The continuum requires a loss
The continuum requires a loss
When a digital-to-analog converter outputs a signal, it does not produce a smooth, continuous wave. It outputs a rigid, jagged staircase of discrete voltage levels, holding each step flat until the next clock tick.
If you look at the frequency spectrum of this staircase, you do not just see the intended signal. You see infinite, high-frequency spectral images—ghostly replicas of the baseband signal echoing endlessly upward into the ultrasonic range. The sharp edges of the staircase are constructed entirely out of these high-frequency echoes.
To recover the true, smooth analog wave, the system must pass the staircase through a reconstruction filter. This low-pass analog filter acts as a guillotine, violently slicing away every frequency above the Nyquist limit.
By destroying the high-frequency replicas, the filter destroys the sharp edges of the steps. The rigid corners melt away, and the discrete jumps blur back into a continuous, flowing curve.
The smoothness of the final analog wave is not something the filter adds. The smoothness is what remains after the high-frequency violence has been removed. The continuum is achieved through an act of deletion.