Signals and Networks: Low-pass filter

A low-pass filter is a filter that passes signals with a frequency lower than a certain cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency.

Low-pass filters provide a smoother form of a signal, removing the short-term fluctuations, and leaving the longer-term trend.

1st Order Filter

A first-order low-pass filter (with one pole) can be described in Laplace notation as:

\frac{\text{Output}}{\text{Input}} = K \frac{1}{\tau s + 1}

where s is the Laplace transform variable, τ is the filter time constant, and K is the gain of the filter in the passband.

The gain-magnitude frequency response of a first-order (one-pole) low-pass filter. Power gain is shown in decibels (i.e., a 3 dB decline reflects an additional half-power attenuation).

RC filter

(Passive, first order low-pass RC filter)

One simple low-pass filter circuit consists of a resistor in series with a load, and a capacitor in parallel with the load.

The capacitor exhibits reactance, and blocks low-frequency signals. At higher frequencies the reactance drops, and the capacitor effectively functions as a short circuit.

$ V_{out} = \frac{X_c}{X_c + R} V_{in}$

Putting $X_c = \frac{1}{sC}$, we get

$\Rightarrow \frac{V_{out}}{V_{in}} = \frac{1}{1 + sRC}$, which is a first order filter.

The combination of resistance and capacitance gives the time constant of the filter $\tau = RC$. The break frequency or cutoff frequency (in hertz), is determined by the time constant:

$ f_c = \frac{1}{2\pi \tau} = \frac{1}{2\pi RC}$

Active Low Pass Filter

It uses an Op-amp to realize an active filter.

The cutoff frequency (in hertz) is defined as:

$ f_c = \frac{1}{2\pi R_2 C}$

The gain in the passband is −R₂/R₁, and the stopband drops off at −6 dB per octave (that is −20 dB per decade) as it is a first-order filter.

Other resources:
http://www.sensorsmag.com/sensors/electric-magnetic/an-introduction-analog-filters-1023