# Phase Lock Loop – Explained

Phase Lock Loops (PLLs) are an important component of communication systems, where they are used for carrier phase and frequency synchronization. They are also used in test and measurement equipment such as in Signal Generators and Vector Network Analyzers (VNAs) for frequency synthesis. Although not discussed here in detail but PLLs are also quite adept at generating multiples of a base frequency e.g. if you have a reference signal at 10MHz then a PLL can be used to generate a 100MHz signal (X=10) or even a 1GHz signal (X=100). In fact, you can also divide the frequency to get low frequency signals. In the first case the feedback frequency is divided by X and in the second case the reference or input frequency is divided by X.

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# KAY’s Single Frequency Estimator

As previously discussed, finding the frequency of a complex sinusoid embedded in noise is a classical problem in Signal Processing. The problem is compounded by the fact that number of samples available is usually quite small. So far, we have discussed Zero Crossing, FFT, MUSIC and ESPRIT methods of frequency estimation. Zero Crossing method is simplest of the above four but it can detect only one sinusoid at a time. Advantage of Zero Crossing method is that it is computationally not that complex. It does not require complex matrix manipulations as some of the other methods do.

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# A Comparison of FFT, MUSIC and ESPRIT Methods of Frequency Estimation

As discussed in previous posts it is frequently required in communications and signal processing to estimate the frequency of a signal embedded in noise and interference. The problem becomes more complicated when the number of observations (samples) is quite limited. Typically, the resolution in the frequency domain is inversely proportional to the window size in the time domain. Sometimes the signal is composed of multiple sinusoids where the frequency of each needs to be estimated separately. Simple techniques such as Zero Crossing Estimator fail in such a scenario.  Even some advanced techniques such as MATLAB function “pwelch” fail to distinguish closely spaced sinusoids.

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# Frequency Estimation Using Zero Crossing Method

A sinusoidal signal is the most fundamental type of signal that exists in communication systems, power systems, navigation systems etc. It is controlled by three parameters which are the amplitude, phase and frequency. The last two, that is phase and frequency, are interconnected. As discussed in my previous post Instantaneous Frequency (IF) is nothing but the rate of change of phase. This can be mathematically described as:

IF=Δφ/Δt

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# Modeling Phase and Frequency Synchronization Error

Carrier phase or frequency synchronization is a common problem in wireless communication systems. These two problems are interrelated as instantaneous frequency is just the rate of change of phase. The problem of carrier frequency offset might appear due to one of two reasons. Either the oscillators at the transmitter and receiver are not aligned in the frequency domain or there is a Doppler shift introduced by the channel (remember that a moving object in the wireless environment introduces a Doppler shift). In the case of the former the frequency misalignment is given in parts per million (ppm). A typical value for commercially available oscillators is ±20 ppm. Assuming that there is maximum frequency error at both the transmitter and receiver the error increases to ±40 ppm. At 1GHz this translates to 40*1,000,000,000/1,000,000 = 40kHz.

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# Index Modulation Explained

Wireless researchers are continuously exploring ways to increase the spectral efficiency (bits/sec/Hz) and energy efficiency (bits/Joule) of wireless communication systems [1]. Spectral efficiency can generally be improved by using larger constellations or by using multiple antennas at the transmitter and receiver, better known as MIMO. But increasing energy efficiency is not that straightforward. Let’s consider this in bit more detail.