by: Dan Doberstein, President DKD Instruments.
Understanding Correlation and how it is used in SS systems is the key that unlocks the inner workings of SS systems. It is the most asked question about SS systems and the hardest to explain.
This article is an attempt to explain correlation as used in SS systems. A simple PLL model is presented to demonstrate correlation at its simplest. This model is extended to the sliding correlator and the related topic of signal retrieval from a negative SNR environment. Negative signal to noise ratio is discussed.
Correlation is at the heart of Spread Spectrum systems. Many articles discuss complicated correlation techniques and their theoretical properties but few are directed at the fundamentals. In particular the following questions; What is Correlation in SS systems? Why is it used in SS Systems? How about a simple real world correlator that can get me going? What does Negative Signal to Noise Ratio mean? How can a signal be retrieved from a negative SNR situation?
Lets start with "what is correlation in SS systems?". Figure one shows a simple SS baseband correlator. As you can see it is just a multiplier and an averager. The averager can be replaced by a simple low pass filter. The multiplier can be approximated by a simple EX-OR gate for digital data, Mixer/Chopper for analog type signals or software operations in sampled data systems. Carrier based correlation requires a little more hardware, a bandpass filter and a detector, but the principles are the basically same.
Two signals are applied to the multiplier. One signal is the received signal the other is our "copy" of this signal provided by our own generator. This is the key concept of correlation - by correlating the received signal with what we know the signal should be , i.e. our "copy" , we can recover the orignal signal that was sent. Our copy may not be exact, in that is phase or frequency may be off a bit, but with the proper tracking system we can null these differences out and our "copy" will be in near perfect synchronism with the received signal. We cannot retrieve the data, which is what we want, until we have "correlation" or ,put another way, synchronism of our copy with the received signal.
But what about the data? Doesn't our copy also have to have the data on it to be a true "copy"? This is the clever part of SS systems. The data is modulated on the transmitted signal such that for correlation and tracking purposes the data part of the signal can be essentially ignored. Once we have correlation we can get that data because the multiplication part of the correlation process strips off our noise like signal and leaves just the data that was hidden inside this signal.
Lets look at how the correlator works. The correlator is essentially comparing the two signals. If the two signals are completely misaligned, in time and frequency, the output of the averager/filter is essentially low or zero volts. When the two signals are in perfect synchronism , essentially identical, the averager/filter output will be at its maximum. For situations between these two extremes the output of the averager/filter will be between zero and the maximum. What we now have is a way of measuring how close we are to perfect synchronization. By simply monitoring the output voltage we can determine if our copy is exactly "lined up" with the received signal or slightly off. With a little more circuitry one can build a system that uses the output of the correlator to automatically recognize correlation. This same system can be made to keep our copy "lined up" even if things like doppler and other disturbances change the received signal or our "copy" generator has internal disturbances. In SS systems this is know as "tracking".
So why is correlation needed? It has to do with the very nature of the signals used in SS systems. SS systems generally transmit noise like signals with the data we want to receive embedded in them. Without the correlation tool we could not strip away the noise like portion of the signal to reveal the data. These noise like signals have very desirable properties. They are interference tolerant. With the radio spectrum ever more crowded interference is ever on the rise. SS signals can share the same frequency. In other words multiple transmitters can transmit on the same frequency and if the system is designed properly not interfere with each other. This property is called CDMA for direct sequence systems. This allows more channels for a given piece of radio spectrum. In low signal level applications SS can allow signal retrieval when the desired signal is BELOW the noise floor. More on this later.
So what about a simple example of a correlator? Well one of the simplest correlation receivers is the PLL. Though usually not thought of or analyzed in these terms the PLL is essentially a correlation machine. Though we can't send data with this simple model we can demonstrate the basics of correlation receivers and sliding correlators.
Figure 2 shows our basic PLL system. It is composed of a multiplier followed by filter which performs the averaging function. The actual multiplier used is usually a special one that allows frequency and phase differences to create error signals for tracking purposes. The two inputs to the multiplier are the received signal and our "copy" of the received signal. In the PLL case our "copy" signal and generator are extremely simple - it is just a sinewave that is generated by a VCO. The received signal is assumed to have this sine wave component buried somewhere in it. The PLL's job is to recover it, both in phase and frequency. The VCO is controlled by the output of the correlator, which forms a closed loop tracking system.
Assuming the VCO frequency is close enough to the received signal we have partial correlation and the system "pushes" the VCO frequency so as to achieve lock. In lock the VCO is now producing a near exact replica of the signal we are trying to receive, our received sinewave. Therefore when the PLL locks onto that sinewave we have essentially received or recovered our desired signal.
PLL receivers can recover sinewaves that are buried in noise signals so deep you can't see them. This is called negative SNR. What it means is if you look at the received signal with an o'scope you will swear it is just noise with no discernable sinewave component. Or equivalently a spectrum analyzer would show no "spike" just an elevated noise floor. **
A problem arises when trying to receive negative SNR sinewaves using our PLL receiver. In order to retrieve these signals we must make the effective bandwidth of the PLL progressively narrower as SNR decreases. The effective bandwidth is also called the "Loop Bandwidth" and is related to the correlator filter bandwidth, VCO gain and other factors. As we narrow our effective bandwidth the PLL "pull in range" becomes smaller and smaller. The "pull in range" of a PLL is the maximum difference in frequency that our "copy" sinewave can be from the desired sinewave we are trying to recover. If our VCO frequency is to far off from the desired frequency lock will never occur.
A solution to this problem is the so called sliding correlator. See figure 3. This system uses a ramp generator to "slide" the VCO frequency until it falls into the "pull in range" window of the PLL receiver. This is the search mode. Once lock or correlation is detected the ramp is held at this voltage and the system is allow to track the desired sinewave, i.e. the track mode. Using this method large negative SNR signals/sinewaves can be recovered.
Once you understand this model it is just a simple step to the more complicated waveforms that are correlated/recovered in actual SS systems. The principles are exactly the same. The sliding correlator is one of the most common correlation techniques used in SS systems. For Direct Sequence (DS) systems that use a sliding correlator the transmitted "noise like" waveform is a Pseudo random sequence , or P/N code, of 0's and 1's. Our copy of the P/N code is slid in time, not frequency, to search for correlation. Instead of a frequency "pull in range window" for lock in the PLL case, we now have a time alignment window of +/-1 code bits. When the received and copy codes are within +/-1 code bits of alignment correlation is declared, tracking is initiated, and the search mode terminated.
I hope this article helps in understanding the correlation process and how correlation is used in SS systems.
** A very good analyzer may be able to see something using a narrow RBW, but SNR is expressed in terms of the receivers bandwidth, which would be wider in this case.
It is important to distiguish between the random noise that our sinewave is buried in and the non-random , "noise like", signals used in SS systems. The former is not predictable nor can it be recreated exactly by any known generator.