Help with digital signals (discrete fourier transform)

In summary, the problem involves calculating the Discrete Fourier Transform (DFT) of a 1Hz cosine wave sampled 4 times per second for 1 second. The formula for the DFT is X(K) = \sum_{n=0}^{N-1} x(n)e^{-j*2*\pi*\frac{k*n}{n}}, and the solution involves finding the values for x[n] and then plugging them into the formula and adding the outputs together.
  • #1
danhamilton
9
0
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
 
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  • #2
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
danhamilton said:
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
MATLABdude said:
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
danhamilton said:
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
danhamilton said:
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]=
 

1. What is a discrete Fourier transform (DFT)?

A discrete Fourier transform is a mathematical technique used to analyze a signal in the frequency domain. It converts a discrete signal, such as a digital signal, into a representation of its component frequencies.

2. Why is the discrete Fourier transform important?

The discrete Fourier transform is important because it allows us to analyze digital signals and understand their frequency components. This is useful in many applications, such as signal processing, data compression, and image processing.

3. How is the discrete Fourier transform different from the continuous Fourier transform?

The discrete Fourier transform is a discrete version of the continuous Fourier transform. The main difference is that the discrete Fourier transform operates on a finite set of data points, while the continuous Fourier transform operates on a continuous function. In other words, the discrete Fourier transform is used for discrete signals, while the continuous Fourier transform is used for continuous signals.

4. What types of signals can the discrete Fourier transform analyze?

The discrete Fourier transform can analyze any type of discrete signal, including digital signals, time series data, and sampled analog signals. It is commonly used in fields such as signal processing, telecommunications, and audio engineering.

5. How is the discrete Fourier transform calculated?

The discrete Fourier transform is calculated using a mathematical algorithm called the Fast Fourier Transform (FFT). The FFT algorithm breaks down the signal into smaller components and calculates the frequency spectrum of each component. This process is repeated until the entire signal is analyzed, resulting in a representation of the signal in the frequency domain.

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