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G.E. Ivey
science forum Guru

Joined: 29 Apr 2005
Posts: 308

Posted: Sun Jan 15, 2006 6:06 pm    Post subject: Re: pre-fourier piecewise continuous function simple misunderstanding

 Quote: I am beginning to learn about piecewise continuous functions as an intro to Fourier series yet have a few queries. Hopefully someone would be kind to answer. The book gives the following two examples. 1) y = 0 for 0<= t< T/2 y = sin t for T/2 <= t < T,period T Waveform 0 for the first half period and a sine wave of amplitude 1 over second. Waveform is 0 then raises from 0 at T/2 to 1 in a sin fashion (upward hump) then down to 0 at T. (Half wave rectified signal) 2) y = 2 sin t for 0 <= t < T/2, y = 2 sin t for T/2 = t < T Waveform is two half sine waves both going from 0 to max 1. Two upwards humps. I initially thought that if the time t, was running continuously then at t = T/2 the sine wave in example 1 would be a negative hump rather than the positive one given. This also assumes the period of the sine wave is the same period as that of the created piecewise function, doesnt it? Is this fair to assume? No, it isn't. It is "fair", and the only correct "assumption", to assume that the period of sin(t) is 2 pi! If you want the period of the sin wave to be the same as the period of the piecewise function (T), you will need to use sin(2pi t/T).

Are you sure you wrote these correctly? Since T is just some arbitrary number, there will probably be a discontinuity at t= T. Only if T is some multiple of 2pi will sin(T/2)= 0 so that the function is continuous.

 Quote: Also, similarly, if time was running continuosly wouldnt the example 2) just be a y = 2sint for 0

y= 2sin t for 0< t< T! Are you sure you have copied it properly?

 Quote: It seems as tho the time, resets when a function is introduced yet another example given is: I'm not sure what you mean by that but, no the independent variable does not go back to 0: that's why they say "for T/2< t< T". y= 0 for 0 <= t <2, y =t for 2<= t<3, period 3 where at t=2, y becomes t not 0. Yes, that correct. since y= 0 for t< 2, not "t<= 2" that formula does not apply at t= 2. Instead the value of y(2) is 2, not 0.
Chris1171

Joined: 01 Oct 2005
Posts: 50

Posted: Sun Jan 15, 2006 6:06 pm    Post subject: pre-fourier piecewise continuous function simple misunderstanding

I am beginning to learn about piecewise continuous functions as an intro to
Fourier series yet have a few queries. Hopefully someone would be kind to
answer. The book gives the following two examples.

1) y = 0 for 0<= t< T/2

y = sin t for T/2 <= t < T,period T

Waveform 0 for the first half period and a sine wave of amplitude 1 over
second.

Waveform is 0 then raises from 0 at T/2 to 1 in a sin fashion (upward hump)
then down to 0 at T. (Half wave rectified signal)

2) y = 2 sin t for 0 <= t < T/2, y = 2 sin t for T/2 <= t < T

Waveform is two half sine waves both going from 0 to max 1. Two upwards
humps.

I initially thought that if the time t, was running continuously then at t =
T/2 the sine wave in example 1 would be a negative hump rather than the
positive one given. This also assumes the period of the sine wave is the
same period as that of the created piecewise function, doesnt it? Is this
fair to assume? Also, similarly, if time was running continuosly wouldnt the
example 2) just be a y = 2sint for 0<t<T. It seems as tho the time, resets
when a function is introduced yet another example given is:

y= 0 for 0 <= t <2, y =t for 2<= t<3, period 3 where at t=2, y becomes t not
0.

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