User:DDadyka/Leaky integrator

A graph of a solution to a leaky integrator; the input changes at T=5.

In mathematics, a leaky integrator equation is a specific differential equation, used to describe a component or system that takes the integral of an input, but gradually leaks a small amount of input over time. It appears commonly in hydraulics, electronics, and neuroscience where it can represent either a single neuron or a local population of neurons.

Equation

The equation is of the form

${\frac {dx}{dt}}=-Ax+u(t)$

where $u(t)$ is the input and $A$ is the rate of the 'leak' which can be considered as the integrator inverse time constant $\tau =1/A$.

General solution

Output signal of an ideal and leaky integrator given a constant input signal.

The equation is a nonhomogeneous first-order linear differential equation. For constant $u(t)=C$ its solution is

$x(t)=ke^{-At}+{\frac {C}{A}}$

where $k$ is a constant encoding the initial condition. For zero initial condition $x(0)=0$

$k=-{\frac {C}{A}},x(t)={\frac {C}{A}}(1-e^{-At})$.

For a time much shorter than the filter time constant $t\ll \tau$ the linear expansion of the exponential function can be used: $e^{-At}\approx 1-At$ so $x(t)\approx Ct$ which corresponds to the integration of a constant input signal $C$ over time $t$.

For a time significantly greater than time constant $t\gg \tau$ the value of the output signal approaches a constant value. Equating the derivative to zero ${\frac {dx}{dt}}=0$ simply yields $x_{\infty }=C/A$.

Frequency-domain considerations

Comparison of the frequency characteristics of an ideal integrator and a leaky integrator.

The frequency-domain characteristics of a leaky integrator can be determined from its equation. Let the input signal be a complex harmonic signal $u(t)=e^{j\omega t}$, the desired solution $x$ can be written as $x=H(\omega )e^{j\omega t}$, where $H(\omega )$ - the complex transfer function. Then, by substitution

$j\omega H(\omega )e^{j\omega t}=-AH(\omega )e^{j\omega t}+e^{j\omega t},$

therefore the complex transfer function is

$H(\omega )={\frac {1}{A+j\omega }}$.

This transfer function corresponds(up to normalization) to a first order low-pass filter with a cutoff frequency $\omega _{0}=A$ For frequencies significantly exceeding cutoff frequency $\omega \gg A$ the frequency response of an leaky integrator becomes close the frequency response of an ideal integrator

$H(\omega )\to {\frac {1}{j\omega }}$

For frequencies significantly lower than the cutoff frequency $\omega \ll A$ the transfer function becomes independent of frequency and the signal is transmitted through the integrator scaled by $1/A$ and with its spectrum unchanged.

$H(\omega )\to {\frac {1}{A}}$

Discrete-time realization

Replacing the derivative with a finite difference ${\frac {dx}{dt}}={\frac {x_{t}-x_{t-1}}{\Delta t}}$ and solving for $x_{t}$ gives recursive formula $x_{t}={\frac {1}{1+A\Delta t}}x_{t-1}+{\frac {A\Delta t}{1+A\Delta t}}u_{t}.$

Defining $\alpha ={\frac {\Delta t}{\tau +\Delta t}}$ the recursion becomes $x_{t}=(1-\alpha )x_{t-1}+\alpha u_{t}/A$ which is the standard recursive form of an exponential moving average.

Examples

Many physical, biological , and technical systems are described by a model in which the accumulation of a certain quantity is counteracted by losses that depend on the system's current state.

Integrating RC circuit

An integrating RC circuit is a good example of a leaky integrator. Assume the voltage $V_{C}$ across the capacitor at the initial time moment is zero. Until the voltage on the capacitor has time to change significantly, the current through the resistor is $I\approx V_{in}/R$ so

${\frac {dV_{C}}{dt}}={\frac {V_{in}}{RC}}$

$V_{C}\approx {\frac {1}{RC}}\int _{0}^{t}V_{\mathrm {in} }\,dt$,

the voltage across the capacitor acts approximately like an integrator of the input voltage. But for a slowly changing input voltage (relative to the RC time constant) the output voltage has time to change - the difference between the input and output voltages decreases which reduces the current $I=(V_{in}-V_{c})/R$, where $-V_{c}/R$ term can be formally considered as a leakage current $I_{leak}$ (the capacitor self-discharg through a resistor R and the input circuit).

See also

• RC circuit
• Low-pass filter
• Exponential smoothing
• Integrator
• Op amp integrator