Why spectral transformation is required

Spectral (frequency) transformation converts a filter of one type or cut-off into another. It is required because:

• One prototype, many filters. Design theory (Butterworth, Chebyshev) is worked out for a low-pass prototype with a normalised cut-off. Transformations turn this into a low-pass filter with another cut-off, or a high-pass, band-pass or band-stop filter, so no separate design procedure is needed for each type.
• Saves design effort. An existing, well-tested low-pass design can be reused by changing the cut-off or type.
• Simple, systematic method. In the analog domain we replace s; in the digital domain we replace z⁻¹ by an all-pass function, for example z⁻¹ → −(z⁻¹ + α)/(1 + αz⁻¹) for low-pass to high-pass.
• Needed with impulse invariance. High-pass and band-stop analog filters are not band-limited, so impulse invariance would alias them badly. Instead, a digital low-pass filter is designed first and then converted to the required type by a digital-domain transformation.