Exploring the 21 Point DFT in MATLAB

[impervious]

Hello, I was trawling through some MATLAB work, and came across a question that bugged me, and I hope that someone might be able to give a suitable explanation.

The question was explain why the 21 point DFT has so many values equal to zero. Both DFT's are sinc functions where:

sinc(L)= sin(L * theta)/sin(L)

Heres my MATLAB code :

>> nn=0:20;
>> x=zeros(21,1);
>> x(1:7)=boxcar(7);
>> X=fft(x);
>> subplot(2,2,1),stem(nn,x);
>> subplot(2,2,2),stem(nn,abs(X));
>> subplot(2,2,3),stem(nn,angle(X));

The output of this is attached.

I understand that the basic sinc function is widely used in signals due to its zero crossings, however neither the input of the output look much like sinc functions I have seen before.

Any suggestions much appreciated.

 

[vela]

Your boxcar isn't centered, so it's introducing a complex phase. Try setting x to {1,1,1,1,0,...,0,1,1,1}. You'll then find the imaginary parts of X are essentially 0. Plot the real part of X, and it'll look like a sinc function.

 

[impervious]

Unfortunately I think I am using the boxcar they want me to use, and are talking in relation to the six zero points in my DFT.
Since they occur at sampling integers of 3, I can't help but feel this is somehow an important point that I don't understand.

 

[vela]

Your first plot of X isn't of the plain sinc function; it's a plot of its absolute value. Can you see why your plot looks the way it does now?

On the phase plot, you might notice that the phase of every third element is essentially ±π. If you ignore those, you can see the remaining phases follow a nice pattern. That pattern arises because your boxcar isn't centered in the time domain.

 

[DeeNos]

Hi there,

Thank you for sharing your MATLAB code and question. As a fellow scientist, I am happy to help provide an explanation for why there are so many values equal to zero in the 21 point DFT.

First, let's define what the 21 point DFT is. The Discrete Fourier Transform (DFT) is a mathematical algorithm that converts a time-domain signal into its frequency-domain representation. The 21 point DFT means that we are taking a signal with 21 data points and converting it into its frequency-domain representation. This is commonly used in signal processing and analysis.

Now, onto the question of why there are so many values equal to zero in the 21 point DFT. This can be explained by the properties of the DFT and the input signal that you have chosen. The DFT is a complex-valued function, meaning that it has both real and imaginary components. In your code, you have defined the input signal as a boxcar function, which is a rectangular pulse. When we take the DFT of a rectangular pulse, it results in a sinc function in the frequency domain. This is because the DFT of a rectangular pulse is a series of sinc functions with different frequencies.

In your code, you can see that the output of the DFT (X) has both magnitude and phase components, as shown in the third subplot. The magnitude component shows the amplitude of each frequency component, while the phase component shows the phase shift of each frequency component. Since the input signal is a rectangular pulse, which has a flat frequency spectrum, the DFT of this signal will also have a flat magnitude spectrum. This means that most of the values in the DFT will be close to zero, with only a few non-zero values representing the sinc function peaks.

In conclusion, the reason for so many values being equal to zero in the 21 point DFT is due to the properties of the DFT and the input signal being a rectangular pulse. I hope this explanation helps to clarify your question. If you have any further questions or need more clarification, please don't hesitate to ask. Happy exploring!