(New page: == Definition of fourier transform for DT signal == These are the fourier coefficients, which must be calculated from the function in this case, rather than vice versa in CT signals. <ma...) |
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The primary importance in the DT example, is making sure that a constant K exists, so that the signal can be forced to be periodic. | The primary importance in the DT example, is making sure that a constant K exists, so that the signal can be forced to be periodic. | ||
| − | We will make a relatively low period, say <math>\,N= | + | We will make a relatively low period, say <math>\,N=4</math> |
| − | <math>\,x[n]= | + | <math>\,x[n]=6cos(4\pi n)+2cos(\pi n)</math> |
We have to make the periods make sense in DT, so we must make sure that <math>\,\frac{2\pi }{\omega_0}</math> is a whole number for both functions, so multiply it in this fashion: | We have to make the periods make sense in DT, so we must make sure that <math>\,\frac{2\pi }{\omega_0}</math> is a whole number for both functions, so multiply it in this fashion: | ||
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<math>N_1=\frac{2\pi }{4\pi}k=\frac{1}{2}k</math>, and | <math>N_1=\frac{2\pi }{4\pi}k=\frac{1}{2}k</math>, and | ||
| − | <math>N_2=\frac{2\pi }{ | + | <math>N_2=\frac{2\pi }{\pi}k=\frac{2}{1}k</math> |
so <math>\,k=2</math> makes them both integers: | so <math>\,k=2</math> makes them both integers: | ||
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<math>\,N_1=1</math> | <math>\,N_1=1</math> | ||
| − | <math>\,N_2= | + | <math>\,N_2=4</math> |
| − | and <math>\,N_2</math> is the lowest usable period(ie the highest) and <math>\,N= | + | and <math>\,N_2</math> is the lowest usable period(ie the highest) and <math>\,N=4</math> |
| − | Now we have our period, and thus the limits of our sum. The limit goes to N-1, so we need to find <math>\,x[0]</math> | + | Now we have our period, and thus the limits of our sum. The limit goes to N-1, so we need to find <math>\,x[0]</math> through <math>\,x[3]</math>: |
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| + | <math>\,x[0]=8</math> | ||
| + | |||
| + | <math>\,x[1]=4</math> | ||
| + | |||
| + | <math>\,x[2]=8</math> | ||
| + | |||
| + | <math>\,x[3]=4</math> | ||
Revision as of 13:04, 25 September 2008
Definition of fourier transform for DT signal
These are the fourier coefficients, which must be calculated from the function in this case, rather than vice versa in CT signals.
$ a_k = \frac{1}{N} \sum^{N-1}_{n=0} x[n] e^{-jk\frac{2\pi}{N} n} $
Where N is the period of the function.
Example of a periodic DT signal
The primary importance in the DT example, is making sure that a constant K exists, so that the signal can be forced to be periodic.
We will make a relatively low period, say $ \,N=4 $
$ \,x[n]=6cos(4\pi n)+2cos(\pi n) $
We have to make the periods make sense in DT, so we must make sure that $ \,\frac{2\pi }{\omega_0} $ is a whole number for both functions, so multiply it in this fashion:
$ N_1=\frac{2\pi }{4\pi}k=\frac{1}{2}k $, and
$ N_2=\frac{2\pi }{\pi}k=\frac{2}{1}k $
so $ \,k=2 $ makes them both integers:
$ \,N_1=1 $
$ \,N_2=4 $
and $ \,N_2 $ is the lowest usable period(ie the highest) and $ \,N=4 $
Now we have our period, and thus the limits of our sum. The limit goes to N-1, so we need to find $ \,x[0] $ through $ \,x[3] $:
$ \,x[0]=8 $
$ \,x[1]=4 $
$ \,x[2]=8 $
$ \,x[3]=4 $
