A light-emitting diode (LED) luminaire adopts LED light sources (LEDs) and a corrugated light exit window. Light rays emitting at different angles from the LEDs launch onto the corrugated light exit window with numerous light images of the LEDs. The light images are so heavily overlapped that smooth out hot spots with dark spots, an effect of averaging, creating a uniform illumination output. The LED luminaire is capable of averaging white light emissions from a plurality of LEDs, mixing light emissions from various white LEDs at different correlated color temperatures (CCTs), or mixing light emissions from various white LEDs at a specific CCT with emissions from LEDs having saturated colors, rendering a uniform illumination with a consistent intensity or color hue within viewing angles.