Half wave symmetry and integrals

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SUMMARY

A periodic wave x(t) with half-wave symmetry satisfies the condition x(t + T0/2) = -x(t), where T0 is the wave's period. This property leads to the conclusion that the integral of the wave, X(t), also exhibits half-wave symmetry, expressed as X(t + T0/2) = -X(t). The integration of the symmetry condition confirms this relationship, validating the conclusion across various examples discussed.

PREREQUISITES
  • Understanding of periodic functions and their properties
  • Knowledge of integral calculus and its applications
  • Familiarity with wave symmetry concepts
  • Basic proficiency in mathematical notation and terminology
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  • Explore the properties of Fourier series and their relation to wave symmetry
  • Study the implications of half-wave symmetry in signal processing
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Mathematicians, engineers, and students studying wave mechanics or signal processing who seek to understand the implications of half-wave symmetry in periodic functions and their integrals.

fernlund
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If I have a periodic wave x(t) with half-wave symmetry, it means that:

x(t + T0/2) = -x(t)

where T0 is the period of the wave. Would this automatically lead to the conclusion that

X(t + T0/2) = -X(t)

where X'(t) = x(t), i.e X(t) is the integral of x(t).

?
 
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fernlund said:
If I have a periodic wave x(t) with half-wave symmetry, it means that:

x(t + T0/2) = -x(t)

where T0 is the period of the wave. Would this automatically lead to the conclusion that

X(t + T0/2) = -X(t)

where X'(t) = x(t), i.e X(t) is the integral of x(t).

?

I think so. Can you integrate $x(t+T_0/2)=-x(t)$ on both sides w.r.t. $t$, and obtain your result?
 
Ackbach said:
I think so. Can you integrate $x(t+T_0/2)=-x(t)$ on both sides w.r.t. $t$, and obtain your result?

Yeah, it's correct for all of the examples I can think of. Thanks!
 

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