A Uniformly Convergent Approximation of Complex Half-Band Filters

Gagan Mirchandani and Mohamed Elfataoui

In contrast to the well known and classic frequency domain method for generating a discrete-time analytic signal, we show that the same signal can also be generated using a real-time complex half-band FIR filter. More significantly, we show that the spectrum of the $N$ length filter ($N$ a multiple of 4) converges uniformly to the ideal complex half-band spectrum as $N \rightarrow \infty$, except away from the discontinuities at $0$ and $\pi$, where it converges pointwise. The filter design, in contrast to some other methods, is easily scalable and stable. Furthermore, we derive the closed form expression of the filter frequency response. For evaluating filter performance, we focus on the spacial shiftability property of the filter and compare it with that of other filter designs. Using a total variation measure for determining function variation, we see that shiftability is excellent for an impulse input and better on average, with other inputs.