain?

 

[mcfetridges]

Ya in the frequency domain it becomes autocorrelation. But is that the only way?

 

[LeBrad]

If autocorrelation means multiplication, then yes, that is correct. Convolution in one domain goes to multiplication in the other.

Maybe there's an easier way to do it, but I don't think this way is that hard.

A triangle is a box convolved with another box. A box is a sinc(x) function, so a triangle would be sinc(x)^2.

The Fourier transform of an impulse train is another impulse train, but with different spacing and weights.

Multiply the two together, and you get a sampled sinc(x)^2 function. Yes?