Bandlimited Impulse Train (BLIT) Synthesis

• The ideal anti-aliasing filter, applied to a continuous-time signal before sampling, consists of a rectangular window applied in the frequency domain with a cutoff at half the sample rate.

• The inverse Fourier transform of this filter is a sinc function with a zero-crossing interval of one sample:
  $$h_s(t) = \mbox{sinc}(F_st) \ensuremath{\stackrel{\Delta}{=}}\frac{sin(\pi F_s t)}{\pi F_s t}.$$

• The ideal unit-amplitude impulse train with period $T_1$ seconds is given by
  $$x(t) = \sum_{l=-\infty}^{\infty} \delta(t + lT_1).$$

• In order to bandlimit the ideal impulse train, we can apply the ideal anti-aliasing filter $h_s$ to it as:
  $$\begin{eqnarray*} x_f(t) \ensuremath{\stackrel{\Delta}{=}}(x \ast h_s)(t) &=& \s... ... lT_1) \\ &=& \sum_{l=-\infty}^{\infty} \mbox{sinc}(t/T_s + lP) \end{eqnarray*}$$

  where $P = T_1 / T_s$ is the period in samples.

• Since $x_f(t)$ is now bandlimited, it can be sampled without aliasing as
  $$y[n] \ensuremath{\stackrel{\Delta}{=}}x_f(nT_s) = \sum_{l=-\infty}^{\infty} \mbox{sinc}(n + lP).$$

• This infinite summation can be reduced to an expression of the form (Stilson and Smith, 1996)
    $$y[n] = (M/P) \mbox{Sinc}_M( M n / P )$$ (2)
  where the digital sinc function is given by
  $$\mbox{Sinc}_M(x) \ensuremath{\stackrel{\Delta}{=}}\frac{\sin(\pi x)}{M \sin(\pi x / M)}$$

• $M$ is the number of harmonics and is always odd. It cannot exceed the period $P$ in samples.

• Equation (2) is a closed-form expression that can be used to generate a bandlimited impulse train in a manner similar to the use of DSFs.

  Figure 10: The time and frequency magnitude response of an impulse train waveform (440 Hz fundamental) created using the BLIT sinc function of Stilson & Smith.

  

• Note that similar numerical instabilities will arise with this expression as was found with the DSFs. L'Hopital's rule can be used to evaluate the limit of the $\mbox{Sinc}_M$ function as its denominator approaches a value of zero, resulting in
  $$\mbox{Sinc}_M(x) = \frac{\cos(\pi x)}{\cos(\pi x / M)},$$
  which converges to a value of 1.0 for $x$ = 0.