What is a half-band filter and why is it efficient?

Half-band filter is a lowpass FIR filter with cutoff at fs/4 (half of Nyquist frequency). Special property: every other coefficient (except center) is zero, reducing computation by nearly half. Design: For symmetric FIR with odd length, impulse response h[n] = 0 for n = even (except n=0). Results from frequency response symmetry: H(e^jw) + H(e^j(pi-w)) = 1. Applications: Efficient 2x decimation/interpolation, cascade for higher rate changes, and quadrature mirror filter banks. Half-band filters are fundamental building blocks in multirate systems.