Convolutions Flashcards
What are convolutions?
A mathematical process which takes two input functions f(x) and g(x) produces a third output function
In physics:
- f(x) represents the response of a measurement system
- g(x) is the quantity being measured
- convolution gives what we actually measure
What do convolutions tell us?
Tells us how the properties of our measurement
system distort the quantity being measured
What is greater the width of the output of the two original functions?
Width of output is always greater than widths of original two functions
How do you get the convolution function?
The signal function is given by g(x) and the system response function by f(x).
Introduce a dummy variable u which enters the two
functions as follows: f(u) and g(x-u).
For a given value of x calculate the ‘overlap’ between
the two functions f(u) and g(x-u).
Integrating the product of the two functions between -∞ and +∞ with respect to the variable u
IT IS A SYMMETRICAL PROCESS
What is the convolution function?
𝑐(𝑥)=𝑓∗𝑔=∫(−∞) (∞)𝑓(𝑢)𝑔(𝑥−𝑢)d𝑢
What is the convolution of a delta function?
𝑔(𝑥)=𝑔(𝑥)∗𝛿(𝑥)
CONVOLUTION OF ANY FUNCTION WITH A DELTA FUNCTION GIVES THE ORIGINAL FUNCTION
What is the convolution theorem?
The FT of a convolution is the product of the FTs of the original functions (with the extra numerical factor)
𝐶(𝑘)=√2𝜋 𝐹(𝑘)𝐺(𝑘)
𝑐(𝑥)=𝑓∗𝑔=∫_(−∞)(∞)𝑓(𝑢)𝑔(𝑥−𝑢)d𝑢
How do you find the the FT of the observed signal, c(x), and of the resolution function?
𝐹(𝑘)=(𝐶(𝑘))/(𝐺(𝑘)√2𝜋)
𝑓(𝑥)=1/2𝜋 ∫(−∞ to ∞)𝑒^𝑖𝑘𝑥 (𝐶(𝑘))/(𝐺(𝑘)) d𝑘
When will there be an intensity profile be equal to one?
Only if the detector has zero width
What happens if we have a non-zero detector width?
A distortion to the measured profile
How to sketch the convolution of two functions?
- Draw functions on the same axis
- Define x as the centre to centre value
- Find the values just before the over lap, completely overlapped, just before coming out and just after the overlap.
-Draw new graph
What happens to the convolution of a function with very different widths?
The shape of the convolution is similar to the shape of the wider original function
What if the convolution of two Gaussian Functions?
It is the same as one Fourier transform of a single one with a different width
c(x)=√Δ/π e^(-Δx^2/2)
