Transfer functions are commonly used in the analysis of systems such as single-input single-output filters in the fields of signal processing, communication theory, and control theory. The term is often used exclusively to refer to linear time-invariant (LTI) systems. Most real systems have non-linear input/output characteristics, but many systems, when operated within nominal parameters (not "over-driven") have behavior close enough to linear that LTI system theory is an acceptable representation of the input/output behavior.

Continuous-time edit

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 edit

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:

This is often referred to as the pulse-transfer function.^{[citation needed]}

Direct derivation from differential equations edit

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^{[clarification needed]} between complex vs. real values, which is traditionally influenced^{[clarification needed]} 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]

Gain, transient behavior and stability edit

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.

Let ${\displaystyle x(t)}$  be the input to a general linear time-invariant system, and ${\displaystyle y(t)}$  be the output, and the bilateral Laplace transform of ${\displaystyle x(t)}$  and ${\displaystyle y(t)}$  be

${\displaystyle {\begin{aligned}X(s)&={\mathcal {L}}\left\{x(t)\right\}\ {\stackrel {\mathrm {def} }{=}}\ \int _{-\infty }^{\infty }x(t)e^{-st}\,dt,\\Y(s)&={\mathcal {L}}\left\{y(t)\right\}\ {\stackrel {\mathrm {def} }{=}}\ \int _{-\infty }^{\infty }y(t)e^{-st}\,dt.\end{aligned}}}$ 

Then the output is related to the input by the transfer function ${\displaystyle H(s)}$  as

${\displaystyle Y(s)=H(s)X(s)}$ 

and the transfer function itself is therefore

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

In particular, if a complex harmonic signal with a sinusoidal component with amplitude ${\displaystyle |X|}$ , angular frequency ${\displaystyle \omega }$  and phase ${\displaystyle \arg(X)}$ , where arg is the argument

${\displaystyle x(t)=Xe^{j\omega t}=|X|e^{j(\omega t+\arg(X))}}$ 

where ${\displaystyle X=|X|e^{j\arg(X)}}$ 

is input to a linear time-invariant system, then the corresponding component in the output is:

${\displaystyle {\begin{aligned}y(t)&=Ye^{j\omega t}=|Y|e^{j(\omega t+\arg(Y))},\\Y&=|Y|e^{j\arg(Y)}.\end{aligned}}}$ 

Note that, in a linear time-invariant system, the input frequency ${\displaystyle \omega }$  has not changed, only the amplitude and the phase angle of the sinusoid has been changed by the system. The frequency response ${\displaystyle H(j\omega )}$  describes this change for every frequency ${\displaystyle \omega }$  in terms of gain:

${\displaystyle G(\omega )={\frac {|Y|}{|X|}}=|H(j\omega )|}$ 

and phase shift:

${\displaystyle \phi (\omega )=\arg(Y)-\arg(X)=\arg(H(j\omega )).}$ 

The phase delay (i.e., the frequency-dependent amount of delay introduced to the sinusoid by the transfer function) is:

${\displaystyle \tau _{\phi }(\omega )=-{\frac {\phi (\omega )}{\omega }}.}$ 

The group delay (i.e., the frequency-dependent amount of delay introduced to the envelope of the sinusoid by the transfer function) is found by computing the derivative of the phase shift with respect to angular frequency ${\displaystyle \omega }$ ,

${\displaystyle \tau _{g}(\omega )=-{\frac {d\phi (\omega )}{d\omega }}.}$ 

The transfer function can also be shown using the Fourier transform which is only a special case of the bilateral Laplace transform for the case where ${\displaystyle s=j\omega }$ .

Common transfer function families edit

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

In control engineering and control theory the transfer function is derived using the Laplace transform.

The transfer function was the primary tool used in classical control engineering. However, it has proven to be unwieldy for the analysis of multiple-input multiple-output (MIMO) systems, and has been largely supplanted by state space representations for such systems.^{[citation needed]} In spite of this, a transfer matrix can always be obtained for any linear system, in order to analyze its dynamics and other properties: each element of a transfer matrix is a transfer function relating a particular input variable to an output variable.

A useful representation bridging state space and transfer function methods was proposed by Howard H. Rosenbrock and is referred to as Rosenbrock system matrix.

In optics, modulation transfer function indicates the capability of optical contrast transmission.

For example, when observing a series of black-white-light fringes drawn with a specific spatial frequency, the image quality may decay. White fringes fade while black ones turn brighter.

The modulation transfer function in a specific spatial frequency is defined by

${\displaystyle \mathrm {MTF} (f)={\frac {M(\mathrm {image} )}{M(\mathrm {source} )}},}$ 

where modulation (M) is computed from the following image or light brightness:

${\displaystyle M={\frac {L_{\max }-L_{\min }}{L_{\max }+L_{\min }}}.}$ 

In imaging, transfer functions are used to describe the relationship between the scene light, the image signal and the displayed light.

Transfer functions do not properly exist for many non-linear systems. For example, they do not exist for relaxation oscillators;^[6] however, describing functions can sometimes be used to approximate such nonlinear time-invariant systems.

• ECE 209: Review of Circuits as LTI Systems — Short primer on the mathematical analysis of (electrical) LTI systems.