Signals Energy of 2 signals - Integral limits correct?

In summary, if signals x(t) and y(t) are orthogonal, then the expected value of z(t) = x(t) + y(t) is equal to the sum of the expected values of x(t) and y(t). This is because the integral of x(t)y(t) with respect to t is equal to zero due to their orthogonality. Without more information about x(t) and y(t), it is not possible to determine the exact integral of x(t)y(t).
  • #1
thomas49th
655
0
If signals x(t) and y(t) are orthogonal and if z(t) = x(t) + y(t) then
E_{z} = E_{x} + E_{y}:


Proof:

[tex] E_{z} => \int^{\infty}_{-\infty} {(x(t) + y(t))^{2}} dt
=> \int {(x(t) + y(t))^{2}}^{2} dt
=> \int (x^{2}(t)) + \int(y^{2}(t))dt + \int x(t)y(t)dt
=> E_{x} + E_{y}
[/tex]

because [tex]\int x(t)y(t)dt[/tex] = 0 because of integration by parts:

u = x(t) dv/dt = y(t)
u' = dx/dt, v = [tex]frac{y^{2}(t)}{2}[/tex]

so [tex]x(t)\frac{y^{2}(t)}{2} - \int {\frac{y^{2}(t)}{2}\frac{dx}{dt}}dt[/tex]
[tex]x(t)\frac{y^{2}(t)}{2} - \int {\frac{y^{2}(t)}{2}}dx[/tex]
we can treat y^2(t) as a constant so:

[tex]x(t)\frac{y^{2}(t)}{2} - \int^{\infty}_{-\infty} {\frac{y^{2}(t)}{2}}dx[/tex]
[tex]x(t)\frac{y^{2}(t)}{2} - } [{\frac{y^{2}(t)x}{2}}]^{\infty t}_{-\infty t}[/tex]

but the problem is that the limits were destined for integrating with respect to time. I'm not integrating with respect to x.

Any suggestions?
Thanks
Thomas
 
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  • #2
The integral of x(t)y(t) isn't zero because of some bogus 'integration by parts' argument. It's zero because that's what 'orthogonal' means.
 
  • #3
Ofcourse! Execellent. May I ask, out of interest alone what the integral of x(t)y(t) with respect to t should be?

Thanks
Thomas
 
  • #4
thomas49th said:
Ofcourse! Execellent. May I ask, out of interest alone what the integral of x(t)y(t) with respect to t should be?

Thanks
Thomas

There's really nothing in particular you can say about it without knowing more about x(t) and y(t). y(t)dt can't be integrated to y(t)^2/2. That's y(t)dy(t). So integration by parts isn't useful.
 

1. What is the significance of the integral limits in signals energy?

The integral limits in signals energy refer to the range over which the energy of a signal is being calculated. This range is typically defined by the start and end time of the signal. The integral limits are important because they determine the total energy contained within the signal and can affect the accuracy of energy calculations.

2. Are the integral limits always correct in signal energy calculations?

No, the integral limits may not always be correct in signal energy calculations. This can happen if the start or end time of the signal is not accurately defined, or if the signal is not sampled at a high enough frequency to capture all of its energy. It is important to ensure the integral limits are correct in order to obtain accurate energy measurements.

3. How do the integral limits affect the energy of a signal?

The integral limits directly affect the energy of a signal. A larger range of integration will result in a larger energy measurement, while a smaller range will result in a smaller energy measurement. This is because the integral limits determine the amount of time over which the signal's energy is being calculated.

4. Can the integral limits be adjusted to improve the accuracy of signal energy calculations?

Yes, the integral limits can be adjusted to improve the accuracy of signal energy calculations. If the start or end time of the signal is not accurately defined, adjusting the integral limits to encompass the entire signal can result in a more accurate energy measurement. Similarly, increasing the sampling frequency can also improve the accuracy of energy calculations.

5. How can I calculate the energy of a signal if the integral limits are not known?

If the integral limits are not known, the energy of a signal can still be calculated using other methods such as the power spectral density or the autocorrelation function. However, these methods may not provide an accurate measurement of the total energy contained within the signal. It is best to accurately define the integral limits in order to obtain the most accurate energy measurement.

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