topic strange values for the FFT of an odd function in Intel® oneAPI Math Kernel Library

strange values for the FFT of an odd function

vincent-torri — Fri, 08 Oct 2010 15:13:44 GMT

I want to compute the Fourier coefficients of the 2-periodic function that is equal to x |---> x on [-1,1]. When I display them, they have the correct imaginary value, but a real value which is not equal to 0 (like it should be), but which is equal to -1.

Here is the code:

[cpp]#include 

#include 
#include 

float fct_der(float x)
{
  return x;
}


int main()
{
  float _Complex x_in[32];
  float _Complex x_out[32];
  DFTI_DESCRIPTOR_HANDLE my_desc1_handle;
  MKL_LONG status;
  int i;

  for (i = 0; i < 32; i++)
    {
      float x;

      x = -1.0f + (float)(2 * i) / 32.0f;
      x_in = fct_der(x);
    }

  status = DftiCreateDescriptor( &my_desc1_handle, DFTI_SINGLE, DFTI_COMPLEX, 1, 32);
  status = DftiSetValue( my_desc1_handle, DFTI_PLACEMENT, DFTI_NOT_INPLACE);
  status = DftiCommitDescriptor( my_desc1_handle );
  status = DftiComputeForward( my_desc1_handle, x_in, x_out);
  status = DftiFreeDescriptor(&my_desc1_handle);

  for (i = 0; i < 32; i++)
    {
      printf ("%f + i %f\n", creal(x_out), cimag(x_out));
    }

  return 0;
}[/cpp]

And here is the output:

[plain]-1.000000 + i 0.000000
-1.000000 + i 10.153171
-1.000000 + i 5.027339
-1.000000 + i 3.296558
-1.000000 + i 2.414214
-1.000000 + i 1.870868
-1.000000 + i 1.496606
-1.000000 + i 1.218504
-1.000000 + i 1.000000
-1.000000 + i 0.820679
-1.000000 + i 0.668179
-1.000000 + i 0.534511
-1.000000 + i 0.414214
-1.000000 + i 0.303347
-1.000000 + i 0.198912
-1.000000 + i 0.098491
-1.000000 + i 0.000000
-1.000000 + i -0.098491
-1.000000 + i -0.198912
-1.000000 + i -0.303347
etc...[/plain]

Does someone see why the real part is not 0 ?

thank you

strange values for the FFT of an odd function

Todd_R_Intel — Fri, 08 Oct 2010 18:41:20 GMT

I'm trying to read your program and understand what you're trying to do. You mention a real function and in your program you initialize your variable of type float with real data, but then you assign it to your float _Complex data variable and perform a complex FFT.

Is this intended? Is there a reason you don't just use a real data type for x_in and perform a real FFT?

Todd

strange values for the FFT of an odd function

vincent-torri — Fri, 08 Oct 2010 21:20:47 GMT

it is intended, indeed. It is the first time that i use the Intel compiler and the MKL. So I am doing tests. Especially tests that can theoretically be computed. It works for an even function (I tried x^2-1) and the imaginary part is 0. But not with that simple odd function.

In the future, i will use the FFT on a complex valued function (more precisely, i want to solve a non linear Schrdinger equation). So I prefer getting familiar with the complex API.

strange values for the FFT of an odd function

Dmitry_B_Intel — Sat, 09 Oct 2010 03:31:01 GMT

Hi vtorri,

The function you take FFT of is not odd. An odd function must have x_in[0]==0.
Instead of x_in = [-1 + (0:31)*2/32] try [0, -1 + (0:30)*2/30].

Thanks
Dima

strange values for the FFT of an odd function

vincent-torri — Sat, 09 Oct 2010 15:42:35 GMT

Of course the function that i took is odd. I discretized x |--> x on [-1,1] which is obviously odd. On the other hand, your remark raises a good point: must I "shift" the vector values, with something like:

int shift = N/2 + 1;
int i;
for (i = 0; i < N; i++)
out2 = out[i+shift)%N];

so that the discretized value of x=0 is the first value of my array ?

strange values for the FFT of an odd function

Dmitry_B_Intel — Mon, 11 Oct 2010 02:23:44 GMT

Right, the input x[0:N-1] to FFT is expecpected to bedirectly mapped frominterval[0,P] rather than from [-P/2,P/2], so that f(0) is the first value of the input array.

Thanks
Dima

strange values for the FFT of an odd function

vincent-torri — Mon, 11 Oct 2010 10:57:38 GMT

I shifted the values:

[bash]  for (i = 0; i < 32; i++)
    {
      float x;

      x = -1.0f + (float)(2 * i) / 32.0f;
      tmp = fct_der(x);
    }

  shift = 32 / 2;
  for (i = 0; i < 32; i++)
    {
      x_in = tmp[(i+shift)%32];
    }[/bash]

but the result is still wrong (1st column : i, 2nd : x_in, 3rd and 4th x_out) :

[bash]0 ** 0.000000 ** -1.000000 + i 0.000000
1 ** 0.062500 ** 1.000000 + i -10.153171
2 ** 0.125000 ** -1.000000 + i 5.027339
3 ** 0.187500 ** 1.000000 + i -3.296558
4 ** 0.250000 ** -1.000000 + i 2.414214
5 ** 0.312500 ** 1.000000 + i -1.870868
6 ** 0.375000 ** -1.000000 + i 1.496606
7 ** 0.437500 ** 1.000000 + i -1.218504
8 ** 0.500000 ** -1.000000 + i 1.000000
9 ** 0.562500 ** 1.000000 + i -0.820679
10 ** 0.625000 ** -1.000000 + i 0.668179
11 ** 0.687500 ** 1.000000 + i -0.534511
12 ** 0.750000 ** -1.000000 + i 0.414214
13 ** 0.812500 ** 1.000000 + i -0.303347
14 ** 0.875000 ** -1.000000 + i 0.198912
15 ** 0.937500 ** 1.000000 + i -0.098491
16 ** -1.000000 ** -1.000000 + i 0.000000
17 ** -0.937500 ** 1.000000 + i 0.098491
18 ** -0.875000 ** -1.000000 + i -0.198912
19 ** -0.812500 ** 1.000000 + i 0.303347
20 ** -0.750000 ** -1.000000 + i -0.414214
21 ** -0.687500 ** 1.000000 + i 0.534511
22 ** -0.625000 ** -1.000000 + i -0.668179
23 ** -0.562500 ** 1.000000 + i 0.820679
24 ** -0.500000 ** -1.000000 + i -1.000000
25 ** -0.437500 ** 1.000000 + i 1.218504
26 ** -0.375000 ** -1.000000 + i -1.496606
27 ** -0.312500 ** 1.000000 + i 1.870868
28 ** -0.250000 ** -1.000000 + i -2.414214
29 ** -0.187500 ** 1.000000 + i 3.296558
30 ** -0.125000 ** -1.000000 + i -5.027339
31 ** -0.062500 ** 1.000000 + i 10.153171
[/bash]

Any idea ?

strange values for the FFT of an odd function

Dmitry_B_Intel — Mon, 11 Oct 2010 11:13:59 GMT

Hi vtorri,

To be odd, the discretization should obey the property:
x_in[0] = 0
x_in = -x_in[N-k], k=1...N-1
In your case this property doesn't hold for k=16.

Thanks
Dima

strange values for the FFT of an odd function

vincent-torri — Mon, 11 Oct 2010 11:22:44 GMT

Ok. So the shift is not what should be done, actually. It's more a symetry. And it's certainly why it works without problem with en even function. I'll try that.

strange values for the FFT of an odd function

vincent-torri — Mon, 11 Oct 2010 11:52:53 GMT

it works well for an even function because the number of coefficients are even. The problem is the number of coefficient for an odd function if I have to satisfy : x_in[0]=0 and x_in=x_in[N-i] i=1...N-1 (the number of coef is then odd)

strange values for the FFT of an odd function

Dmitry_B_Intel — Mon, 11 Oct 2010 12:17:05 GMT

Consider even N = 2L.

For even function this should hold:
x_in[0] = any
x_in = x_in[N-k]
In particular, x_in = x_in which means x_in is arbitrary.

For odd function this should hold:
x_in[0] = any
x_in = -x_in[N-k]
In particular, x_in = -x_in which means x_in must be zero.

Thanks
Dima

strange values for the FFT of an odd function

vincent-torri — Mon, 11 Oct 2010 21:52:09 GMT

I have found my error. The way I discretized x |----> x is wrong.The value ax x=-1 should be 0 and not -1. Thank you for your answers !