Fourier Transforms: Evaluating Periodic Triangle Pulses

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Homework Help Overview

The discussion revolves around evaluating the Fourier transform of a periodic train of triangle-shaped pulses and subsequently determining the power spectral density. Participants explore the relationship between the Fourier transform of individual triangles and their periodic arrangement.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to understand how to evaluate the Fourier transform for a periodic signal, considering the use of Fourier series and relationships between coefficients. Some participants discuss the concept of convolution and its implications in the frequency domain, questioning if there are alternative methods to approach the problem.

Discussion Status

The discussion is active, with participants sharing insights about convolution and its relationship to multiplication in the frequency domain. There is an exploration of different methods, including the use of autocorrelation and the properties of sinc functions, but no consensus has been reached on the best approach.

Contextual Notes

Participants are navigating the complexities of periodic signals and their transforms, with some uncertainty regarding the implications of convolution and the specific characteristics of the triangle pulse in the context of Fourier analysis.

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

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