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Transfer Function

Definition: Transfer function is the ratio of the $ y=y(t) output response function and the $ f=f(t) input function in the extended complex frequency domain (i.e. their Laplace transformation) for any linear shift invariant system
$ f(t)=\begin{cases} & \varphi(t), \quad \text{for} \quad 0\leq t \\ & 0, \quad \text{for} \quad t<0 \end{cases} \quad \text{and} \quad \int_{-\infty}^{\infty} |f(t)|dt<\infty
Transfer function $ w(s)=\frac{\mathcal{L} \left \{ y(t) \right \}}{\mathcal{L} \left \{ f(t) \right \}} =\frac{Y(s)}{F(s)} (let’s see Figure 31.)

Image
Figure 31.

 
Consequence of definition:
If the transfer function of a linear shift invariant system is known, then the output response can be determined by any input functions as follow:

$ w(s) =\frac{Y(s)}{F(s)}

$ Y(s)=w(s)F(s)

$ y(t)=\mathcal{L}^{-1} \left \{ w(s)F(s) \right \}=\mathcal{L}^{-1} \left \{ w(s) \mathcal{L}^{-1} \left \{ f(t) \right \} \right \}

Based on the previously described principle the transfer function of a linear shift invariant system can be expressed by polynomial rational function:
$ w(s)=\frac{b_0+b_1s+b_2s^2+ \tex{...} +b_ms^m}{ a_0+a_1s+a_2s^2+\tex{...}+a_ns^n }=\frac{\sum_{i=1}^mb_is^i}{\sum_{k=1}^na_ks^k }=\frac{P(s)}{N(s)};\quad m<n \quad .

Let’s write down both the numerator and the denominator into fully-factored form:
$ P(s)=B\prod_{i=1}^m(s-z_i) ; where $ z_i is the root of numerator, i.e the zero of $w(s).
$ N(s)=A\prod_{k=1}^n(s-p_k) ; where $ p_k is the root of the denominator, i.e. pole of $w(s).

$ w(s)=\frac{B}{A}\frac{\prod_{i=1}^m(s-z_i) }{\prod_{k=1}^n(s-p_k)}=H \frac{\prod_{i=1}^m(s-z_i) }{\prod_{k=1}^n(s-p_k)} , as the Figure 32. shows, the transfer function can be presented on the complex plane by the fully-factored form expression of transfer function.

Image
Figure 32.

 
Next problem is to determine the transfer function in the real parameter space (domain).

Let’s execute the inverse Laplace transformation with the following conditions: $ \forall p_k poles should be single order pole.

$ w(t)=\mathcal{L}^{-1} \left \{ w(s) \right \}=\mathcal{L}^{-1} \left \{ H \frac{\prod_{i=1}^m(s-z_i) }{\prod_{k=1}^n(s-p_k)} \right \}=H_0\delta(t)+\sum_{k=1}^nH_ke^{p_kt}

In case of linear invariant systems the poles and zeros can be described as follow: $ p_k=\sigma_k or $ p_{l,l+1} = \sigma_{l} \pm j\omega_l , i.e. $ p_{l} = \sigma_{l} + j\omega_l and $ p_{l+1} = \sigma_{l} - j\omega_l are created complex conjugate pairs. Similar things can be written for the zeros.

With help of Euler’s relation $w(t) can be written as follow:

$ w(t)= H_0\delta(t)+\sum_{i}D_ie^{\sigma_{i} t} \text{cos}(\omega_it+\varphi_i).

The following important definitions and theorem can be written for the transfer function by the pole-zero arrangement of a linear invariant system.

Minimum phase system:
Definition: If all the zeros of transfer function of a linear invariant system are on the left side of the complete complex domain, i.e. $ \forall\text{Re}(z_k)<0 is called minimal phase system.

All pass system:
Definition: If the transfer function of the linear invariant system satisfies the following statement: if all poles are located on the left side of the complete complex domain i.e. $ \forall\text{Re}(p_k)<0 as well as all the zeros are on the right side of the complete complex domain i.e. $ \forall\text{Re}(z_k)>0 , and all the poles and zeros are each other reflected images by the imaginary axis (i.e. $ \forall p_k = -\forall (z_k)^* ) then it is called all pass system.

Theorem:
Transfer function of a linear invariant system can be separated into a minimal phase and all pass transfer function.

 


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