Lock-In Length-Reference Cross-Scan - Signals and Noise in Physics Measurements Step by Step

The lock-in scan multiplies each signal sample by a reference sample and then adds the visible products. With four matching samples, every product is 1 and the correlation sum is 4; changing only the reference can weaken, cancel, or invert that sum. Real lock-in measurements depend on reference phase, sampling length, noise, bandwidth, calibration, finite records, and uncertainty.

Lock-in correlation is scanned across sample length and reference choice. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

A matching reference adds one product per sample

With four matching samples, every product is one and the correlation sum is four.

1+1+1+1=4

A longer matching run raises the correlation sum

When the reference matches every sample, correlation grows with the number of visible products.

\begin{array}{c|c|c|c}n&\text{products}&\text{corr}&\text{accepted}\\2&\left(1,1\right)&2&1\\4&\left(1,1,1,1\right)&4&1\\6&\left(1,1,1,1,1,1\right)&6&1\\\end{array}

A wrong reference can cancel or invert the sum

Length is fixed at four here. Changing only the reference changes the product row and the accepted bit.

\begin{array}{c|c|c|c}n&\text{products}&\text{corr}&\text{accepted}\\4&\left(1,1,1,1\right)&4&1\\4&\left(1,1,-1,-1\right)&0&0\\4&\left(-1,-1,-1,-1\right)&-4&0\\\end{array}