# Digital Signal Processing/Analog Mappings

There are a number of different mappings from the analog domain to the digital domain, and vice versa. All mappings must satisfy the following rules:

- Stability must be preserved in both domains. This means that the interior of the unit circle in the digital domain must correspond to the left-half plane in the analog plane.
- The mapping must be one-to-one. every point in one domain must correspond to a unique point in the other domain. This relationship should hold both ways.

The relationship between the analog and digital domains is highly nonlinear, and there is no way to perfectly reproduce an analog filter in a digital system, or to reproduce a digital filter in an analog system. However, it is possible to faithfully reproduce (within a certain tolerance) the magnitude response of the filter. It is rarely possible to reproduce the phase response of a filter across the transformation, however.