Phase response of FFT in practice

Question

As I know if the impulse response is symmetric around sample zero phase response should be entirely zero.

The code below just set a rectangular window for vector "in"

const int N = 10;

// in = 0 , 0 , 0 , 0 , 1 , 1 , 1 , 0 , 0 , 0 , 0
std::vector< std::complex<double> > in (N);
std::vector< std::complex<double> > out (N);
std::vector< std::complex<double> > polarOut (N);

auto middleElem = in.begin() + in.size()/2;
std::fill( middleElem - 1, middleElem + 2, 1);

fftw_plan my_plan  = fftw_plan_dft_1d(N, reinterpret_cast<fftw_complex*>(&in[0]),
                                         reinterpret_cast<fftw_complex*>(&out[0]), FFTW_FORWARD, FFTW_ESTIMATE);
fftw_execute(my_plan);

std::transform( out.begin(), out.end(), polarOut.begin(),
                []( auto& in ){
                                return  std::complex<double>( std::abs(in), std::arg(in) );
                              } );

I getting back a sinc like result in magnitude, But I am so curious why I have values different than zero in phase :

    polarOut    <11 items>  std::vector<std::complex<double>>
        [0] (3.000000, 0.000000)    std::complex<double>
        [1] (2.682507, -2.855993)   std::complex<double>
        [2] (1.830830, 0.571199)    std::complex<double>
        [3] (0.715370, -2.284795)   std::complex<double>
        [4] (0.309721, -1.999195)   std::complex<double>
        [5] (0.918986, 1.427997)    std::complex<double>
        [6] (0.918986, -1.427997)   std::complex<double>
        [7] (0.309721, 1.999195)    std::complex<double>
        [8] (0.715370, 2.284795)    std::complex<double>
        [9] (1.830830, -0.571199)   std::complex<double>
        [10]    (2.682507, 2.855993)    std::complex<double>

Any explanation about the phase response of behaviour of this fft will be great.

Answer 1

The problem is that your input vector is not symmetric when considering the definition of the DFT:

$$X[k]=\sum_{n=0}^{N-1}x[n]e^{-j2\pi nk/N}\tag{1}$$

Note that from (1) the time indices start at $n=0$. If you periodically continue the signal $x[n]$ in your example you get

n: ... -5 -4 -3 -2 -1 | 0 | 1 2 3 4 5 6 7 8 9 10

x[n]: ... 1 0 0 0 0 | 0 | 0 0 0 1 1 1 0 0 0 0

If $x[n]$ were symmetric, the values for negative $n$ would need to be the same as the values for positive $n$. For $n=\pm 4$ this is not the case. If you defined your signal instead as

x[n] = 0 0 0 0 1 1 1 0 0 0

i.e. a signal of length $N=10$, then it would be symmetric and, consequently, its DFT (FFT) would be real-valued, apart from numerical errors.

Answer 2

Those are complex conjugates, which when you add give only , the real part. Intuitively exp(jwo)+exp(-jwo). so the resulting phase has the form invtan(imaginary_part/real_part) (invtan is tan inverse function), where b is o. so the phase spectrum will only have zeros.

That is the angles obtained, from a+jb is theta, and from that of a-jb is -theta. Both will sum up to give zero.