A major use of cross correlation is in the detection of signals buried in noise. Typical applications involve radar, sonar, ultrasound imaging, and other reflective ranging techniques. The waveshape of the signal is generally known and the problem is to detect the presence and location of the return echo even though it is not visible inside the noise envelope. The top trace in Figure 1 is a simulated ultrasound application. The transmitted and return signals are contained in the waveform, which has a very low signal-to-noise ratio. A copy of the pulse, shown in trace M1 in Figure 1, is used as a reference waveform. The reference is crosscorrelated with the waveform in trace C1, as shown in math function trace F1. The existence and location of the return echo is easily determined in the F1 trace even though it was invisible in the original signal.
The correlation process incrementally slides the reference waveform over the signal being processed looking for a matching signal. Signals that are not related to the reference waveform result in a correlation value of 0. If a matching signal is found, the correlation value increases to a maximum value of 1 for a perfect match and -1 for a matching but inverted waveform. The correlation function for each time delay is plotted against the time delay to form the correlation function as shown in trace F1.
While averaging can be used to extract a signal from random noise, as shown in trace F2 in Figure 1, correlation can detect a signal in the presence of a deterministic signal such as a sinusoid. In Figure 2, a 20-MHz sine wave has been added to the ultrasound pulse. As in the previous example, trace F1 contains the cross-correlation of traces C1 and M1. It clearly shows the location of the echo. Trace F2, displaying the summed average of trace C1, shows that the interfering sine wave is not attenuated by the averaging process and is unable to extract the echo from the input signal.
This application is an excellent example of the power of the correlation function to detect signals which have suffered severe degradation. The correlation function can also be used to characterize signal-propagation paths and provide information on path delay.
Figure 3 shows a simple example of using correlation to determine the propagation delay of a pulse. The 22-ns delay between the pulses acquired in channels 1 and 2, as measured with the delta time at level parameters, corresponds to the peak of the cross-correlation function measured using relative cursors. In more complex studies, multiple correlation measurements can be used to locate signal sources or to track down reflection paths even in the presence of interfering signals. This type of measurement is commonly used in acoustic and seismic studies.
Correlation is but one of many advanced signal-processing tools available in Teledyne LeCroy oscilloscopes.