**4. Applications in system identification**

Hilbert transform relates the real part and the imaginary part of the transfer function of any *physically viable* linear time-invariant system. By "physical viability" we mean a system should be stable and causal. Stability requires the systems to produce bounded output if the input is bounded [so called the bounded-input bounded-output criterion, (Oppenheim & Schafer, 2010)]. Causality prohibits the system from producing responses before any stimulus comes in. Denote the impulse response as *h*(*t*) and its Laplace transform as *H*(*s*). The above conditions requires that


The two conditions above ensure that *H*(*s*) converges and is analytic in the entire right half-plane, and in particular on the imaginary axis *s* = *jω*. Therefore, the real and imaginary part of *H*(*jω*) = *HR*(*ω*) + *jHI*(*ω*) are inter-dependent in term of the *Kramers-Krönig relations* (King, 2009b):

$$H\_I(\omega) = \frac{1}{\pi} \int\_{-\infty}^{\infty} \frac{H\_R(u) du}{\omega - u} \tag{17}$$

$$H\_R(\omega) = -\frac{1}{\pi} \int\_{-\infty}^{\infty} \frac{H\_I(u) du}{\omega - u} \tag{18}$$

which is basically Hilbert transform in its time-frequency dual form.

The Kramers-Krönig relations govern how physical viable transfer functions can vary in frequency. For instance, the real and imaginary part of an electromagnetic wave propagation function defines the attenuation and the wavenumber per unit length as a function of frequency. The frequency dependence of both functions is referred to as *dispersion*, for in optics it describes how red light travels faster than violet light in water — so we behold the beauty of rainbows after raining. It is now intriguing to realize that the attenuation and velocity of light are two inter-dependent functions of frequency. More physics-inclined readers can refer to King's thorough discussion on dispersion relations in electrodynamics and optics (King, 2009b).

To a certain extent, the concept that the real and the imaginary parts are inter-dependent similarly applies to the magnitude and phase of transfer functions of a physically viable system. Note that any transfer function *H*(*jω*) can be decomposed logarithmically into magnitude and phase:

$$
\log H(j\omega) = \log|H(j\omega)| + j\angle H(j\omega).
$$

This shows that the log-magnitude and the phase are real and imaginary parts of the log-spectrum, respectively. It might appear that they must satisfy the Kramers-Krönig relations. Unfortunately, this is a wishful thinking since apparently *H*˜ (*jω*) = exp(−*jωτ*)*H*(*jω*), where *τ* is a constant, would have the same magnitude as *H*(*jω*) but a different phase response.6

It turns out that, for any given magnitude response, the uniqueness of phase response can be established if the transfer function satisfies a *minimum-phase* criterion; the criterion requires

<sup>6</sup> in fact, ˜ *h*(*t*) = *h*(*t* − *τ*).

8 Will-be-set-by-IN-TECH

Hilbert transform relates the real part and the imaginary part of the transfer function of any *physically viable* linear time-invariant system. By "physical viability" we mean a system should be stable and causal. Stability requires the systems to produce bounded output if the input is bounded [so called the bounded-input bounded-output criterion, (Oppenheim & Schafer, 2010)]. Causality prohibits the system from producing responses before any stimulus comes in. Denote the impulse response as *h*(*t*) and its Laplace transform as *H*(*s*). The above

The two conditions above ensure that *H*(*s*) converges and is analytic in the entire right half-plane, and in particular on the imaginary axis *s* = *jω*. Therefore, the real and imaginary part of *H*(*jω*) = *HR*(*ω*) + *jHI*(*ω*) are inter-dependent in term of the *Kramers-Krönig relations*

> ∞ −∞

 ∞ −∞

*HR*(*u*)*du*

*HI*(*u*)*du*

*<sup>ω</sup>* <sup>−</sup> *<sup>u</sup>* (17)

*<sup>ω</sup>* <sup>−</sup> *<sup>u</sup>* (18)

• All sigularities of *H*(*s*) are located in the left half-plane (stability).

*HI*(*ω*) = <sup>1</sup>

*HR*(*ω*) = <sup>−</sup> <sup>1</sup>

which is basically Hilbert transform in its time-frequency dual form.

*π*

*π*

The Kramers-Krönig relations govern how physical viable transfer functions can vary in frequency. For instance, the real and imaginary part of an electromagnetic wave propagation function defines the attenuation and the wavenumber per unit length as a function of frequency. The frequency dependence of both functions is referred to as *dispersion*, for in optics it describes how red light travels faster than violet light in water — so we behold the beauty of rainbows after raining. It is now intriguing to realize that the attenuation and velocity of light are two inter-dependent functions of frequency. More physics-inclined readers can refer to King's thorough discussion on dispersion relations in electrodynamics and optics (King,

To a certain extent, the concept that the real and the imaginary parts are inter-dependent similarly applies to the magnitude and phase of transfer functions of a physically viable system. Note that any transfer function *H*(*jω*) can be decomposed logarithmically into

log *H*(*jω*) = log |*H*(*jω*)| + *j*∠*H*(*jω*). This shows that the log-magnitude and the phase are real and imaginary parts of the log-spectrum, respectively. It might appear that they must satisfy the Kramers-Krönig relations. Unfortunately, this is a wishful thinking since apparently *H*˜ (*jω*) = exp(−*jωτ*)*H*(*jω*), where *τ* is a constant, would have the same magnitude as *H*(*jω*) but a

It turns out that, for any given magnitude response, the uniqueness of phase response can be established if the transfer function satisfies a *minimum-phase* criterion; the criterion requires

**4. Applications in system identification**

conditions requires that

(King, 2009b):

2009b).

<sup>6</sup> in fact, ˜

magnitude and phase:

different phase response.6

*h*(*t*) = *h*(*t* − *τ*).

• *h*(*t*) = 0 for all *t* < 0 (causality)

that all zeros and poles of the transfer function *H*(*s*) to be located in the left-half plane. This criterion ensures that all the singularities of log *H*(*s*) are located in the left-half plane so the real and imaginary parts of log *H*(*s*) become a Hilbert transform pair. Otherwise, any transfer function can be uniquely factorized as a product of a minimum-phase function *M*(*jω*) and an all-pass function *P*(*jω*). It is noteworthy that the system whose transfer function is *M*(*jω*) has the minimal energy delay among all linear time-invariant systems of the same magnitude response. Further readings are recommended in Chapter 5 and 12 of Oppenheim & Schafer (2010).
