Not getting the exact data when reversing FFT

Question

Okay, what I'm trying to achieve is simple. Apply FFT on some random data and then apply the reverse algorithm on the output to get back the input. I'm using kissFFT library for this.

Code:

const int fft_siz = 512;
const int inverse = 1;

kiss_fft_cpx* in = (kiss_fft_cpx*)malloc(sizeof(kiss_fft_cpx) * fft_siz);
kiss_fft_cpx* out = (kiss_fft_cpx*)malloc(sizeof(kiss_fft_cpx) * fft_siz);
kiss_fft_cpx* rec = (kiss_fft_cpx*)malloc(sizeof(kiss_fft_cpx) * fft_siz);
kiss_fft_cfg cfg = kiss_fft_alloc(fft_siz, !inverse, NULL, NULL);
kiss_fft_cfg icfg = kiss_fft_alloc(fft_siz, inverse, NULL, NULL);

srand((unsigned int)time(NULL));
for(int i = 0; i < fft_siz; i++)
{
    in[i].r = rand() % 256;
    in[i].i = rand() % 256;
}

kiss_fft(cfg, in, out);

// scaling
for(int i = 0; i < fft_siz; i++)
{
    out[i].r /= fft_siz;
    out[i].i /= fft_siz;
}

kiss_fft(icfg, out, rec);

unsigned int count = 0;
for(int i = 0; i < fft_siz; i++)
    if(in[i].r != rec[i].r)
    {
        count++;
        printf( "in[%3d].r does not match rec[%3d].r :: %3d :: %f\n",
                i, i, count, in[i].r - rec[i].r);
    }
    else if(in[i].i != rec[i].i)
    {
        count++;
        printf( "in[%3d].i does not match rec[%3d].i :: %3d :: %f\n",
                i, i, count, in[i].i - rec[i].i);
    }

free(in);
free(out);
free(rec);
free(cfg);
free(icfg);

kiss_fft_cleanup();

Output:

in[  0]:     71.000000       85.000000 -- out[  0]:    127.095703      124.541016
in[  1]:    248.000000       27.000000 -- out[  1]:     -7.083314        0.072701
in[  2]:     64.000000       18.000000 -- out[  2]:     -3.770610        2.682554
in[  3]:      6.000000       96.000000 -- out[  3]:     -7.929140       -2.897723
in[  4]:     98.000000       23.000000 -- out[  4]:     -0.719621       -5.854260
in[  5]:    250.000000      188.000000 -- out[  5]:      0.397226       -1.248124
in[  6]:    231.000000        3.000000 -- out[  6]:     -7.934285       -2.367196
in[  7]:      6.000000      105.000000 -- out[  7]:     -0.317480       -2.955601
in[  8]:    172.000000      143.000000 -- out[  8]:     -4.236186        3.911616
in[  9]:     16.000000      134.000000 -- out[  9]:     -0.162577       -5.353521
in[ 10]:    230.000000      112.000000 -- out[ 10]:     -4.703711        7.791993
in[ 11]:      5.000000       26.000000 -- out[ 11]:     -2.636305        0.188381
in[ 12]:     16.000000      127.000000 -- out[ 12]:      1.137413        4.576081
in[ 13]:    112.000000       86.000000 -- out[ 13]:      0.978051       -0.408992
in[ 14]:     40.000000       23.000000 -- out[ 14]:      5.231920       -2.347566
in[ 15]:     75.000000       26.000000 -- out[ 15]:      0.009981       -2.091559

note                                ::count::difference
--------------------------------------------------------
in[  1].r does not match rec[  1].r ::   1 :: -0.000031
in[  3].r does not match rec[  3].r ::   2 :: -0.000015
in[  4].i does not match rec[  4].i ::   3 :: -0.000004
in[  6].i does not match rec[  6].i ::   4 :: -0.000008
in[  7].r does not match rec[  7].r ::   5 :: -0.000002
in[  9].r does not match rec[  9].r ::   6 :: -0.000015
in[ 11].r does not match rec[ 11].r ::   7 :: -0.000015
in[ 12].r does not match rec[ 12].r ::   8 :: -0.000015
in[ 13].i does not match rec[ 13].i ::   9 :: -0.000008
in[ 14].i does not match rec[ 14].i ::  10 :: 0.000008
in[ 15].r does not match rec[ 15].r ::  11 :: -0.000015

Debug. If you go to the bottom, you'll see that there's 317 mismatches. I'm also outputting the difference between values i.e. (in[].r - rec[].r) or (in[].i - rec[].i).

What I'm showing next is the input data where white dots represent the real part and red dots the imaginary part.

This is the output data of FFT represented in purple along with the reconstructed data in white and red.

Notice the small difference? I'm guessing this is related to floating point precision. How can I overcome this problem to get the exact same input data that I used FFT on ?

Edit:

I noticed that the range off error in my case is something like ]0 , 0.002]. So as a workaround, I rounded the reconstructed data and got a good result. But still... This only works if the fractional part of my numbers is 0.0.

Answer 1

You're guess is correct; floats have a limited precision, and some of the input precision will be lost in the process. If you want the exact input, you'll need some sort of arbitrary-precision floating-point library (for example, the GNU MP library).