A channel whose frequency response varies across the signal band can distort pulses and create intersymbol interference. An equalizer applies another filter intended to compensate for that distortion.

If the channel response is $H(f)$, an ideal zero-forcing idea would choose

$$G(f)=\frac{1}{H(f)}$$

so that

$$G(f)H(f)=1.$$

This works poorly where $|H(f)|$ is very small: the equalizer then requires very large gain and can strongly amplify noise.

Practical equalization therefore balances distortion removal against noise enhancement rather than blindly inverting the channel.

Equalizers can be implemented in continuous or discrete time and may be fixed from a known channel model or adapted from observed data.

The essential role is the same: pulse shaping controls the waveform intentionally created by the transmitter, while equalization compensates for unwanted linear distortion introduced after transmission.