# Is a leaky integrator the same thing as a low pass filter?

The equation governing a leaky integrator (according to Wikipedia at least) is

$\frac{d\mathcal{O}}{dt} + A\mathcal{O}(t) = \mathcal{I}(t)$.

Is a continuous-time leaky integrator thus the same thing as a low pass filter with time-constant $A$, up to some scaling of the input?

• Yes, but be sure to check the definition of time constant. – Dilip Sarwate Aug 22 '12 at 21:52

A so-called leaky integrator is a first-order filter with feedback. Let's find its transfer function, assuming that the input is $x(t)$ and the output $y(t)$:

$$\frac{dy(t)}{dt} + Ay(t) = x(t)$$

$$\mathcal{L}\left\{\frac{dy(t)}{dt} + Ay(t)\right\} = \mathcal{L}\left\{x(t)\right\}$$

where $\mathcal{L}$ denotes application of the Laplace transform. Moving forward:

$$sY(s) + AY(s) = X(s)$$

$$H(s) = \frac{Y(s)}{X(s)} = \frac{1}{s + A}$$

(taking advantage of the Laplace transform's property that $\frac{dy(t)}{dt} \Leftrightarrow sY(s)$, assuming that $y(0) = 0$).

This system, with transfer function $H(s)$, has a single pole at $s = -A$. Remember that its frequency response at frequency $\omega$ can be found by letting $s=j\omega$:

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

To get a rough view of this response, first let $\omega \to 0$:

$$\lim_{\omega \to 0} H(\omega) = \frac{1}{A}$$

So the system's DC gain is inversely proportional to the feedback factor $A$. Next, let $w \to \infty$:

$$\lim_{\omega \to \infty} H(\omega) = 0$$

The system's frequency response therefore goes to zero for high frequencies. This follows the rough prototype of a lowpass filter. To answer your other question with respect to its time constant, it's worth checking out the system's time-domain response. Its impulse response can be found by inverse-transforming the transfer function:

$$H(s) = \frac{1}{s+A} \Leftrightarrow e^{-At}u(t) = h(t)$$

where $u(t)$ is the Heaviside step function. This is a very common transform that can often be found in tables of Laplace transforms. This impulse response is an exponential decay function, which is usually written in the following format:

$$h(t) = e^{-\frac{t}{\tau}}u(t)$$

where $\tau$ is defined to be the function's time constant. So, in your example, the system's time constant is $\tau = \frac{1}{A}$.

• Thanks for the answer! So it seems as transfer functions $\frac{1}{1+i \omega \tau}$ and $\frac{1}{\tau + i \omega}$ are different... – Kris Aug 23 '12 at 2:45

The frequency response is the same, yes, but the application is different:

• With a low-pass filter, your signal is in the passband. The cutoff frequency of the filter is set above the highest frequency you want to keep in your signal.
• With a leaky integrator, your signal is in the stopband. The cutoff frequency of the filter is set below the lowest frequency in your signal.

Also, integrators are always first-order, while low-pass filters can be any order.

• Same response except for the DC-gain... – Arnfinn Sep 16 '16 at 8:24