Solving Fourier Sinusoids using DFT and ω = π/6 | Homework Statement

In summary: It is kind of a strange question, but I think the point is to practice using Fourier series and understanding how they can be used to represent different types of signals. It's also a good exercise in recognizing symmetries and utilizing them in Fourier analysis.
  • #1
DmytriE
78
0

Homework Statement


The entries of the time-domain vector:
x(1) = [2 1 -1 -2 -1 1 2 1 -1 -2 -1 1] ; N = 12

are given by 2cos(ωn) where n = 0:11. what is the value of ω? express x(1) as the sum of two Fourier sinusoids. By considering the appropriate columns of the Fourier matrix V, determine the DFT X(1).


Homework Equations


ω = (2π/N)


The Attempt at a Solution


I know that ω = π/6

But when determining the Fourier sinusoid I can only express it as the sum of 7 different parts.

1/6 + 1/6cos(pi*n/6) - 1/6cos(pi/3*n)...1/6(-1)^n.

But I says to express it as 2 Fourier sinusoids. I don't know how to simplify it or how to decide which columns would lead me to a solution.

When I fft the above equation (partial shown) It get x(1) back. So the equation is right but does not fully answer the question. Any help would be greatly appreciated!

DmytriE
 
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  • #2
UPDATE:
ω = π/3 because the fundamental period of the time vector is 6.

What is meant by "express x(1) as the sum of two Fourier sinusoids."
 
  • #3
DmytriE said:
UPDATE:
ω = π/3 because the fundamental period of the time vector is 6.

What is meant by "express x(1) as the sum of two Fourier sinusoids."

Correct on w = pi/3.

I don't understand why express x(1) as a sum of two sinusoids when the original wave is already a pure sinusoid ... 2cos(nπ/3), n = 0, 1,2, ...
 
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  • #4
rude man said:
2cos(nπ/3), n = 1,2, ...

Using Matlab I did fft of the function you gave above and it does not reproduce the time vector. I am looking for the sinusoidal equivalent I guess using only two sinusoidal functions to show this.
 
  • #5
t = 1/6 + 1/6*cos(pi/6*n) - 1/6*cos(pi/3*n) - 1/3*cos(pi/2*n) - 1/6*cos(2*pi/3*n)+ 1/6*cos(5*pi/6*n) +(-1).^n/6

^This the the equation I get that will give me the full time vector. But as far as I can tell it is made up of more than 2 sinusoidal functions...
 
  • #6
DmytriE said:
Using Matlab I did fft of the function you gave above and it does not reproduce the time vector. I am looking for the sinusoidal equivalent I guess using only two sinusoidal functions to show this.

I meant n = 0, 1, 2, ..

The Fourier series of cos(wt) is cos(wt)! So the Fourier series of 2cos(nπ/3) is 2cos(nπ/3), seems like.

I guess I'm missing something here.
 
  • #7
DmytriE said:
But when determining the Fourier sinusoid I can only express it as the sum of 7 different parts.

1/6 + 1/6cos(pi*n/6) - 1/6cos(pi/3*n)...1/6(-1)^n.

But I says to express it as 2 Fourier sinusoids. I don't know how to simplify it or how to decide which columns would lead me to a solution.
I have done no Fourier work since I was a student, so don't take this too seriously...

EDIT [strike]Doesn't that 1/6 term indicate DC? There is no DC component here.[/strike] I hadn't read to the end of your series. http://physicsforums.bernhardtmediall.netdna-cdn.com/images/icons/icon11.gif [Broken]

Perhaps you are being asked to approximate that stepwise wave using just 2 sinusoids? You'll need the fundamental, together with another to account for the step noise, and by the nature of the symmetry you can see it must be an odd harmonic.
 
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  • #8
If you look at x, you can see that it is symmetric across number 2. Since the real value x is symmetric, then X must show conjugate circular symmetric. Remember that x will be symmetric at N/2. Therefore the elements n and N-n are the same distance from the center of symmetry. this can be expressed as x = e^(m) + e^(m-N). At least this is my understanding of this problem. You can refer to pages 176 through 180 of the following text if that helps.
 
  • #9
polaris90 said:
You can refer to pages 176 through 180 of the following text if that helps.

Which text are you referring to?
 
  • #10
Yes, thank you Polaris. I will take a look at the text. Haha. I just realized which text you were talking about.
 
  • #12
Is this a sum of two Fourier Sinusoid?
n = 0:11;
e^(j * pi/3*n) + e^(-j*pi/3*n).

I hope so since I cannot seem to write two sinusoids that will produce the time domain vector.
 
  • #13
DmytriE said:
Is this a sum of two Fourier Sinusoid?
n = 0:11;
e^(j * pi/3*n) + e^(-j*pi/3*n).

I hope so since I cannot seem to write two sinusoids that will produce the time domain vector.

Looks good to me if by pi/3*n you mean nπ/3.

The whole question seems obtuse to me, frankly. A set of discrete numbers is not a sinusoid per se. A sinusoid can generate discrete numbers, to be sure, as in this case. But what is the point of the question, other than to present arcane math?
 
  • #14
rude man said:
Looks good to me if by pi/3*n you mean nπ/3.

Yup, that's exactly what I mean. :approve:
 

1. What is a Fourier Sinusoid?

A Fourier Sinusoid is a mathematical function that can be used to represent a periodic signal. It is made up of a series of sine and cosine functions with different amplitudes, frequencies, and phases.

2. How is a Fourier Sinusoid used in signal processing?

In signal processing, Fourier Sinusoids are used to decompose a complex signal into simpler sine and cosine waves. This allows for a better understanding and analysis of the signal's frequency and amplitude components.

3. What is the difference between a Fourier Sinusoid and a Fourier Transform?

A Fourier Transform is a mathematical operation that converts a signal from the time domain to the frequency domain. It represents the entire spectrum of a signal. On the other hand, a Fourier Sinusoid only represents a single frequency component of a signal.

4. Can Fourier Sinusoids be used to analyze non-periodic signals?

No, Fourier Sinusoids can only be used to analyze periodic signals. Non-periodic signals, such as a single pulse or a random noise signal, do not have distinct frequency components and therefore cannot be represented by a Fourier Sinusoid.

5. What is the relationship between Fourier Sinusoids and the Fourier Series?

The Fourier Series is a mathematical representation of a periodic signal using a combination of sine and cosine functions. A Fourier Sinusoid is a single term in the Fourier Series. In other words, the Fourier Series is a sum of multiple Fourier Sinusoids.

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