I have an issue using the FFTPACK5.1 in Fortran 90 which contains subroutines to compute discrete Fourier transforms. I manage to install it and use the routines but when I'm checking if everything is ok with a simple sine wave with a frequency A I get a non zero coefficient not at A (in frequency space, in the spectrum) but at 2A. There is a shift in the spectrum and I don't understand why. I'm almost sure (but I have doubts) that I compute correctly the frequency axis steps:
With N being the number of points of my original sine wave, and Fech my sample frequency I compute the frequency axis steps as df(i)=Fech(i-1)/N.
I'm using the rfft1f routine, so if someone has an experience with it and knows my problem I will be really greatful to understand what is wrong here.
Here is my code:
! n: number of samples in the discret signal
integer ( kind = 4 ), parameter :: n = 4096
real, parameter :: deuxpi=6.283185307
!frequence is the frequence of the signal
!fech is the frequence of sampling
real :: frequence,fech
integer :: kk
! r is the signal i want to process
! t is the built time and f is the built frequency
real ( kind = 4 ) r(n),t(n),f(n)
!Arrays routines need to work (internal recipe):
real ( kind = 4 ), allocatable, dimension ( : ) :: work
real ( kind = 4 ), allocatable, dimension ( : ) :: wsave
!size of arrays wsave and work for internal recipe
lensav = n + int ( log ( real ( n, kind = 4 ) ) / log ( 2.0E+00 ) ) + 4
lenwrk = n
allocate ( work(1:lenwrk) )
allocate ( wsave(1:lensav) )
! initializes rttft1f, wsave array
call rfft1i ( n, wsave, lensav, ier )
frequence=0.5
fech=20
! I built here the signal
do kk=1,n
t (kk) = (kk-1) / (fech)
f (kk) = fech*(kk-1) / n
r (kk) = sin( deuxpi * t(kk) * frequence )
end do
!and I call the rfft1f to build the Discrete Fourier Transform:
call rfft1f ( n, inc, r, lenr, wsave, lensav, work, lenwrk, ier )
!I get back r which contains now the fourier coefficients and plot it
I'm expecting with a simple sine wave to have a dirac at the frequency at 0.5 (cf code) but instead I get a dirac at 1., in the frequency domain. Besides, the spectrum looks odd... Here is what I get:
As is typical of routines computing discrete fourier transform of real-valued sequences, the resulting complex valued spectrum is returned for only half the spectrum. To fit the values into the original N-element array, only the real-part of the first value (which is always real) is returned. Similarly in the case of even values of n, the real-part of the n/2-th complex value (which is also always real) is returned. For all other complex values, the real and imaginary parts are interleaved.
So for even n, the corresponding frequency is given by:
r(1) -> 0
r(2), r(3) -> Delta
r(4), r(5) -> 2*Delta
...
r(n) -> (n/2)*Delta
And for odd n:
r(1) -> 0
r(2), r(3) -> Delta
r(4), r(5) -> 2*Delta
...
r(n-1),r(n) -> ((n-1)/2) * Delta
where Delta is given as fech/n.
To plot the complex values, you'd probably want to plot either the real (indices 1,2,4,6,...) & imaginary (indices 3,5,7,...) parts as two separate graphs, or the amplitude & phase (again as two separate graphs).
Related
Program Foucault
IMPLICIT NONE
REAL,DIMENSION(:),ALLOCATABLE :: t, x,y
REAL,PARAMETER :: pi=3.14159265358979323846, g=9.81
REAL :: L, vitessea, lat, h, omega, beta
INTEGER :: i , zeta
zeta=1000
Allocate(x(zeta),y(zeta),t(zeta))
L=67.
lat=49/180*pi
omega=sqrt(g/L)
h=0.01
Do i= 1,zeta
IF(i==1 .OR. i==2) THEN
t(1)=0.0
t(2)=0.0
x(1)=0.1
x(2)=1
y(1)=0.0
y(2)=0.0
ELSE
t(i+1)=real(i)*h
x(i+1)=(-omega**2*x(i)+2.0*((y(i)-y(i-1))/h)*latang(lat))*h**2+2.0*x(i)-x(i-1)
y(i+1)=(-omega**2*y(i)-2.0*((x(i)-x(i-1))/h)*latang(lat))*h**2+2.0*y(i)-y(i-1)
END IF
WRITE(40,*) t(i), x(i)
WRITE(60,*) t(i), y(i)
WRITE(50,*) x(i), y(i)
END DO
Contains
REAL Function latang(alpha)
REAL, INTENT(IN) :: alpha
REAL :: sol
latang=2*pi*sin(alpha)/86400
END FUNCTION
End Program Foucault
I'm trying to code the original Foucault Pendulum in Paris. My code seems to be working but so far, I could only get the below right graphic, "the flower" evolution. Therefore, I changed my parameters constantly to get the left graphic but I couldn't.
I took parameters of Foucault Pendulum installed in Paris with L=67, angular velocity of earth =2*pi/86400 and latitude of 49/180*pi.
My initial conditions are as written in the code. I tried a way range of parameters varying all of my initial conditions, my latitude and angular velocity but i couldn't get the left desired results.
I used Foucault differential equations as below : i coded them with Finite difference method (more simple than Runge-Kutta) by replacing the 2nd order derivation by its central finite difference. And the first order one by it's backward finite difference. By then, i build my loop by isolating x(i+1) and y(i+1) in both equations.
My code is very sensitive to parameters such as h (=derivation step), earth angular velocity and latitude (which is normal). I tried to change a way big range of parameters from a big h step to a small one, to a minimal and high latitude, initial conditions...etc but i couldn't ever get the left graphic which i rather need.
What could be made to get the left one ?
I was able to get the two charts, by speeding up the earth's rotation 120× fold, and allowing the simulation to run for 32 swings of the pendulum. Also, I noticed that Euler integration added energy to the system making for bad results, so I reverted to a standard RK4 implementation.
and here is the code I used to solve this ODE:
program FoucaultOde
implicit none
integer, parameter :: sp = kind(1.0), dp = kind(1d0)
! Constants
real, parameter :: g=9.80665, pi =3.1415926536
! Variables
real, allocatable :: y(:,:), yp(:), k0(:),k1(:),k2(:),k3(:)
real :: lat, omega, h, L, earth, period
real :: t0,x0,y0,vx0,vy0
integer :: i, zeta, f1, swings
! Code starts here
swings = 32
zeta = 400*swings
L = 67
lat = 49*pi/180
period = 24*60*60 ! period = 86400
earth = (2*pi*sin(lat)/period)*120 !120 multiplier for roation
omega = sqrt(g/L)
allocate(y(5,zeta))
allocate(yp(5), k0(5),k1(5),k2(5),k3(5))
! make pendulum complete 'swings' cycles in 'zeta' steps
h = swings*2*pi/(omega*zeta)
t0 = 0
x0 = 0.5 ! Initial displacement
y0 = 0
vx0 = 0
vy0 = 0
! Initial conditions in the state vector Y
Y(:,1) = [t0,x0,y0,vx0,vy0]
do i=2, zeta
! Euler method (single step)
! Yp = ode(Y(:,i-1))
! Runge-Kutta method (four steps)
k0 = ode(Y(:,i-1))
k1 = ode(Y(:,i-1) + h/2*k0)
k2 = ode(Y(:,i-1) + h/2*k1)
k3 = ode(Y(:,i-1) + h*k2)
Yp = (k0+2*k1+2*k2+k3)/6
! Take a step
Y(:,i) = Y(:,i-1) + h*Yp
end do
open( newunit=f1, file='results.csv', status = 'replace', pad='no')
! write header
write (f1, '(a15,a,a15,a,a15,a,a15,a,a15)') 't',',', 'x',',','y',',', 'vx',',','vy'
! write rows of data, comma-separated
do i=1, zeta
write (f1, '(g,a,g,a,g,a,g,a,g)') y(1,i),',',y(2,i),',',y(3,i),',',y(4,i),',',y(5,i)
end do
close(f1)
contains
function ode(Y) result(Yp)
real, intent(in) :: Y(5)
real :: Yp(5), t,px,py,vx,vy,ax,ay
! Read state vector Y to component values
t = Y(1)
px = Y(2)
py = Y(3)
vx = Y(4)
vy = Y(5)
! Reference paper:
! http://www.legi.grenoble-inp.fr/people/Achim.Wirth/final_version.pdf
ax = -(omega**2)*px + 2*vy*earth ! (equation 53)
ay = -(omega**2)*py - 2*vx*earth ! (equation 54)
! State vector rate. Note, rate of time is aways 1.0
Yp = [1.0, vx, vy, ax, ay]
end function
end program FoucaultOde
The resulting file results.csv looks like this for me (for checking)
t, x, y, vx, vy
.000000 , 5.000000 , .000000 , .000000 , .000000
.4105792E-01, 4.999383 , .1112020E-06, -.3004657E-01, .8124921E-05
.8211584E-01, 4.997533 , .8895339E-06, -.6008571E-01, .3249567E-04
.1231738 , 4.994450 , .3001796E-05, -.9011002E-01, .7310022E-04
.1642317 , 4.990134 , .7114130E-05, -.1201121 , .1299185E-03
.2052896 , 4.984587 , .1389169E-04, -.1500844 , .2029225E-03
.2463475 , 4.977810 , .2399832E-04, -.1800197 , .2920761E-03
.2874054 , 4.969805 , .3809619E-04, -.2099106 , .3973353E-03
...
from which I plotted the 2nd and 3rd columns in one chart, and the 4th and 5th for the second chart.
There is one thing that may be wrong depending on how you manage different step sizes, and an observation on the physics of the real-world example. With the initialization of the arrays, you imply an initial velocity of about 0.9/0.01=90 [m/s] in x direction away from the center. To get compatible results for different step sizes, you would need to adapt the calculation of x(2). However, in the graphs the plot starts from a point with zero velocity. This you can implement to first order by setting x(2)=x(1)=1. As the used integration method is also first order, this is sufficient.
For the second point, note that one can write the system using complex coordinates z=x+iy as
z'' = -w^2*z - 2*i*E*z', E = Omega*sin(theta)
This is a linear ODE with constant coefficients, the solution of it is
z(t) = exp(-i*E*t) * (A*cos(w1*t)+B*sin(w1*t)), w1 = sqrt(w^2+E^2)
This describes a pendulum motion of frequency w1 whose plane rotates with frequency E clockwise. The grand rotation has period T=2*pi/E, during which w1*T/(2*pi)=w1/E pendulum swings occur.
Now insert your numbers, w=sqrt(g/L)=0.383 and E=2*pi*sin(49°)/86400=5.49e-05, so that essentially w1=w. The number of pendulum cycles per full rotation is w/E=6972, so that you can expect a densely filled circle in the plot. Or a very narrow double wedge if only a few cycles are plotted. As each cycle takes 2*pi/w=16.4 [s], and the integration goes 1000 steps of step size 0.01, in the plot as it is you can expect a swing forth and part of the swing back.
To be more realistic, set the initial velocity to zero, that is, the pendulum is taken to its start position and then let go. Also increase the time to 30 [s] to have more than one pendulum cycle in the plot.
It from this we can see that the solutions converge, and with some imagination, that they converge linearly.
To get a plot like in the cited images, one needs a much smaller fraction of w/E, counting the swings, it has to be around 15. Note that you can not get this ratio anywhere on earth with a realistically scaled pendulum. So set w=pi, E=pi/16 and integrate over 15 time units using the first order method.
This detoriorates really fast, even for the smallest step size with 40 points in a pendulum cycle.
For a better result, increase the local truncation order to the next higher by using the central difference in the first derivative approximation.
z(i+1) - 2*z(i) + z(i-1) = -w^2*z(i)*dt^2 - i*E*(z(i+1)-z(i-1))*dt
z(i+1) = ( 2*z(i) - z(i-1) - w^2*z(i)*dt^2 + i*E*z(i-1)*dt ) / (1+i*E*dt)
The division by the complex number can also be easily carried out in the real components of the trajectory,
! x(i+1)-2*x(i)+x(i-1) = h^2*(-omega**2*x(i)) + h*earth*(y(i+1)-y(i-1))
! y(i+1)-2*y(i)+y(i-1) = h^2*(-omega**2*y(i)) - h*earth*(x(i+1)-x(i-1))
t(i) = t(i-1) + h
cx = (2-(h*omega)**2)*x(i) - x(i-1) - h*earth*y(i-1)
cy = (2-(h*omega)**2)*y(i) - y(i-1) + h*earth*x(i-1)
den = 1+(h*earth)**2
x(i+1) = (cx + h*earth*cy)/den
y(i+1) = (cy - h*earth*cx)/den
Now to respect the increased order, also the initial points need to have an order of accuracy more, using again zero initial speed, this gives in the second order Taylor expansion
z(2) = z(1) - 0.5*w^2*z(1)*dt^2
All the step sizes that gave deviating and structurally deteriorating results in the first order method now give a visually identical, structurally stable results in this second order method.
I am trying to write a FORTRAN code to evaluate the fast Fourier transform of the Gaussian function f(r)=exp(-(r^2)) using FFTW3 library. As everyone knows, the Fourier transform of the Gaussian function is another Gaussian function.
I consider evaluating the Fourier-transform integral of the Gaussian function in the spherical coordinate.
Hence the resulting integral can be simplified to be integral of [r*exp(-(r^2))*sin(kr)]dr.
I wrote the following FORTRAN code to evaluate the discrete SINE transform DST which is the discrete Fourier transform DFT using a PURELY real input array. DST is performed by C_FFTW_RODFT00 existing in FFTW3, taking into account that the discrete values in position space are r=i*delta (i=1,2,...,1024), and the input array for DST is the function r*exp(-(r^2)) NOT the Gaussian. The sine function in the integral of [r*exp(-(r^2))*sin(kr)]dr resulting from the INTEGRATION over the SPHERICAL coordinates, and it is NOT the imaginary part of exp(ik.r) that appears when taking the analytic Fourier transform in general.
However, the result is not a Gaussian function in the momentum space.
Module FFTW3
use, intrinsic :: iso_c_binding
include 'fftw3.f03'
end module
program sine_FFT_transform
use FFTW3
implicit none
integer, parameter :: dp=selected_real_kind(8)
real(kind=dp), parameter :: pi=acos(-1.0_dp)
integer, parameter :: n=1024
real(kind=dp) :: delta, k
real(kind=dp) :: numerical_F_transform
integer :: i
type(C_PTR) :: my_plan
real(C_DOUBLE), dimension(1024) :: y
real(C_DOUBLE), dimension(1024) :: yy, yk
integer(C_FFTW_R2R_KIND) :: C_FFTW_RODFT00
my_plan= fftw_plan_r2r_1d(1024,y,yy,FFTW_FORWARD, FFTW_ESTIMATE)
delta=0.0125_dp
do i=1, n !inserting the input one-dimension position function
y(i)= 2*(delta)*(i-1)*exp(-((i-1)*delta)**2)
! I multiplied by 2 due to the definition of C_FFTW_RODFT00 in FFTW3
end do
call fftw_execute_r2r(my_plan, y,yy)
do i=2, n
k = (i-1)*pi/n/delta
yk(i) = 4*pi*delta*yy(i)/2 !I divide by 2 due to the definition of
!C_FFTW_RODFT00
numerical_F_transform=yk(i)/k
write(11,*) i,k,numerical_F_transform
end do
call fftw_destroy_plan(my_plan)
end program
Executing the previous code gives the following plot which is not for Gaussian function.
Can anyone help me understand what the problem is? I guess the problem is mainly due to FFTW3. Maybe I did not use it properly especially concerning the boundary conditions.
Looking at the related pages in the FFTW site (Real-to-Real Transforms, transform kinds, Real-odd DFT (DST)) and the header file for Fortran, it seems that FFTW expects FFTW_RODFT00 etc rather than FFTW_FORWARD for specifying the kind of
real-to-real transform. For example,
! my_plan= fftw_plan_r2r_1d( n, y, yy, FFTW_FORWARD, FFTW_ESTIMATE )
my_plan= fftw_plan_r2r_1d( n, y, yy, FFTW_RODFT00, FFTW_ESTIMATE )
performs the "type-I" discrete sine transform (DST-I) shown in the above page. This modification seems to fix the problem (i.e., makes the Fourier transform a Gaussian with positive values).
The following is a slightly modified version of OP's code to experiment the above modification:
! ... only the modified part is shown...
real(dp) :: delta, k, r, fftw, num, ana
integer :: i, j, n
type(C_PTR) :: my_plan
real(C_DOUBLE), allocatable :: y(:), yy(:)
delta = 0.0125_dp ; n = 1024 ! rmax = 12.8
! delta = 0.1_dp ; n = 128 ! rmax = 12.8
! delta = 0.2_dp ; n = 64 ! rmax = 12.8
! delta = 0.4_dp ; n = 32 ! rmax = 12.8
allocate( y( n ), yy( n ) )
! my_plan= fftw_plan_r2r_1d( n, y, yy, FFTW_FORWARD, FFTW_ESTIMATE )
my_plan= fftw_plan_r2r_1d( n, y, yy, FFTW_RODFT00, FFTW_ESTIMATE )
! Loop over r-grid
do i = 1, n
r = i * delta ! (2-a)
y( i )= r * exp( -r**2 )
end do
call fftw_execute_r2r( my_plan, y, yy )
! Loop over k-grid
do i = 1, n
! Result of FFTW
k = i * pi / ((n + 1) * delta) ! (2-b)
fftw = 4 * pi * delta * yy( i ) / k / 2 ! the last 2 due to RODFT00
! Numerical result via quadrature
num = 0
do j = 1, n
r = j * delta
num = num + r * exp( -r**2 ) * sin( k * r )
enddo
num = num * 4 * pi * delta / k
! Analytical result
ana = sqrt( pi )**3 * exp( -k**2 / 4 )
! Output
write(10,*) k, fftw
write(20,*) k, num
write(30,*) k, ana
end do
Compile (with gfortran-8.2 + FFTW3.3.8 + OSX10.11):
$ gfortran -fcheck=all -Wall sine.f90 -I/usr/local/Cellar/fftw/3.3.8/include -L/usr/local/Cellar/fftw/3.3.8/lib -lfftw3
If we use FFTW_FORWARD as in the original code, we get
which has a negative lobe (where fort.10, fort.20, and fort.30 correspond to FFTW, quadrature, and analytical results). Modifying the code to use FFTW_RODFT00 changes the result as below, so the modification seems to be working (but please see below for the grid definition).
Additional notes
I have slightly modified the grid definition for r and k in my code (Lines (2-a) and (2-b)), which is found to improve the accuracy. But I'm still not sure whether the above definition matches the definition used by FFTW, so please read the manual for details...
The fftw3.f03 header file gives the interface for fftw_plan_r2r_1d
type(C_PTR) function fftw_plan_r2r_1d(n,in,out,kind,flags) bind(C, name='fftw_plan_r2r_1d')
import
integer(C_INT), value :: n
real(C_DOUBLE), dimension(*), intent(out) :: in
real(C_DOUBLE), dimension(*), intent(out) :: out
integer(C_FFTW_R2R_KIND), value :: kind
integer(C_INT), value :: flags
end function fftw_plan_r2r_1d
(Because of no Tex support, this part is very ugly...) The integral of 4 pi r^2 * exp(-r^2) * sin(kr)/(kr) for r = 0 -> infinite is pi^(3/2) * exp(-k^2 / 4) (obtained from Wolfram Alpha or by noting that this is actually a 3-D Fourier transform of exp(-(x^2 + y^2 + z^2)) by exp(-i*(k1 x + k2 y + k3 z)) with k =(k1,k2,k3)). So, although a bit counter-intuitive, the result becomes a positive Gaussian.
I guess the r-grid can be chosen much coarser (e.g. delta up to 0.4), which gives almost the same accuracy as long as it covers the frequency domain of the transformed function (here exp(-r^2)).
Of course there are negative components of the real part to the FFT of a limited Gaussian spectrum. You are just using the real part of the transform. So your plot is absolutely correct.
You seem to be mistaking the real part with the magnitude, which of course would not be negative. For that you would need to fftw_plan_dft_r2c_1d and then calculate the absolute values of the complex coefficients. Or you might be mistaking the Fourier transform with a limited DFT.
You might want to check here to convince yourself of the correctness of you calculation above:
http://docs.mantidproject.org/nightly/algorithms/FFT-v1.html
Please do keep in mind that the plots on the above page are shifted, so that the 0 frequency is in the middle of the spectrum.
Citing yourself, the nummeric integration of [r*exp(-(r^2))*sin(kr)]dr would have negative components for all k>1 if normalised to 0 for highest frequency.
TLDR: Your plot is absolute state of the art and inline with discrete and limited functional analysis.
I'm attempting to test a program that calculates the discrete Fourier transform of a signal, namely a sine wave. To test it, I need to plot my results. However, the result is an array of size N (currently at 400) and is filled with complex numbers of the form z = x + iy. So I know that to test it I need to plot these results, and that to do this I need to plot |z|. Here's my program:
program DFT
implicit none
integer :: k, N, x, y, j, r, l, istat, p
integer, parameter :: dp = selected_real_kind(15,300)
real, allocatable,dimension(:) :: h
complex, allocatable, dimension(:) :: rst
complex, dimension(:,:), allocatable :: W
real(kind=dp) :: pi
p = 2*pi
!open file to write results to
open(unit=100, file="dft.dat", status='replace')
N = 400
!allocate arrays as length N, apart from W (NxN)
allocate(h(N))
allocate(rst(N))
allocate(W(-N/2:N/2,1:N))
pi = 3.14159265359
!loop to create the sample containing array
do k=1,N
h(k) = sin((2*pi*k)/N)
end do
!loop to fill the product matrix with values
do j = -N/2,N/2
do k = 1, N
W(j,k) = EXP((2.0_dp*pi*cmplx(0.0_dp,1.0_dp)*j*k)/N)
end do
end do
!use of matmul command to multiply matrices
rst = matmul(W,h)
print *, h, w
write(100,*) rst
end program
So my question is how do I take the magnitude of all the individual complex numbers in the array?
The ABS intrinsic function returns the magnitude of a complex number in Fortran. It is an elemental function as well, so for an array of type complex simply ABS( array ) will return a real array with the same kind as the original containing the results you want.
I'm attempting to take the inverse transform of a complex 1D arrays forward transform in Fortran 90 and fftw. However, the output I receive from the inverse transform is at times completely different from the original input, whereas some values possess an incorrect real section but a correct imaginary part and a few match the original values perfectly.
I've noticed that this issue disappears if dx (the spacing between x values) is reduced to 0.01. Increasing n to compensate for this reduction in x's range then results in the issue resurfacing.
At this point, I believe the issue lies in the 1/cosh segment of the input array as I've been able to replace this with other complex inputs with no issues.
This code is adapted from a MATLAB file in which the form of the input only differs due to MATLAB using sech instead of 1/cosh.
Fortran isn't my 'go to' language so I'm wondering if I've made some normally obvious mistake due to my familiarity with python/matlab .
As for more specifics on the outputs,
The matlab version of this code produces the same values for the in array but the operation of the forward transform and the inverse transform produce different results,
Matlab
out2(2) = 5.5511e-17 + 6.9389e-18i
out2(3) = 5.5511e-17 - 1.3878e-17i
out2(4) = 5.5511e-17 + 2.7756e-17i
out2(1024) = 0.9938 + 0.0994i
out2(2048) = 0 - 1.3878e-17i
Fortran
out2(2) = -5.5511151231257827E-017 - 6.9388939039072284E-018i
out2(3) = 0.0000000000000000 + 1.3877787807814457E-017i
out2(4) = 0.0000000000000000 + 0.0000000000000000i
out(1024) = 0.99380163159683255 + 9.9410098890158616E-002i
out2(2048) = -5.5511151231257827E-017 - 6.9388939039072284E-018i
PROGRAM FFTEXAMPLE
implicit none
include 'fftw3.f'
INTEGER :: n, j, nindex, i
REAL :: dx
DOUBLE COMPLEX, ALLOCATABLE :: in(:), out(:), in2(:), out2(:)
REAL(kind = 8), ALLOCATABLE :: x(:)
INTEGER*8 :: plan, plan2
nindex = 11
n = 2 ** nindex
dx = 0.05 ! Spacing between x array values
allocate( in(n), out(n), x(n), in2(n), out2(n) )
CALL dfftw_plan_dft_1d( plan, n, in, out, FFTW_FORWARD, FFTW_ESTIMATE )
CALL dfftw_plan_dft_1d( plan2, n, in2, out2, FFTW_BACKWARD, FFTW_ESTIMATE )
x = (/ (-dx*n/2 + (i-1)*dx, i=1, n) /) ! Seeds x array from -51.2 to 51.15
! Create values for the input array
DO j = 1, n, 1
in(j) = 1/cosh ( x(j)/1.0040 ) * exp( (0.0, -1.0) * 1.9940 * x(j) )
END DO
CALL dfftw_execute_dft( plan, in, out ) ! FWD transform
!DO j = 1, n, 1
! in2(j) = cmplx(REAL(out(j)), AIMAG(out(j)))
!END DO
in2 = out
CALL dfftw_execute_dft( plan2, in2, out2 ) ! Inverse transform
out2 = out2/n ! Divide output by n to normalise
CALL dfftw_destroy_plan( plan )
CALL dfftw_destroy_plan( plan2 )
END PROGRAM
I am trying to write a programme to calculate an absorption band model for seismic waves. The whole calculation is based on 3 equations. If interested, see equations 3, 4, 5 on p.2 here:
http://www.eri.u-tokyo.ac.jp/people/takeuchi/publications/14EPSL-Iritani.pdf
However, I have debugged this programme several times now but I do not seem to get the expected answer. I am specifically trying to calculate Q_1 variable (seismic attenuation) in the following programme, which should be a REAL positive value on the order of 10^-3. However, I am getting negative values. I need a fresh pair of eyes to take a look at the programme and to check where I have done a mistake if any. Could someone please check? Many thanks !
PROGRAM absorp
! Calculate an absorption band model and output
! files for plotting.
! Ref. Iritani et al. (2014), EPSL, 405, 231-243.
! Variable Definition
! Corners - cf1, cf2
! Frequency range - [10^f_strt, 10^(f_end-f_strt)]
! Number of points to be sampled - n
! Angular frequency - w
! Frequency dependent Attenuation 1/Q - Q_1
! Relaxation times - tau1=1/(2*pi*cf1), tau2=1/(2*pi*cf2)
! Reference velocity - V0 (km/s)
! Attenuation (1/Q) at 1 Hz - Q1_1
! Frequency dependent peak Attenuation (1/Qm) - Qm_1
! Frequency dependent velocity - V_w
! D(omega) numerator - Dw1
! D(omega) denominator - Dw2
! D(omega) - D_w
! D(2pi) - D_2pi
IMPLICIT NONE
REAL :: cf1 = 2.0e0, cf2 = 1.0e+5
REAL, PARAMETER :: f_strt=-5, f_end=12
INTEGER :: indx
INTEGER, PARAMETER :: n=1e3
REAL, PARAMETER :: pi=4.0*atan(1.0)
REAL, DIMENSION(1:n) :: w, Q_1
REAL :: tau1, tau2, V0, freq, pow
REAL :: Q1_1=0.003, Qm_1
COMPLEX, DIMENSION(1:n) :: V_w
COMPLEX, PARAMETER :: i=(0.0,1.0)
COMPLEX :: D_2pi, D_w, Dw1, Dw2
! Reference Velocity km/s
V0 = 12.0
print *, "F1=", cf1, "F2=", cf2, "V0=",V0
! Relaxation times from corners
tau1 = 1.0/(2.0*pi*cf1)
tau2 = 1.0/(2.0*pi*cf2)
PRINT*, "tau1=",tau1, "tau2=",tau2
! Populate angular frequency array (non-linear)
DO indx = 1,n+1
pow = f_strt + f_end*REAL(indx-1)/n
freq=10**pow
w(indx) = 2*pi*freq
print *, w(indx)
END DO
! D(2pi) value
D_2pi = LOG((i*2.0*pi + 1/tau1)/(i*2.0*pi + 1/tau2))
! Calculate 1/Q from eq. 3 and 4
DO indx=1,n
!D(omega)
Dw1 = (i*w(indx) + 1.0/tau1)
Dw2 = (i*w(indx) + 1.0/tau2)
D_w = LOG(Dw1/Dw2)
!This is eq. 5 for 1/Qm
Qm_1 = 2.0*pi*Q1_1*IMAG(D_w)/ &
((Q1_1**2-4)*IMAG(D_w)**2 &
+ 4*Q1_1*IMAG(D_w)*REAL(D_w))
!This is eq. 3 for Alpha(omega)
V_w(indx) = V0*(SQRT(1.0 + 2.0/pi*Qm_1*D_w)/ &
REAL(SQRT(1.0 + 2.0/pi*Qm_1*D_2pi)))
!This is eq. 4 for 1/Q
Q_1(indx) = 2*IMAG(V_w(indx))/REAL(V_w(indx))
PRINT *, w(indx)/(2.0*pi), (V_w(indx)), Q_1(indx)
END DO
! write the results out
100 FORMAT(F12.3,3X,F7.3,3X,F8.5)
OPEN(UNIT=1, FILE='absorp.txt', STATUS='replace')
DO indx=1,n
WRITE(UNIT=1,FMT=100), w(indx)/(2.0*pi), REAL(V_w(indx)), Q_1(indx)
END DO
CLOSE(UNIT=1)
END PROGRAM
More of an extended comment with formatting than an answer ...
I haven't checked the equations you refer to, and I'm not going to, but looking at your code makes me suspect misplaced brackets as a likely cause of errors. The code, certainly as you've shown it here, isn't well formatted to reveal its logical structure. Whatever you do next invest in some indents and some longer lines to avoid breaking too frequently.
Personally I'm suspicious in particular of
!This is eq. 5 for 1/Qm
Qm_1 = 2.0*pi*Q1_1*IMAG(D_w)/ &
((Q1_1**2-4)*IMAG(D_w)**2 &
+ 4*Q1_1*IMAG(D_w)*REAL(D_w))