The use of filters in enhancing signal-to-noise ratio is unsuitable if wide band operation is desired. Alteration of the shape of the signal is unavoidable with this approach. It is rare that a phenomenon producing a signal only occurs once, although it must be recognized that even if the process is repetitive the set of signals produced may not be identical even in the absence of noise. It is unlikely for example, that nerve impulses observed from an axon will be exactly reproduced. However, it is always possible to idealize the situation so that a perfect repetitive signal exists. Departures are then considered as fluctuations about this ideal signal. These fluctuations can then be incorporated into the unavoidable noise contribution.
In this situation an alternative is to collect a set of signals and synthesize the ideal signal by averaging. This
procedure is accomplished with efficiency using the arrangement shown above. The set of signals can be represented
as a set of time functions xi(t).
In order that the signals be repetitive it is assumed that they are of finite duration T, each has a defined start time, and the entire set is generated by translation of the ideal signal by shifts greater than T, so that one signal has died out before the second begins. The system is triggered by the start point of each signal providing a zero time for each signal. The control flip flop (FF)is set by a trigger pulse Tr generated at the signal zero time. The output of the FF acts as a start measurement command, STRT and a series of samples of the signal are taken at equal intervals tr = r/fc. The sample taken at tr on the ith signal will be designated Xi(r). The procedure by which an average signal is obtained corresponds to averaging each of the corresponding samples taken at tr, r = 1,M over a set of signals i = 1,N. The situation is indicated in the diagram below.
During acquisition the average after the nth signal for the rth sample can be written
The sum in the second line corresponds to (n-1)Yn-1(r) so the expression becomes
The last form is the desired expression in that it expresses the averaging process in a recursive relation that is physically realizable as demonstrated by the arrangement above. The difference amplifier is operated in the linear range with a gain of unity. To appreciate the situation more clearly it is necessary to follow the first pair of measurements. As with the transient analyser the memory address register counts the clock pulses so that memory address r corresponds to the sample at tr. A second counter is driven by the trigger and hence counts the number of signals n.
Before the first signal all registers are clear. Consider the situation for any sample r of the first signal n = 1. Then the non-inverting input of the amplifier is X1(r), while the inverting input is zero. The output fed to the ADC is also X1(r). The content of the signal counter is n = 1, so the ADC reference voltage is v0, the quantizing voltage of the DAC to which the counter is connected. It should be recalled that the digitized output depends inversely on the reference voltage. The digitized value of the sample is added to zero and stored in memory as the first estimate of the average Y1(r). When the same time point is analysed for the second signal, n = 2, the negative input of the amplifier is the analogue version of Y1(r), since the memory contents of address r are connected to the DAC coupled to this input. The amplifier output is now X2(r)-Y1(r). The ADC reference voltage is now 2v0 At the end of the conversion the ADC input to the adder is then (X2(r)-Y1(r))/2. The second input to the adder is Y1(r) so at the write enable (also serving as the LOAD signal for MDR) the sum is stored updating the average estimate. Repetition in the procedure continues the averaging procedure, so that the algorithm is performed as set out in the last recursive equation. Of course the parameters of the ADC and DACs must be carefully matched so that the constants of proportionality cancel. The completion of sampling on a single signal is indicated by MAR overflow setting the carry bit,c. The carry bit is reset by the trigger at the start of each cycle.
The effect of averaging on the mean square fluctuation (variance) relies upon the fluctuations being independent random quantities. In this case it is well known that the mean square fluctuation in the sum of two random variables is the sum of their mean square fluctuations. Thus if the variance of an individual sample is s 2 then given
for a completed measurement involving N signals, the variance in the sum is Ns2. But the quantity ZN(r) = NYN(r) so the variance in the average is
Thus the signal to noise ratio following the averaging of N signals is a factor of sqrt(N) greater than that of the individual signals.
In order for the procedure to work successfully a suitable trigger must be available. Since the trigger time essentially determines the effective start time for each signal, variation in synchronization between the trigger and the actual signal means that corresponding samples will not occur at exactly the same phase point of the signal. It is sometimes possible to generate the trigger from the signal itself, as with internal triggering on an oscilloscope. If, however, the signal is essentially buried in the noise it is usually not possible to obtain a suitable trigger.
A common practice is a stimulus-response study in which the response of a system evoked by an externally applied stimulus is observed. Since the generation of the stimulus is under external control, it is always possible to generate a well defined trigger signalling the time at which the stimulus occurs. The response signal to be studied may be very weak and masked by noise, but its time of occurrence is independently established by the stimulus. Thus the stimulus and response are correlated, as are the trigger and response signal. In such studies signal averaging can achieve truly remarkable results, and responses completely concealed by noise may be clearly delineated. An important general feature of this approach is that advantage is taken of a phase relationship. Recall that the Fourier representation of a signal involves a complex function, which when written in polar form may be decomposed into an amplitude and phase. The use of filters makes use of amplitude information by selecting the region of frequencies in which the amplitude is large and rejecting those regions which have no significant amplitude. In the stimulus-response type of study it is possible to make use of phase information. This general approach is classified as a correlation method and its general treatment will be undertaken in the next section.
The terminology of stimulus-response immediately brings to mind biological applications, but the method is quite general and examples of physical systems studied using this approach are not uncommon. In both nuclear magnetic and electron spin resonance, the system responds to an externally applied electromagnetic field. If pulse techniques are used, then this is the type of stimulus response approach discussed. Continuous wave studies still represent correlation studies since there is a well-defined phase relationship between the detected signal and the externally imposed field. A more obvious example is that of photo- and radiation chemistry. Here the lifetimes of extremely unstable chemical species are studied. The approach is to expose a material system to a short burst of light from a laser, or radiation from an accelerator. Photodetectors are arranged to detect characteristic photo-absorption as a measure of the concentration of the species created. The absorptivity is measured as a function of the lag after the pulse.