Signal Measurement in Noisy Environments
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, 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.
Furthermore, the response width (effective bandwidth) of detected signal depends on the amplitude of the modulation. Generally, linewidth/modulation function has a monotonically increasing, non-linear behavior.
Read more about this topic: Lock-in Amplifier
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