Question on discrete fourier transform

[mngeow]

I have this question in Discrete Fourier Transform which I don't really get.Its not so much the calculation but more on the theory of it.

Here's the question :

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and here's the answer to the question :

**broken link removed**

What I don't get is that how come there is only 1 AC component?Is it because the result of the DFT only has one non-zero component?But if that was the case,I'm still confused,I learnt that for this 4 point DFT, X(0) would be the DC component,and X(1) would be a mirror image of X(3),and X(2) would be the highest frequency component,as such there would be 2 AC components.

So does that mean that in theory we expected there to be 2 AC components but the result of the DFT only produced one?

Or does the signal X(n) contain 2 AC components,but it became only 1 AC component after the DFT?

I'm really confused about this.
Thanks for the help guys!

 

[moffy]

The signal is a square wave with amplitude +/- 1. So that's the reason for no X(0), it has no DC. It has many frequency components, all multiples of the fundamental frequency, but the 4 point DFT has limited resolution, because it has only 4 points. It is only picking up the fundamental.

 

[MrAl]

The signal is a square wave with amplitude +/- 1. So that's the reason for no X(0), it has no DC. It has many frequency components, all multiples of the fundamental frequency, but the 4 point DFT has limited resolution, because it has only 4 points. It is only picking up the fundamental.

Hi moffy,

Very good answer i would say! I was going to say something similar because of the fact that x(n) alternates regularly. However, i would not try to force x(n) to be a square wave. It's a sampled wave with samples at four intervals equally spaced, but that means the only thing definite about it is that it has impulses at those intervals...everything else in between is undefined. It could be a triangle wave, a sine wave, just about any other wave that goes plus and minus at regular intervals like that.

 

[moffy]

Yeah, good point.