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									Lock-in amplifier
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A lock-in amplifier from SIGNAL RECOVERY




...and one from Stanford Research Systems.

A lock-in amplifier (also known as a phase-sensitive detector) is a type of amplifier
that can extract a signal with a known carrier wave from extremely noisy environment
(S/N ratio can be -60 dB or even less[citation needed]). It is essentially a homodyne with an
extremely low pass filter (making it very narrow band). Lock-in amplifiers use mixing,
through a frequency mixer, to convert the signal's phase and amplitude to a DC—actually
a time-varying low-frequency—voltage signal.

The device is often used to measure phase shift, even when the signals are large and of
high signal-to-noise ratio, and do not need further improvement.

In order that signals be recovered at low signal-to-noise ratios, it is essential that a strong,
clean reference signal is available at the same frequency as the signal to be measured.
This is often not the case in many experiments, so the instrument is not a magic solution
which can recover signals buried in the noise in every circumstance, but only in a limited
set of circumstances.

The lock-in amplifier was invented by Princeton University physicist Robert H. Dicke
who founded the company Princeton Applied Research (PAR) to market the product.
Contents
      1 Basic principles
      2 Application to signal measurements in a noisy environment
      3 External links
      4 Articles about lock-in amplifiers



[edit] Basic principles
Operation of a lock-in amplifier relies on the orthogonality of sinusoidal functions.
Specifically, when a sinusoidal function of frequency ν is multiplied by another
sinusoidal function of frequency μ not equal to ν and integrated over a time much longer
than the period of the two functions, the result is zero. In the case when μ is equal to ν,
and the two functions are in phase, the average value is equal to half of the product of the
amplitudes.

In essence, a lock-in amplifier takes the input signal, multiplies it by the reference signal
(either provided from the internal oscillator or an external source), and integrates it over a
specified time, usually on the order of milliseconds to a few seconds. The resulting signal
is an essentially DC signal, where the contribution from any signal that is not at the same
frequency as the reference signal is attenuated essentially to zero, as well as the out-of-
phase component of the signal that has the same frequency as the reference signal
(because sine functions are orthogonal to the cosine functions of the same frequency),
and this is also why a lock-in is a phase sensitive detector.

For a sine reference signal and an input waveform Uin(t), the DC output signal Uout(t)
can be calculated for an analog lock-in amplifier by:




where φ is a phase that can be set on the lock-in (set to zero by default).

Practically, many applications of the lock-in only require recovering the signal amplitude
rather than relative phase to the reference signal; a lock-in usually measures both in-
phase (X) and out-of-phase (Y) components of the signal and can calculate the magnitude
(R) from that.

[edit] Application to signal measurements in a noisy
environment
The essential idea in signal recovery is that noise tends to be spread over a wider
spectrum, often much wider than the signal. In the simplest case of white noise, even if
the root mean square of noise is 106 times as large as the signal to be recovered, if the
bandwidth of the measurement instrument can be reduced by a factor much greater than
106 around the signal frequency, then the equipment can be relatively insensitive to the
noise. In a typical 100 MHz bandwidth (e.g. an oscilloscope), a bandpass filter with width
much narrower than 100 Hz would accomplish this.

In summary, even when noise and signal are indistinguishable in the time domain, if the
signal has a definite frequency band and there is no large noise peak within that band,
noise and signal can be separated sufficiently in the frequency domain.

If the signal is either slowly varying or otherwise constant (essentially a DC signal), then
1/f noise typically overwhelms the signal. It may then be necessary to use external means
to modulate the signal. For example, when detecting a small light signal against a bright
background, the signal can be modulated either by a chopper wheel, acousto-optical
modulator, photoelastic modulator at a large enough frequency so that 1/f noise drops off
significantly, and the lock-in amplifier is referenced to the operating frequency of the
modulator. In the case of an atomic force microscope, in order to achieve nanometer and
piconewton resolution, the cantilever position is modulated at a high frequency, to which
the lock-in amplifier is again referenced.

When the lock-in technique is applied, care must be taken to calibrate the signal, because
lock-in amplifiers generally detect only the root-mean-square signal of the operating
frequency. For a sinusoidal modulation, this would introduce a factor of      between the
lock-in amplifier output and the peak amplitude of the signal, and a different factor for
non-sinusoidal modulation. In the case of extremely nonlinear systems, it may in fact be
advantageous to use a higher harmonic for the reference frequency, because of
frequency-doubling that takes place in a nonlinear medium.

								
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