Sensitivity (electronics)

The sensitivity of a microphone is usually expressed as the sound field strength in decibels (dB) relative to 1 V/Pa (Pa = N/m²) or as the transfer factor in millivolts per pascal (mV/Pa) into an open circuit or into a 1 kiloohm load.^{[citation needed]} The sensitivity of a hydrophone is usually expressed as dB relative to 1 V/μPa.^[7]

The sensitivity of a loudspeaker is usually expressed as dB / 2.83 V_RMS at 1 metre.^{[citation needed]} This is not the same as the electrical efficiency; see Efficiency vs sensitivity. This is an example where sensitivity is defined as the ratio of the sensor's response to the quantity measured. One should realize that when using this definition to compare sensors, the sensitivity of the sensor might depend on components like output voltage amplifiers, that can increase the sensor response such that the sensitivity is not a pure figure of merit of the sensor alone, but of the combination of all components in the signal path from input to response.

Sensitivity in a receiver, such a radio receiver, indicates its capability to extract information from a weak signal, quantified as the lowest signal level that can be useful.^[8] It is mathematically defined as the minimum input signal ${\displaystyle S_{i}}$ required to produce a specified signal-to-noise S/N ratio at the output port of the receiver and is defined as the mean noise power at the input port of the receiver times the minimum required signal-to-noise ratio at the output of the receiver:

${\displaystyle S_{i}=k(T_{a}+T_{rx})B\;\cdot \;{\frac {S_{o}}{N_{o}}}}$

where

${\displaystyle S_{i}}$ = sensitivity [W]

${\displaystyle k}$ = Boltzmann constant

${\displaystyle T_{a}}$ = equivalent noise temperature in [K] of the source (e.g. antenna) at the input of the receiver

${\displaystyle T_{rx}}$ = equivalent noise temperature in [K] of the receiver referred to the input of the receiver

${\displaystyle B}$ = bandwidth [Hz]

${\displaystyle {\frac {S_{o}}{N_{o}}}}$ = Required SNR at output [-]

The same formula can also be expressed in terms of noise factor of the receiver as

${\displaystyle S_{i}=N_{i}\;\cdot \;F\;\cdot \;SNR_{o}=kT_{a}B\;\cdot \;F\;\cdot \;SNR_{o}}$

where

${\displaystyle F}$ = noise factor

${\displaystyle N_{i}}$ = input noise power

${\displaystyle SNR_{o}}$ = required SNR at output.

Because receiver sensitivity indicates how faint an input signal can be to be successfully received by the receiver, the lower power level, the better. Lower input signal power for a given S/N ratio means better sensitivity since the receiver's contribution to the noise is smaller. When the power is expressed in dBm the larger the absolute value of the negative number, the better the receive sensitivity. For example, a receiver sensitivity of −98 dBm is better than a receive sensitivity of −95 dBm by 3 dB, or a factor of two. In other words, at a specified data rate, a receiver with a −98 dBm sensitivity can hear (or extract useable audio, video or data from) signals that are half the power of those heard by a receiver with a −95 dBm receiver sensitivity.^{[citation needed]}.

For electronic sensors the input signal ${\textstyle S_{i}}$ can be of many types, like position, force, acceleration, pressure, or magnetic field. The output signal for an electronic analog sensor is usually a voltage or a current signal ${\textstyle S_{o}}$. The responsivity of an ideal linear sensor in the absence of noise is defined as ${\textstyle R=S_{o}/S_{i}}$, whereas for nonlinear sensors it is defined as the local slope ${\displaystyle \mathrm {d} S_{o}/\mathrm {d} S_{i}}$. In the absence of noise and signals at the input, the sensor is assumed to generate a constant intrinsic output noise ${\textstyle N_{oi}}$. To reach a specified signal to noise ratio at the output ${\displaystyle SNR_{o}=S_{o}/N_{oi}}$, one combines these equations and obtains the following idealized equation for its sensitivity^[5] ${\displaystyle S}$, which is equal to the value of the input signal ${\textstyle S_{i,SNR_{o}}}$ that results in the specified signal-to-noise ratio ${\displaystyle SNR_{o}}$ at the output:

${\displaystyle S=S_{i,SNR_{o}}={\frac {N_{oi}}{R}}SNR_{o}}$

This equation shows that sensor sensitivity can be decreased (=improved) by either reducing the intrinsic noise of the sensor ${\textstyle N_{oi}}$ or by increasing its responsivity ${\textstyle R}$. This is an example of a case where sensivity is defined as the minimum input signal required to produce a specified output signal having a specified signal-to-noise ratio.^[2] This definition has the advantage that the sensitivity is closely related to the detection limit of a sensor if the minimum detectable SNR_o is specified (SNR). The choice for the SNR_o used in the definition of sensitivity depends on the required confidence level for a signal to be reliably detected (confidence (statistics)), and lies typically between 1-10. The sensitivity depends on parameters like bandwidth BW or integration time τ=1/(2BW) (as explained here: NEP), because noise level can be reduced by signal averaging, usually resulting in a reduction of the noise amplitude as ${\displaystyle N_{oi}\propto 1/{\sqrt {\tau }}}$ where ${\displaystyle \tau }$ is the integration time over which the signal is averaged. A measure of sensitivity independent of bandwidth can be provided by using the amplitude or power spectral density of the noise and or signals (${\displaystyle S_{i},S_{o},N_{oi}}$) in the definition, with units like m/Hz^1/2, N/Hz^1/2, W/Hz or V/Hz^1/2. For a white noise signal over the sensor bandwidth, its power spectral density can be determined from the total noise power ${\displaystyle N_{oi,\mathrm {tot} }}$ (over the full bandwidth) using the equation ${\displaystyle N_{oi,\mathrm {PSD} }=N_{oi,\mathrm {tot} }/BW}$. Its amplitude spectral density is the square-root of this value ${\displaystyle N_{oi,\mathrm {ASD} }={\sqrt {N_{oi,\mathrm {PSD} }}}}$. Note that in signal processing the words energy and power are also used for quantities that do not have the unit Watt (Energy (signal processing)).

In some instruments, like spectrum analyzers, a SNR_o of 1 at a specified bandwidth of 1 Hz is assumed by default when defining their sensitivity.^[2] For instruments that measure power, which also includes photodetectors, this results in the sensitivity becoming equal to the noise-equivalent power and for other instruments it becomes equal to the noise-equivalent-input^[9] ${\displaystyle NEI=N_{oi,ASD}/R}$. A lower value of the sensitivity corresponds to better performance (smaller signals can be detected), which seems contrary to the common use of the word sensitivity where higher sensitivity corresponds to better performance.^[6][10] It has therefore been argued that it is preferable to use detectivity, which is the reciprocal of the noise-equivalent input, as a metric for the performance of detectors^[9][11] ${\displaystyle D=R/N_{oi}}$.

As an example, consider a piezoresistive force sensor through which a constant current runs, such that it has a responsivity ${\displaystyle R=1.0~\mathrm {V} /\mathrm {N} }$. The Johnson noise of the resistor generates a noise amplitude spectral density of ${\displaystyle N_{oi,{\textrm {ASD}}}=10~\mathrm {nV} /{\sqrt {\mathrm {Hz} }}}$. For a specified SNR_o of 1, this results in a sensitivity and noise-equivalent input of ${\displaystyle S_{i,ASD}=NEI=10~\mathrm {nN} /{\sqrt {\mathrm {Hz} }}}$ and a detectivity of ${\displaystyle (10~\mathrm {nN} /{\sqrt {\mathrm {Hz} }})^{-1}}$, such that an input signal of 10 nN generates the same output voltage as the noise does over a bandwidth of 1 Hz.