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What is an alternative to filters when the signal and noise are similar frequencies?

Signal averaging, autocorrelation and cross correlation

All carried out in the time domain


What is signal averaging?

You take several sweeps of the signal measuring several points in time each time.

A point in time is selected and averaged using the values from each sweep.

This produces a cleaner signal


For signal averaging to be effective what conditions must be met?

- The signal in each sweep must be coherent and synchronised (lined up in time)

- The noise should be uncorrelated (incoherent/ random) i.e not biological noise


What is the relationship between signal averaging and improvement in the signal to noise ratio?

Improvement in the signal to noise ratio is proportional to the square root of the number of sweeps (n).


What does the relationship of signal average to S2N ratio follow?

Signal averaging follows the law of diminishing returns

i.e 4 sweeps = 2 factor improvement
16 sweeps is 4 times greater but only a 4 factor improvement i.e twice the gain of the 4 sweeps.

Although very noise signal might require many sweeps to extract the signal


What is autocorrelation?

Reveals the existence of repeated structure or periodicity in a signal that might not otherwise be clear.


How does autocorrelation work?

The function is evaluated by taking a copy of a signal an shifting it by t (greek letter not t for time that just represents a shift) relative to the other signal.

Therefore the autocorrelation function is at a peak when the signals most closely overlap. i.e shift = o or one period.

Shift occurs in both negative and positive directions


Does the signal peak at one period of phase shift?

No only because there are no terms in the summation equation to describe this.


In plain english what does the autocorrelation function measure?

Measures the correspondence of a function with a series of time shifted version of itself.


Describe the autocorrelation of a finite and infininte sinusoid;

decaying cosine function and non-decaying cosine function respectively


What happens in an autocorrelation function of random noise?

Large speak at t = 0 but then drops to nothing as noise is random and therefore demonstrates no periodicity.

Thus we describe noise as uncorrelated or incoherent


When using the autocorrelation function what is it important to account for?

Necessary to account for both short data lengths and non-wide band noise that will produce peaks in the autocorrelation spectrum. It is therefore necessary to use statistical techniques to access significance of autocorrelation peaks.


What is cross correlation?

Computationally the same as autocorrelation except comparing two different waves.

Detects the correspondence between different signals in the presence of noise.

Alternatively it can be used to detect hypothesised wave forms in the presence of noise.

Significance of correspondence must be determined.