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==Problem== | ==Problem== | ||
| − | 1. The signal has only two Fourier coefficients ( | + | 1. The signal has only two non-zero Fourier coefficients (-1 and 1). |
| − | 2. The fundamental period of the signal is 8 | + | 2. The fundamental period of the signal is 8. |
| − | 3. | + | 3. The function is sinusoidal. |
| + | |||
| + | 4. Both Fourier coefficients are non-real. | ||
| + | |||
| + | 5. The signal has a maximum value of 0.5 and a minimum of -0.5. | ||
==Solution== | ==Solution== | ||
| − | The answer is <math>{1 \over 2}\sin({pi \over 4}t)</math> | + | Problem shown worked by steps from above: |
| + | |||
| + | 1. We know that the answer must be of the following form | ||
| + | |||
| + | <math>ae^{j\omega_0t} + be^{j\omega_0t}</math> | ||
| + | |||
| + | |||
| + | 2. From this we can find omega naught | ||
| + | |||
| + | <math>\omega_0 = {2\pi \over 8}</math> | ||
| + | |||
| + | Therefore we have <math>ae^{j{\pi \over 4}t} + be^{j{\pi \over 4}t}</math> | ||
| + | |||
| + | |||
| + | 3. The final answer must have the form of c*sin(\omega_0t) or c*cos(\omega_0t) | ||
| + | |||
| + | |||
| + | 4. We now have something like this | ||
| + | <math>aje^{j{\pi \over 4}t} + bje^{j{\pi \over 4}t}</math> | ||
| + | Also, the function is based on sin, because of Euler's formula: <math> \sin(t) = {e^{jt} - e^{-jt} \over 2j}</math> Notice the j in there. | ||
| + | |||
| + | 5. This suggests that the signal is <math>{1 \over 2} \sin(\omega_0t)</math> | ||
| + | |||
| + | With the information above, we can derive the answer: | ||
| + | |||
| + | <math>{1 \over 2}\sin({\pi \over 4}t)</math> | ||
| + | |||
| + | from | ||
| + | |||
| + | <math>{j \over 4}e^{j{\pi \over 4}t} + {-j \over 4}je^{j{\pi \over 4}t}</math> | ||
Latest revision as of 17:07, 25 September 2008
Homework 4 Ben Horst: 4.1 :: 4.3 :: 4.5
Problem
1. The signal has only two non-zero Fourier coefficients (-1 and 1).
2. The fundamental period of the signal is 8.
3. The function is sinusoidal.
4. Both Fourier coefficients are non-real.
5. The signal has a maximum value of 0.5 and a minimum of -0.5.
Solution
Problem shown worked by steps from above:
1. We know that the answer must be of the following form
$ ae^{j\omega_0t} + be^{j\omega_0t} $
2. From this we can find omega naught
$ \omega_0 = {2\pi \over 8} $
Therefore we have $ ae^{j{\pi \over 4}t} + be^{j{\pi \over 4}t} $
3. The final answer must have the form of c*sin(\omega_0t) or c*cos(\omega_0t)
4. We now have something like this
$ aje^{j{\pi \over 4}t} + bje^{j{\pi \over 4}t} $
Also, the function is based on sin, because of Euler's formula: $ \sin(t) = {e^{jt} - e^{-jt} \over 2j} $ Notice the j in there.
5. This suggests that the signal is $ {1 \over 2} \sin(\omega_0t) $
With the information above, we can derive the answer:
$ {1 \over 2}\sin({\pi \over 4}t) $
from
$ {j \over 4}e^{j{\pi \over 4}t} + {-j \over 4}je^{j{\pi \over 4}t} $
