Transfer function

The descriptions below are given in terms of a complex variable, ${\displaystyle s=\sigma +j\cdot \omega }$, which bears a brief explanation. In many applications, it is sufficient to set ${\displaystyle \sigma =0}$ (thus ${\displaystyle s=j\cdot \omega }$), which reduces the Laplace transforms with complex arguments to Fourier transforms with real argument ω. This is common in applications primarily interested in the LTI system's steady-state response (often the case in signal processing and communication theory), not the fleeting turn-on and turn-off transient response or stability issues.

For continuous-time input signal ${\displaystyle x(t)}$ and output ${\displaystyle y(t)}$, dividing the Laplace transform of the output, ${\displaystyle Y(s)={\mathcal {L}}\left\{y(t)\right\}}$, by the Laplace transform of the input, ${\displaystyle X(s)={\mathcal {L}}\left\{x(t)\right\}}$, yields the system's transfer function ${\displaystyle H(s)}$:

${\displaystyle H(s)={\frac {Y(s)}{X(s)}}={\frac {{\mathcal {L}}\left\{y(t)\right\}}{{\mathcal {L}}\left\{x(t)\right\}}}}$

which can be rearranged as:

${\displaystyle Y(s)=H(s)\;X(s)\,.}$

Discrete-time signals may be notated as arrays indexed by an integer ${\displaystyle n}$ (e.g. ${\displaystyle x[n]}$ for input and ${\displaystyle y[n]}$ for output). Instead of using the Laplace transform (which is better-suited for continuous-time signals), discrete-time signals are dealt with using the z-transform (notated using a corresponding capital letter like ${\displaystyle X(z)}$ and ${\displaystyle Y(z)}$), so a discrete-time system's transfer function can be written as:

$${\displaystyle H(z)={\frac {Y(z)}{X(z)}}={\frac {{\mathcal {Z}}\{y[n]\}}{{\mathcal {Z}}\{x[n]\}}}.}$$

This is often referred to as the pulse-transfer function.

Consider a linear differential equation with constant coefficients

${\displaystyle L[u]={\frac {d^{n}u}{dt^{n}}}+a_{1}{\frac {d^{n-1}u}{dt^{n-1}}}+\dotsb +a_{n-1}{\frac {du}{dt}}+a_{n}u=r(t)}$

where u and r are suitably smooth functions of t, and L is the operator defined on the relevant function space, that transforms u into r. That kind of equation can be used to constrain the output function u in terms of the forcing function r. The transfer function can be used to define an operator ${\displaystyle F[r]=u}$ that serves as a right inverse of L, meaning that ${\displaystyle L[F[r]]=r}$.

Solutions of the homogeneous, constant-coefficient differential equation ${\displaystyle L[u]=0}$ can be found by trying ${\displaystyle u=e^{\lambda t}}$. That substitution yields the characteristic polynomial

${\displaystyle p_{L}(\lambda )=\lambda ^{n}+a_{1}\lambda ^{n-1}+\dotsb +a_{n-1}\lambda +a_{n}\,}$

The inhomogeneous case can be easily solved if the input function r is also of the form ${\displaystyle r(t)=e^{st}}$. In that case, by substituting ${\displaystyle u=H(s)e^{st}}$ one finds that ${\displaystyle L[H(s)e^{st}]=e^{st}}$ if we define

${\displaystyle H(s)={\frac {1}{p_{L}(s)}}\qquad {\text{wherever }}\quad p_{L}(s)\neq 0.}$

Taking that as the definition of the transfer function requires careful disambiguation between complex vs. real values, which is traditionally influenced by the interpretation of abs(H(s)) as the gain and −atan(H(s)) as the phase lag. Other definitions of the transfer function are used: for example ${\displaystyle 1/p_{L}(ik).}$[5]

A general sinusoidal input to a system of frequency ${\displaystyle \omega _{0}/(2\pi )}$ may be written ${\displaystyle \exp(j\omega _{0}t)}$. The response of a system to a sinusoidal input beginning at time ${\displaystyle t=0}$ will consist of the sum of the steady-state response and a transient response. The steady-state response is the output of the system in the limit of infinite time, and the transient response is the difference between the response and the steady state response (it corresponds to the homogeneous solution of the above differential equation). The transfer function for an LTI system may be written as the product:

${\displaystyle H(s)=\prod _{i=1}^{N}{\frac {1}{s-s_{P_{i}}}}}$

where s_Pi are the N roots of the characteristic polynomial and will therefore be the poles of the transfer function. Consider the case of a transfer function with a single pole ${\displaystyle H(s)={\frac {1}{s-s_{P}}}}$ where ${\displaystyle s_{P}=\sigma _{P}+j\omega _{P}}$. The Laplace transform of a general sinusoid of unit amplitude will be ${\displaystyle {\frac {1}{s-j\omega _{i}}}}$. The Laplace transform of the output will be ${\displaystyle {\frac {H(s)}{s-j\omega _{0}}}}$ and the temporal output will be the inverse Laplace transform of that function:

${\displaystyle g(t)={\frac {e^{j\,\omega _{0}\,t}-e^{(\sigma _{P}+j\,\omega _{P})t}}{-\sigma _{P}+j(\omega _{0}-\omega _{P})}}}$

The second term in the numerator is the transient response, and in the limit of infinite time it will diverge to infinity if σ_P is positive. In order for a system to be stable, its transfer function must have no poles whose real parts are positive. If the transfer function is strictly stable, the real parts of all poles will be negative, and the transient behavior will tend to zero in the limit of infinite time. The steady-state output will be:

${\displaystyle g(\infty )={\frac {e^{j\,\omega _{0}\,t}}{-\sigma _{P}+j(\omega _{0}-\omega _{P})}}}$

The frequency response (or "gain") G of the system is defined as the absolute value of the ratio of the output amplitude to the steady-state input amplitude:

${\displaystyle G(\omega _{i})=\left|{\frac {1}{-\sigma _{P}+j(\omega _{0}-\omega _{P})}}\right|={\frac {1}{\sqrt {\sigma _{P}^{2}+(\omega _{P}-\omega _{0})^{2}}}},}$

which is just the absolute value of the transfer function ${\displaystyle H(s)}$ evaluated at ${\displaystyle j\omega _{i}}$. This result can be shown to be valid for any number of transfer function poles.

While any LTI system can be described by some transfer function or another, there are certain "families" of special transfer functions that are commonly used.

Some common transfer function families and their particular characteristics are:

• Butterworth filter – maximally flat in passband and stopband for the given order
• Chebyshev filter (Type I) – maximally flat in stopband, sharper cutoff than a Butterworth filter of the same order
• Chebyshev filter (Type II) – maximally flat in passband, sharper cutoff than a Butterworth filter of the same order
• Bessel filter – maximally constant group delay for a given order
• Elliptic filter – sharpest cutoff (narrowest transition between pass band and stop band) for the given order
• Optimum "L" filter
• Gaussian filter – minimum group delay; gives no overshoot to a step function
• Hourglass filter
• Raised-cosine filter