Help with digital signals (discrete fourier transform)

  • #1
I've been working on this problem for around three hours, and I'm getting nowhere... I think it may be that I don't have even the most basic grasp of the material to even get a decent start on the problem, but hopefully someone here will be able to help me...

Homework Statement



Calculate the Discrete Fourier Transform (DFT) of a 1Hz cosine wave
sampled 4 times per second for 1 second.

Homework Equations



[tex]

X(K) = \sum_{n=0}^{N-1} x(n)e^{-j*2*\pi*\frac{k*n}{n}}

[/tex]

The Attempt at a Solution


Honestly, I'm stumped. I don't even know where to start. I'm not asking anyone to do the problem for me, but I'd be forever greatfull if someone could start me in the right direction.

Thanks,
Dan
 

Answers and Replies

  • #2
MATLABdude
Science Advisor
1,655
4
Okay, start with this factoid: a 1 Hz sine wave, sampled 4 times per second for 1 second should give you 5 values (assuming you know the value at 0):
x[0]=
x[1]=
x[2]=
x[3]=
x[4]=

Fill in these values. Now apply the formula you listed above to find the DFT (come on, it's only 5 values!)

X[0]=
X[1]=
X[2]=
X[3]=
X[4]=
 
  • #3
So am I right in saying
x[0]=1
x[1]=0
x[2]=-1
x[3]=0
x[4]=1
for a cosine wave?
 
  • #4
MATLABdude
Science Advisor
1,655
4
So am I right in saying
x[0]=1
x[1]=0
x[2]=-1
x[3]=0
x[4]=1
for a cosine wave?
Yes, that's correct. A sine wave starts at 0, however. I think the answers will differ by an imaginary number in the end (equivalent to a phase shift, if I recall correctly).
 
  • #5
Yes, that's correct. A sine wave starts at 0, however. I think the answers will differ by an imaginary number in the end (equivalent to a phase shift, if I recall correctly).
Why do you find the x[n] values for a sine wave when the question is asking about a cosine wave?
 
  • #6
MATLABdude
Science Advisor
1,655
4
Why do you find the x[n] values for a sine wave when the question is asking about a cosine wave?
Huh. I could swear your original post asked for sine. In that case, carry on!
 
  • #7
So I would plug those values into the forumula, and then add all of the outputs together?
 
  • #8
MATLABdude
Science Advisor
1,655
4
So I would plug those values into the forumula, and then add all of the outputs together?
Yup. And now?

X[0]=
X[1]=
X[2]=
X[3]=
X[4]=
 

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