- #1
roam
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Homework Statement
A certain function ##v(x)## has Fourier transform ##V(\nu)##. The plot of the function is shown in the figure attached below.
For each of these functions give their Fourier transform in terms of ##V(\nu)##. And also state if the FT is Hermitian/anti-Hermitian, even/odd, imaginary/real.
(a) ##v(2x)##
(b) ##v(x)-v(-x)##
(c) ##v(x+2)+v(x-2)##
Homework Equations
Properties of FT.
The Attempt at a Solution
(a) [/B]Using the scaling property of the Fourier transform:
$$FT \Big[ v(2x) \Big] = \frac{1}{|2|} V(\frac{\nu}{2})$$
Now to determine whether this is Hermitian/anti-Hermitian, even/odd, imaginary/real, I have made a table that summarizes the properties (e.g. if a function f if is real F is Hermitian, etc.):
The plot for v(x) looks odd so V(ν) has to be odd. But since we don't have an explicit expression for ##v(x)##, how can we determine if ##v(2x)## is even/odd? Likewise how do we know if his function is real?
(b) Using the symmetry property we find:
$$FT \Big[ v(x)-v(-x) \Big] = V(\nu) - V(-\nu)$$
I think there is a property such that that ##F(-\nu)=F^*(\nu)## (* is the complex conjugate). I am not sure if this is applicable in this case, but if so we have:
$$V(\nu) - V^*(\nu) \implies 2 j \ Im[V(\nu)]$$.
The last step comes from the fact that for any complex number ##z## we have ##z-z^* = 2j \ Im(z)##. I'm not sure how to further write down the FT for this function.
(c) I think maybe we can use the "Interference" property of the FT:
$$f(t-t_0) + f(t+t_0) \longleftrightarrow 2 \ cos (2 \pi \nu t_0) F(\nu)$$
So we find the equation ##2 \ cos (4 \pi \nu) V(\nu).## Is that okay, or do I need to further develop this equation?
Just as in part (a) and (b), I am not sure how to determine if the function is even/odd, real/imaginary. Any help would be greatly appreciated.