Formula for the propagation of complex errors

In summary, the formula for the propagation of complex errors provides a systematic method for determining how uncertainties in measurements affect the overall uncertainty of a calculated result. It incorporates the various sources of error through mathematical relationships, allowing for the combination of multiple variables and their associated uncertainties. The formula typically employs derivatives to assess the sensitivity of the result to each variable, ensuring a comprehensive understanding of how errors propagate through a given calculation. This approach is essential in scientific and engineering applications where precision is critical.
  • #1
accdd
96
20
If I have 2 measurements ##x = (3.0 ± 0.1), y = (-2.0 ± 0.1)## and want to calculate how the error propagates when calculating a function from those values this formula should be used: ##f(x, y) = f(x, y) ± \sqrt {(\frac{\partial f}{\partial x}*\Delta x)^2+(\frac{\partial f}{\partial y}*\Delta y)^2}##
What is the formula for calculating error propagation if x and y are complex (##x = (3 ± 0.1) + (9.5 ± 0.4)ⅈ, y = (2 ± 0.1) - (5 ± 0.4)ⅈ##)?
 
Mathematics news on Phys.org
  • #2
Change to polar coordinates.
 
  • Like
Likes accdd
  • #3
Can you give me an example. Suppose the function is: ##f(x, y) = x + y^2##
In the non-complex case, with the data given in the previous post, I would proceed as follows:
##f(x, y) = x + y^2 = (3+(-2)^2) \pm \sqrt{0.1^2+(2*(-2) *0.1)^2}= 7\pm \sqrt{0.01+0.16}=7\pm0.41##
How can I change to polar coordinates to get the result in case of complex numbers?
The result should be: ##(-18.0 ± 4.0) - (10.5 ± 1.9)ⅈ## (Measurement jl)
 
  • #4
[itex]x+iy\equiv Re^{i\theta} [/itex]where [itex] R=\sqrt{x^{2}+y^{2}}[/itex] and [itex]\theta =\arctan(\frac{y}{x}) [/itex]. This is the easiest representation for complex multiplication (you multiply the argumets and add the angles). Complex addition is easiest in the cartesian notation.
 
  • Like
Likes accdd
  • #5
I got stuck, not able to get the result. Can someone show me how to do it please?
 
  • #6
Svein said:
Change to polar coordinates.
After that, you'd have to know the variance of the amplitude as a function of the variances of the real and imaginary components. If they had the same variance, I think you'd have a Rayleigh distribution. I'm not sure how that generalizes to the case of unequal variances. Does that distribution have a special name?
 

FAQ: Formula for the propagation of complex errors

What is the formula for the propagation of complex errors?

The formula for the propagation of complex errors is derived from the principles of uncertainty propagation. For a function \( f(x_1, x_2, ..., x_n) \) that depends on multiple variables \( x_i \) with associated uncertainties \( \sigma_{x_i} \), the overall uncertainty \( \sigma_f \) can be calculated using the formula: \[\sigma_f = \sqrt{\left(\frac{\partial f}{\partial x_1} \sigma_{x_1}\right)^2 + \left(\frac{\partial f}{\partial x_2} \sigma_{x_2}\right)^2 + ... + \left(\frac{\partial f}{\partial x_n} \sigma_{x_n}\right)^2}\]This formula assumes that the errors are independent and normally distributed.

When should I use the propagation of complex errors?

You should use the propagation of complex errors when you have a function that depends on multiple variables, each with their own uncertainties. This is common in experimental science, engineering, and data analysis, where measurements are combined to calculate derived quantities. The propagation of errors helps in estimating the uncertainty in the final result, ensuring that the derived values reflect the uncertainties of the individual measurements.

What are the key assumptions in the error propagation formula?

The key assumptions in the error propagation formula include: 1. The errors in the measurements are independent of each other. 2. The errors are normally distributed, which implies that they can be described by standard deviations. 3. The function \( f \) is sufficiently smooth around the point of interest, allowing the use of Taylor expansion to approximate the function. These assumptions are important for the validity of the results obtained from the propagation of errors.

How do I apply the error propagation formula in practice?

To apply the error propagation formula in practice, follow these steps: 1. Identify the function \( f \) that relates the measurements. 2. Determine the variables \( x_1, x_2, ..., x_n \) that contribute to \( f \) and their respective uncertainties \( \sigma_{x_i} \). 3. Calculate the partial derivatives \( \frac{\partial f}{\partial x_i} \) for each variable. 4. Substitute the partial derivatives and uncertainties into the error propagation formula to calculate \( \sigma_f \), the uncertainty in the derived quantity. This process allows you to quantify how uncertainties in measurements affect the final result.

What is the difference between systematic and random errors in the context of error propagation?

In the context of error propagation, systematic errors are biases that affect measurements consistently in the same direction, leading

Similar threads

I Error Propagation in Measurements
Replies
3
Views
1K
I Question about partial derivative relations for complex numbers
Replies
9
Views
959
I Partial Differential Equation solved using Products
Replies
2
Views
1K
I Computing the Speed of Traveling Waves in 2 Dimensions
Replies
10
Views
2K
I Calculating the inverse of a function involving the error function
Replies
3
Views
1K
I How to propagate errors through a regression & non-linear model?
Replies
8
Views
2K
I Alternate expressions for the uncertainty propagation
Replies
1
Views
3K
I Newton-Raphson Method With Complex Numbers
Replies
7
Views
2K
I Error propagation of a variable for an integral
Replies
6
Views
4K
I What are the applications of inverses of vector functions?
Replies
17
Views
2K

Hot Threads

Recent Insights

Back
Top