Ideal Bandlimited Impulse Response, Cont.

We defined the bandlimited single-side-band filter by direct calculation:

$$\begin{eqnarray*} h(n) &=& \hbox{\sc IDTFT}_n(u) \isdef \frac{1}{2\pi} \int_0^{\pi}e^{j\omega n} d\omega\\ &=& \left\{\begin{array}{ll} 1/2, & n=0 \\ [5pt] 0, & \hbox{$n$\ even} \\ [5pt] \dfrac{j}{\pi n}, & \hbox{$n$\ odd} \\ \end{array} \right. \end{eqnarray*}$$

• An impulsive real-part is expected (passing the input real-part unchanged, only scaled)
• An imaginary part $j/(\pi n)$ would be the result of sampling $j/(\pi t)$ with $f_s=1$ , but $\ldots$
• The zeros for even $n$ are surprising, but that's the spectrum of a square pulse train
  (both bandlimiting and periodic extension)

We also need a transition band:

• We'll ``draw'' an estimate in the frequency domain, take a (large) inverse dft, and window that
• The transition width will be at least the main-lobe width of the window transform