Correlation techniques in general take advantage of the phase relationship between the signal to be measured
and a reference signal. The approach is exemplified by the arrangement in the accompanying drawing.
The signal is usually first processed by a selective filter. This output is then multiplied by a signal from the processor generated by the reference signal. The product is then integrated and the integral forms the output. Both analogue and digital systems exist.
Both the multiplication and integration processes may in fact be quite simple. For example the output from a linear gate may be looked upon as the product of the analogue input and and logical gate signals. Similarly the simple RC circuit discussed in the section on time invariant systems acts as an approximate integrator if the time constant is large compared to the time scale of the signal. More sophisticated multiplication may be carried out with a multiplying DAC, or if the data is completely digitized, with either a multiplying algorithm, or math coprocessor. Similar an operational amplifier-based integrator or digital summation may be employed.
The quantity calculated is essentially the correlation function, which for finite duration signals may be written
This is equivalent to the autocorrelation function introduced earlier, except that here two different signals are involved. Strictly speaking this is cross correlation, but this is usually evident from the context, and is indicated by the subscripts. The normalization is also different as no sense of averaging is required for the actual signals, unlike the case of noise. In any event, even for noise-like signals finite samples are necessary. The basic principle underlying correlation techniques is that the phase of the noise signal is, like the amplitude, random. Thus while the signal bears a definitive phase relationship with the reference signal the noise cannot and ultimately the contribution will average out. In most cases the correlation is calculated at zero lag, t = 0, only.
13.1 Lock-in Amplifiers.
The most commonly used arrangement employing correlation methodology is the lock-in amplifier. This system is appropriate for continuous wave stimulus response studies at a well-defined frequency. Mathematically, the reference signal is a square wave of unit amplitude 1, ie
the period of which matches that of the response signal, since it is derived from the stimulatory signal. Thus the signal to be measured, the response is locked in phase with the reference.
The effect of the multiplying procedure
is illustrated in the following diagram. The first trace shows the signal overlapped and in phase with the reference
signal. Since the signal and reference are both positive and then negative during precisely the same time intervals
the product is always positive, as indicated in the second trace. Integration essentially produces a positive DC
output. The third trace shows the product when this signal leads the reference by p/2.
In this case the signal excursions are of both polarities during the half period the reference voltage is constant.
Thus in this case the integral is in fact zero. If the signal in the uppermost trace were shifted by one half period,
ie a phase shift of p, the negative portion of the signal would be aligned with the
positive section of the reference. The product would then be the negative of the second trace, giving a negative
output. The lock-in amplifier is thus phase sensitive. Since as discussed above, the phase relationship between
the reference and noise is random, equal contributions of in- and out-of-phase components will occur and integration
will tend to average these out.
The input-output relation may be summarized as
for a single cycle. The output filter is arranged to essentially extract the "DC" level, proportional to the above expression. For an input signal given by Vsin(wt + f) the output is given by (2V/p)cosf. This is directly proportional to the signal component in phase with the reference signal, Vrefsinwt. Note that the important physical point is that the output is proportional to the component in phase with the reference signal. Its representation as sinwt is arbitrary. The system thus measures a combination of the amplitude of the response signal and its phase relative to the stimulus. Both of these may vary in time, but this variation must be sufficiently slow so that its major frequencies lie in the band passed by the low pass and input filters.
The front end of the system consists of an amplifier and filter with relatively narrow pass band centred on the reference signal frequency. The signal is then fanned out into two linear gates enabled by complementary gating signals generated from the reference. The first and second gate are thus open for the first and second halves of the cycle repeatedly. The output of the second gate is inverted and added to the output of the first gate, thereby simulating the multiplication by a single square wave operating between ±1. The sum is sent to a low pass filter. If the system being studied is non-linear, it is possible to operate the lock-in at a harmonic frequency.
This methodology results in excellent signal-to-noise enhancement, taking advantage of both filtering and correlation. It is often used in optical measurements. In this case the incident beam is sent through a shutter operating at a known frequency. It is essential in electron spin and nuclear magnetic resonance studies.
13.2 Real-Time Correlators
Digital systems specifically designed to obtain a continually updated measurement of the correlation between two signals are based upon the concept of a shift register memory.
Consider a system consisting of N shift registers each of length M. The situation in the diagram corresponds to N=3, M=8. This results in an array of logic variables xij;i=1,N;j=1,M. This structure may be used to define M words, each of N-bits, according to
A specific memory location, shown
by the shaded bits in the diagram corresponds to the common bit number of the set of registers, j=5 in this instance.
If the registers are in the shift right mode, the location of each word advances one memory location each time
a data load to the memory is performed. Such a structure forms a hardware stack, or first-in last-out buffer.
For data sampled at discrete times tn = (n-1)T, where T is the sampling period, the calculation can be performed only for discrete lags tm = (m-1)T and the correlation must be approximated by
The above operation is performed in real time, ie as the data is sampled, using the scheme shown.
The output from the y ADC is sent to a shift register memory, so that at any time both the current sample Yn and a set of earlier values Yn-m are captured. At the end of conversion of both ADCs, which receive a common sample command, the contents of the shift register memory are sequentially read onto the bus. At the read-out of Yn-m, the contents of the mth address of a RAM are also read into the memory data register, MDR. The contents of the RAM are then replaced by the sum output, updating the estimate of Rm. As the measurement proceeds the RAM contents are continually being refreshed with the sum over the samples obtained.
An interesting application of real time correlators is the determination of particle size by light scattering. The particles are suspended in solution and undergo random or Brownian motion. At any instant, the set of particles form a specific geometrical scattering structure. The amplitude of scattered light observed in a particular direction is the coherent superposition of the contributions from each particle. The geometrical structure is dynamic because of the Brownian motion and intensity fluctuations are thereby induced. Apparently a detailed analysis reveals that the autocorrelation function of these fluctuations is related to the diffusion constant of the particles. Roughly speaking this is because the diffusion constant limits the speed at which the scattering structure can alter significantly, so the system has a short term memory. The detailed result is
where G = 2q2D. The quantity q is essentially the momentum transfer, ie 2ksinq/2, where k is the light wave vector and q is the scattering angle. D is the diffusion constant and is determined for a given solution with viscosity h at temperature T by the particle diameter d according to
The detailed relations have been included only for completeness, but the important aspect is that the quantity controlling the dependence of the autocorrelation function on lag is inversely related to the particle diameter. For a sample with a distribution of particle diameters it is necessary to introduce the probability density p(G) and the autocorrelation function becomes
Inversion of the equation to obtain the probability density from the observed autocorrelation function is not a trivial task.
13.3 Time-Interval Measurements
A special form of correlation measurement involves the determination of very short life-times say on the order of microseconds or less. In the usual approach involving longer times, a signal corresponding to the variation of an intensity with time is measured. This may be done either by analysis of an analogue signal proportional to intensity, or by successive counting of the quanta emitted. In order to properly delineate the time behaviour samples must be taken on the time scale of the life time and this is not possible in the regime under consideration. Instead a measurement of the distribution in the interval between the formation and termination of the species being studied is determined. A timing marker signifying the formation is referred to as a start event. As a concrete example consider a radioactive atom which emits a beta particle creating a new species. Detection of the beta particle constitutes a start event. Suppose the species is created in an excited state with a life-time of 10 nsec. This state decays to a stable state by emission of a g-particle. Detection of this event represents a stop event, and on average should occur 10 nsec. after the start pulse. In fact the interval between the start and stop times follows an exponential distribution with a mean of 10 ns.
The method used is to convert the time interval to a pulse height in a manner similar to a reverse of the Wilkinson ADC. Consider the situation depicted in the accompanying figure. The action of the RS flip flop is to create a pulse at the input to the integrator of a fixed height and duration equal to the interval between the start and stop pulse edges, t.
The integrator output ramps up at
a slope determined by the RS logic level and time constant, both of which are fixed. Thus the maximum reached,
h, is directly proportional to t. A pulse of this height is then created by enabling the linear gate by the stop
pulse at which time the integrator output is constant, since resetting the flip flop shuts off current to the capacitor.
Such devices are referred to as time-to amplitude converters, or TACs. The TAC output is then fed to a multichannel
analyser so that the pulse height distribution may be acquired.