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pre-fourier piecewise continuous function simple misunderstanding
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G.E. Ivey
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Joined: 29 Apr 2005
Posts: 308

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

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<t<T.
Yes, there is no reason in the world not to write

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.
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Chris1171
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Joined: 01 Oct 2005
Posts: 50

PostPosted: Sun Jan 15, 2006 6:06 pm    Post subject: pre-fourier piecewise continuous function simple misunderstanding Reply with 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? 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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