## UnkleRhaukus 2 years ago laplace transform of a periodic function/

1. UnkleRhaukus

2. UnkleRhaukus

3. UnkleRhaukus

mabye this is better $F(p)=\frac{2}{(1-e^{-4p})p^2}-\frac{2e^{-2p}}{(1-e^{-4p})p^2}-\frac{4e^{-2p}}{(1-e^{-4p})p}$

4. lonliness

yes this one is better but the one that you have solved is much better and simplified , so i dont think you need to do this

5. UnkleRhaukus

@lonliness , what region should i integrate for figure 2

6. UnkleRhaukus

also how did go in this one

7. mahmit2012

|dw:1354803234598:dw|

8. mahmit2012

|dw:1354803478460:dw|

9. mahmit2012

|dw:1354803608097:dw|

10. UnkleRhaukus

but if i go from 0 to 2 pi ( in fig2) the the integral will be messy

11. mahmit2012

wait !

12. UnkleRhaukus

ok

13. mahmit2012

|dw:1354804514349:dw|

14. mahmit2012

|dw:1354804529087:dw|

15. mahmit2012

|dw:1354804592320:dw|

16. mahmit2012

17. UnkleRhaukus

i see that now,

18. mahmit2012

Do you know how I understood my mistake? Because I knew that the bounded signals should have Laplace with bounded limit in s=p=0

19. mahmit2012

|dw:1354805030916:dw|

20. UnkleRhaukus

|dw:1354861124769:dw|

21. mahmit2012

|dw:1354861368611:dw|

22. UnkleRhaukus

the slope is 1 and then -1

23. UnkleRhaukus

yes?

24. UnkleRhaukus

|dw:1354862069169:dw|

25. UnkleRhaukus

@mahmit2012

26. UnkleRhaukus

@TuringTest

27. ali110

@mahmit2012 s=ROC(region of convergence)+jw(frequency responce) so where the ROC of this function?

28. UnkleRhaukus

im not sure , i havent herd that term before, is it a restriction on p?

29. ali110

actually we moved from fourier transform to laplace transform because fourier transform tells only frequency responce of a signal/function but laplace transform can explain stability/unstability of function including with its frequency responce so its easy to design system by using laplace transform

30. ali110

|dw:1354866466108:dw|

31. ali110

$F(s)=\int\limits_{-\infty}^{\infty}f(t)e^-st*dt$

32. UnkleRhaukus

$\delta=\sigma$?

33. ali110

@UnkleRhaukus am i explain it more if u want?

34. UnkleRhaukus

35. ali110

oh yes $s=\sigma+jw$ actually i have not seen it on table before

36. UnkleRhaukus

for some reason my text book has been using $$p$$ for $$s$$ $$x$$ for $$t$$ $$n$$ fr $$\omega$$ $$i$$ for $$j$$

37. ali110

so for moving function(signal) |dw:1354867100786:dw| now for left sided signal |dw:1354867294896:dw| |dw:1354867371794:dw|

38. ali110

i am using book of ALAN.V.OPPENHEIM and also book of SAMARJIT.GHOSH

39. ali110

Have u know before why we use laplace transform?

40. UnkleRhaukus

do we use laplace transform to understand the frequencies of the system?

41. UnkleRhaukus

(also to solve Initial value problems )

42. ali110

yes we use laplace transform to find both frequency responce(jw) as well as its stabality and unstabaliy(sigma) of function s=sigma+jw as see above yes also for initial value problem+modulation problem+final value problem i can give u simple function matlab code if u want!!

43. UnkleRhaukus

yes i would be interested to try out your matlab code , also what is a modulation problem ?

44. ali110

modulation theorem!!!! application of signal and system in communication it is a technique that is use to convey the info over a long distance i.e for voice signal as voice signal dont have high range for long distance actual signal+carrier signal 3400hz+high freq signal(1 Mhz) voice data may die over a long distance if we not use carrier signal signal send for long distance through amplitude modulation as well as frequency modulation we can fix the band width of AM,short band width require if we want to send signal over a lond distance for FM,long band width is needed if we want to send signal for long distance but here there is some cost problem!!i will discuss if u want

45. ali110

but frequency modulation is best as noise signals in atmosphere donot affect on it like mobile phone signals we use frequency modulation technique!!! but still its so costly to use Freq modulation

46. UnkleRhaukus

ok, can you tell me more about the region of convergence for my triangles signal

47. ali110

i ask it for mohan gholami(@mahmit2012)!! he will tell u!!i think he know it!!!

48. ali110

btw fren which book u use for signals and systems can u tell me its authur name?

49. UnkleRhaukus

50. UnkleRhaukus

\begin{align*} \mathcal L\big\{f(x)\big\}&=\int\limits_0^\infty f(x)e^{-px}\cdot\text dx\\ &=\tfrac1{1-e^{-4p}}\int\limits_0^{4}f(x)e^{-px}\cdot\text dx\\ &=\tfrac1{1-e^{-4p}}\left[\int\limits_0^{2}xe^{-px}\cdot\text dx+\int\limits_2^4(4-x)e^{-px}\cdot\text dx\right]\\ &=\tfrac1{1-e^{-4p}}\left[\frac{xe^{-px}}{-p}\Big|_0^2-\int\limits_0^{2}\frac{e^{-px}}{-p}\cdot\text dx+4\int\limits_2^4e^{-px}\cdot\text dx-\int\limits_2^4xe^{-px}\cdot\text dx\right]\\ &=\tfrac1{1-e^{-4p}}\left[\frac{2e^{-2p}}{-p}-\frac{e^{-px}}{p^2}\Big|_0^2+\frac{4e^{-px}}{-p}\Big|_2^4-\frac{xe^{-px}}{-p}\Big|_2^4+\int\limits_2^{4}\frac{e^{-px}}{-p}\cdot\text dx\right]\\ &=\tfrac1{1-e^{-4p}}\left[\frac{-2e^{-2p}}{p}-\frac{e^{-2p}-1}{p^2}+\frac{4e^{-4p}-4e^{-2p}}{-p}-\tfrac{4e^{-4p}-2e^{-2p}}{-p}+\frac{e^{-px}}{p^2}\Big|_2^4\right]\\ &=\tfrac1{1-e^{-4p}}\left[\frac{-2e^{-2p}}{p}+\frac{1-e^{-2p}}{p^2}+\frac{2e^{-2p}}{p}+\frac{e^{-4p}-e^{-2p}}{p^2}\right]\\ &=\frac1{1-e^{-4p}}\left[\frac{1-2e^{-2p}+4e^{-4p}}{p^2}\right]\\ &=\frac{1-2e^{-2p}+4e^{-4p}}{(1-e^{-4p})p^2}\\ \end{align*}

51. UnkleRhaukus

is this right for the wave of triangles?

52. UnkleRhaukus

is there some way to check?

53. UnkleRhaukus

@sirm3d

54. UnkleRhaukus
55. UnkleRhaukus