Why the ifft of a constant signal does not give a delta function with zero DC?

Hi.

Suppose the frequency response is flat (that means a vector of a single constant which I named it 'AAA' or 'BBB' in the code). I am trying to take the ifft of this signal and convert it to the time domain, let's name it 'aaa' or 'bbb' in time domain. In theory I should get a delta function if the constant in frequency domain is from -Inf to +Inf. However the frequency domain signal is limited number of samples . Because of that, when I get the ifft of the constant signal, I see a sinc shape in the time domain instead of a delta function, which is as expected. However, this sinc function does not have a 0 DC value. By that, I mean if you look at the tails of the sinc function, the average is not 0 (it seems they are a bit shifted up and not centered at 0). I know I can control this effect by manipulating the DC component (that is the first sample) in the frequency domain, however that's not a right way to do it. I wanted to know the reason behind this effect and a right way to generate the ifft of a constant signal such that the tails of a generated sinc function in time domain are centered at 0. Any suggestion, idea, or help is really appreciated.

Thanks, ss

    AAA = 2*[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ]; % or 2*ones(1,8)
    BBB = 1*[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]; % or ones(1,16)

    Nfft = 2^14;
    aaa = Nfft * ifft(AAA, Nfft);
    bbb = Nfft * ifft(BBB, Nfft);
    aaa = fftshift(aaa);
    bbb = fftshift(bbb);

    mean(real(aaa(1:6000))) %taking average of the tail.
    mean(real(bbb(1:6000)))

    figure(2)
    plot(real(aaa), 'b')
    hold on
    plot(real(bbb), 'r')
    plot([0,Nfft],[0,0], 'k') % draw a line at 0