Fourier analysis – Determination of frequency components using FFT

The function defined below has three frequencies 2, 4, and 10. How can the Fourier transform (FFT) be used to display these frequencies?

   myfun(t_) = 
 1/(2 Sqrt(
   2)) ((Sqrt)Abs(
     3 + Cos(4 t) - 
      4 Cos(2 t) (-1 + 
         E^(-0.018` t) (Cos(9.99999594999918` t) + 
            0.0009000003645002213` Sin(9.99999594999918` t))^2) + 
      4 E^(-0.018` t) (Cos(9.99999594999918` t) + 
         0.0009000003645002213` Sin(9.99999594999918` t))^2))

nt.number theory – Fourier transform of $ I_Y $, $ Y = { text {numbers with many prime factors} } $

To let $ Y $ Let be the set of integers $ N <n leq 2 N $ with more than $ D log log N $ Prime factors. We can think, say, $ D = ( log log N) ^ {1- epsilon} $,

We have pretty accurate approximations for the size of $ Y $ (I am aware of Chapter II.6 in Tenenbaum's book and the references it contains.) I wonder what work is available there for the Fourier transform $ widehat {1_Y} $ the characteristic function of $ Y $,


I would expect $ widehat {1_Y} $ To have tips on the main arches (ie bows around rations $ a / q $ with a small denominator). That's because $ Y $ is obviously "biased towards divisibility" and should therefore be slightly over-represented in the congruence class $ 0 $ mod $ d $ given for everyone $ d $, relative to other congruence classes mod $ d $,
A back-of-the-envelope calculation suggests that the value at the peak is around $ a / q $ should be roughly proportional $ c ^ { omega (q)} / q $, But what is known?

fa.functional analysis – Characterization of a subset of the Sobolev space $ H ^ k (0.2 pi) $ in Fourier series

To let $ A: = {u in H ^ k (0.2 pi): u ^ {(j)} (0) = u ^ {(j)} (2 pi) mbox {for} j = 0,1, ldots, k-1 } $, from where $ H ^ {k} (0.2 pi) subseteq L ^ 2 (0.2 pi) $ is the Sobolev order room $ k $ on $ (0, 2 pi) $, Can we say that? $ u in A $ iff
$$ sum_ {n = – infty} ^ infty (1 + n ^ 2) ^ k | has {u} (n) | ^ 2 < infty? $$
In the above series $ has {u} (n) $ are the Fourier coefficients of $ u $, We think the answer to this question is positive because we may be able to identify $ A $ with the Sobolev space of the Torus $ H ^ k ( mathbb {T}) $Use this result. But we can not show that there is an isomorphism $ A $ and $ H ^ k ( mathbb {T}) $,

Know a reference for a characterization of $ A $ with Fourier series?

Thank you for any help you can give us.

Fourier analysis – Fourier transform frequency

One possibility is the FourierParameters

  FourierTransform(1, x, w, FourierParameters -> {0, -2*Pi})

Mathematica Graphics

  FourierTransform(Exp(I a x), x, w, FourierParameters -> {0, -2*Pi})

Mathematica Graphics

Compare

funs = {1, DiracDelta(x), Exp(I a x), Cos(a x), Sin(a x)};
result= {#, FourierTransform(#, x, w, FourierParameters -> {0, -2*Pi})}& /@ funs;
Prepend(result, {"f(x)","Fourier transform unitary, ordinary frequency"});
Grid(%, Frame -> All)

Mathematica Graphics

With wikis second column:

Mathematica Graphics

Fourier analysis – spectral solution of elastic waves

As stated in "Wave Propagation in Structures: an FFT-based Spectral Analysis Method" by James F. Doyle:
For the spectral solution of the 1D equation for elastic waves:
$$ widetilde {u} (x, omega) = sum Ce ^ {- i {k_ {n}} x} + De ^ {i {k_ {n}} x}, $$
from where $$ {k_ {n}} = { omega_ {n}} sqrt { rho / E} $$ is the wavenumber. C and D are the indeterminate amplitudes at each frequency. The end of the beam at x = 0 is subjected to a force curve F (t), ie$$ EA frac { partial u (x, t)} { partial x} = F (t) $$
E and A are the modulus of elasticity and the cross-sectional area, respectively. The final solution is the inverse Fourier transform of the following expression:

$$ widetilde {u} (x, omega) = – frac { widetilde {F_ {n}}} {ik_ {n} EA} e ^ {- ik_ {n} x} $$

$ widetilde {F_ {n}} $ is the Fourier transform of the applied force F (t).

The numeric example for the above problem is as follows:

Rod:
diameter=1 inch
density=0.00247 lb/ci
E=10.6e6 lb/si
Pulse, F(t):
0.000000 0
0.001000 0
0.001100 1000
0.001300 0
0.001500 0
(sec) (N)

I wrote the following code in MatLab:

clear all
close all
clc

d=1.0; %inch
A=pi/4*d^2;
rho=0.000247; %lb/inch3
E=10.6e6; %psi
%transform parameters:
n=2^15;
dt=5e-6;
fs=1/dt;
time_fcn = (0:n-1)/fs;
frequency = (0:n-1)*(fs/n);
omega=2*pi*frequency;
F=zeros(1,numel(time_fcn));
nn=find(time_fcn>=0.0011 & time_fcn<=0.0013);
F(nn)=-5e6*(time_fcn(nn)-0.0013);
plot(time_fcn,F)
Fn=fft(F);
plot(omega,Fn)
k=omega*sqrt(0.000247/10.6e6);
A=-Fn(2:numel(omega))./(1i*k(2:numel(omega))*E*A);
x=0;
G(2:numel(omega))= A.*exp(-1i*k(2:numel(omega))*x);
G(1)=simpsons(F,0,max(time_fcn),numel(time_fcn));
U= ifft(G);
plot(time_fcn*1000,U)

The result must be as follows:
Enter image description here

However, I can not achieve the same result as stated in the above book. Can someone tell me where my mistake is?

Thank you all,

Greetings.

Integration – Improper Fourier Transformation

The most common way to check if the Fourier transformation of a function occurs $ f $ is integrable $ (f in L ^ 1 ( mathbb {R})) $ is by demonstrating that the function $ f $ is integrable and at least doubly differentiable. Now I've come across a weaker state where the Fourier transform is not properly integrable. The statement was as follows:

If $ f: mathbb {R} to mathbb {C} $ is a differentiable, integrable function and even, i. $ f (x) = f (-x) $, then the Fourier transformation $ has {f}: mathbb {R} to mathbb {C} $ will be a wrong integral, i. $ has {f} chi _ {(c, d)} $ is integrable for everyone $ c <d $ and the limit $$ lim_ {c to – infty} lim_ {d to + infty} int_c ^ d has {f} $$
will exist. In addition, there would be a formula for this impermissible integral in terms of $ f $?

Can someone give me a hint for this problem? I have no idea how to start?

Can there be an algorithm faster than the fast Fourier transform to square a polynomial?

FFT is a fast algorithm for multiplying two polynomials, but if it is a square (ie, the polynomial multiplied by itself), can we find something better? (I have reached a paper by Jaewook Chung and M. Anwar Hussain and then something known as the Toom-Cook algorithm, but can no longer find any claims.)

real analysis – error estimate for the Fourier series of the fraction of $ x $

Enter image description here

I would like to prove the error estimate for the fractional part of $ x $ as you can read above.

First, I apply the Abel formula and get
$$ left | sum_ {n = N + 1} ^ M frac { sin (2 pi nx)} {n} right | = left | frac { sum_ {n = N + 1} ^ {M} sin (2 pi nx)} {M} + \
sum_ {k = N + 1} ^ {M-1} left ( sum_ {k = 1} ^ n sin (2 pi kx) right) ( frac {1} n- frac {1 } {n + 1}) right | $$

We have that
$$ left | sum_ {n = N + 1} ^ M sin (2 pi n x) right | = left | frac { cos (2 pi Nx) – cos (2 pi Mx)} {2 sin ( pi x)} right | = \
left | frac { sin ( pi (N – M) x) sin ( pi (M + N) x)} { sin ( pi x)} right |. $$

I hope my bill is correct. If so, how can I conclude from these equations that (as $ M to infty $) The mistake is
$ O ((1+ | x | N) ^ {- 1}) $, as $ x $ approaches any integer and $ N to infty $?

For me, I think I can see that, though $ x $ If a fixed number is removed from an integer, then the error is $ O (1 / N) $, I do not know how to handle it $ x $ is considered.

Fourier Transform – Use of FFT in the following convolution in a simulation

I have the following convolution as part of a numerical simulation.

$$ T (r) = int d ^ 3r_2 p (r_2) f (r_2) alpha (r-r_2) $$

My problem is that the analytic expressions for $ f $ and $ p $ exist however, I have the expression for $ alpha $ only in the Fourier domain in the form of $ alpha (k) $, I intended to rate as follows:

  1. Construct a grid with the grid of $ 100 times100 times100 $ With mesh and linspace in numpy
ran = linspace(-1,1,N_r)
x,y,z = meshgrid(ran,ran,ran) #position space
  1. Construct the components xf, yf, zf in the Fourier domain from x, y, z
xf = fftn(x)
yf = fftn(y)
zf = fftn(z)
  1. Find the Fourier transform of $ f (r) times p (r) $ With FFTn in numpy
  2. Multiply it with $ alpha (k) $
  3. Take the inverse Fourier transformation with you ifftn in numpy.

I'm not sure if the above method works, and in fact I could not validate it properly. I've tried using scipy.ndimage.convolve to compare the results with the inverse Fourier transform of the product in the Fourier domain. Is it correct what I do with the code? And is there a way to check if a method works by using a simpler example?

Try to check:

I tried the following to test the theory. Seems like it would not work. I expect the result RES_1 and res_2 be equal. I also used the function real cut off the tiny imaginary part that results from that FFTn and ifftn functions.

x = linspace(-1,1,10)
xf = fftn(x)

def f(x):
    return x**2+x**3*sin(x)

def g(k):
    return k**2+k**3/(3-k**2)

g_k = g(xf)
g_x = real(ifftn(g_k))

res_1 = img_con(g_x,f(x))

res_2 = real(ifftn(g(xf)*fftn(f(x))))

print(res_1)
print(res_2)

Am I doing something wrong?

pde – heat equation and Fourier series

I try to solve the following boundary value problem with the heat equation:
$$
frac { partial {u}} { partial {t}} – frac {1} {4} frac { partial ^ 2 {u}} { partial {x} ^ 2} = 0, ; ; t> 0 text {and} 0 <x < 1, \ u(0,t) = u(1,t) = 0,;; t > 0, \
u (x, 0) = sin {(2 pi x)} – frac {1} {3} sin {(4 pi x)}, ; ; 0 <x <1.
$$

I used the separation of variables $ u (x, t) = X (x) T (t) $ and went downstairs $ u (x, t) = sum_ {k = 1} ^ { infty} let ^ {- frac {1} {4} (k pi) ^ 2 t} sin {(k pi x) } $ for a coefficient b.

My question is, how do I use the initial condition? $ u (x, 0) $ Finish this problem by setting the coefficient & # 39; b & # 39; determine. I'm assuming that I would have to use Fourier series for sine, but I'm not sure how to handle it. Every help is appreciated.