delaynumdelayden

Delaynumdelayden is not a standard term in most references. In wiki-style usage it can be described as the operation of applying separate time-delay operators to the numerator and the denominator of a rational transfer function, often to model or adjust delays in a system.

Concept and mathematical basis

For a discrete-time transfer function G(z) = N(z) / D(z), where N(z) and D(z) are polynomials in z^-1, delaying the numerator by m samples and the denominator by n samples yields G'(z) = z^-m N(z) / z^-n D(z). This simplifies to G'(z) = z^{-(m-n)} [N(z) / D(z)], meaning the overall effect is a net delay of m−n samples, while the magnitude response remains the same. A similar principle holds in continuous time: delaying the numerator by τN and the denominator by τD gives G'(s) = e^{-τN s} N(s) / e^{-τD s} D(s) = e^{-(τN−τD) s} [N(s)/D(s)], i.e., a net delay equal to the difference τN−τD.

Interpretation and use

The operation is mainly a theoretical or modeling tool: it helps to align dynamic elements or to represent transport delays in a way that preserves or isolates certain response characteristics. In practice, equal delays applied to both numerator and denominator cancel, leaving the original transfer function unchanged aside from a possible uniform phase factor; unequal delays yield a net delay. In software environments, the effect is often implemented by shifting coefficients, effectively introducing z^-k factors.

Relation to broader concepts

Delaynumdelayden relates to broader topics such as time-delay modeling, z-transform manipulations, and Padé approximations used to approximate delays in transfer functions. It is typically described in the context of discrete-time systems, but the underlying idea extends to continuous-time representations with exponential delay terms.

See also

Time delay, z-transform, discrete-time systems, continuous-time transfer functions, Padé approximation.