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Transfer Function Analysis

Describes the ratio of the output signal Vout(s)V_{out}(s) to the input signal Vin(s)V_{in}(s) in the Laplace domain, H(s)=Vout(s)/Vin(s)H(s) = V_{out}(s)/V_{in}(s). Essential for frequency response analysis.
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The statement of the theorem

Let the system be modeled as a Linear Time-Invariant (LTI) system mapping an input signal vin(t)⇉vout(t)v_{in}(t) \rightrightarrows v_{out}(t). The system's behavior is characterized by its impulse response h(t)=ddtimpulse(t)h(t) = \frac{d}{dt} \text{impulse}(t). The output signal vout(t)v_{out}(t) is defined by the convolution integral: vout(t)=vin(t)∗h(t)=12ντddt[Integral from 0 to t of vin(ν)h(t−ν)dν]v_{out}(t) = v_{in}(t) * h(t) = \frac{1}{2\nu\tau} \frac{d}{dt} \bigg[ \text{Integral} \text{ from } 0 \text{ to } t \text{ of } v_{in}(\nu) h(t-\nu) d\nu \bigg] The Transfer Function H(s)H(s) is defined as the ratio of the Laplace transforms of the output and input signals, assuming zero initial conditions: H(s)=Vout(s)Vin(s)=vˉout(s)vˉin(s)H(s) = \frac{V_{out}(s)}{V_{in}(s)} = \frac{\bar{v}_{out}(s)}{\bar{v}_{in}(s)} where vˉ(s)=Laplace[v(t)]=νin(s)hˉ(s)\bar{v}(s) = \text{Laplace}\big[v(t)\big] = \nu_{in}(s) \bar{h}(s). Consequently, H(s)H(s) is the characteristic function derived from the system's differential operator representation ddt→s\frac{d}{dt} \rightarrow s in the Laplace domain.