# Frequency Domain Representations

• W. A. Wolovich
Part of the Applied Mathematical Sciences book series (AMS, volume 11)

## Abstract

The reader is undoubtedly familiar with the well known frequency domain methods such as the Nyquist and Bode diagrams, the Nichols chart, and the root locus for analyzing the dynamical behavior of scalar systems. These methods are dependent only on the external (input/output) frequency response characteristics of a system, or equivalently, on the “transfer function” T(s) of a system which is usually expressible as the ratio of two polynomials r(s) and p(s) in the Laplace operator s with real coefficients; i.e. in the scalar case, the transfer function can usually be expressed as:
$${\text{T}}\left( {\text{s}} \right) = \frac{{{\text{r}}\left( {\text{s}} \right)}}{{{\text{p}}\left( {\text{s}} \right)}}$$
(4.1.1)
where the zeros of r(s) and p(s) represent, respectively, the zeros and poles of the system. Frequency domain methods are so named because an evaluation of T(s) at s = jω for any positive real value of ω, yields a complex number T(jω) = α(ω) + j β (ω), the magnitude of which $$\sqrt {{{\alpha ^{2}}\left( w \right) + {\beta ^{2}}\left( w \right)}}$$ represents the ratio of the output to the input amplitude response in steady-state due to a sinusoidal input signal of frequency ω, while $$\theta \; = \;{\text{ta}}{{\text{n}}^{{ - 1}}}\;\frac{{\beta (\omega )}}{{\alpha (\omega )}}$$ represents the difference in phase between the two waveforms.

## Keywords

Transfer Matrix Polynomial Matrix Structure Theorem Equivalence Transformation Companion Form
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