The Radio Link--A tutorial--Part IV

Thermal Noise and Receiver Analysis

All communication systems, wired or unwired, are affected by unwanted signals. These unwanted signals are termed either noise  or interference . There are no clear and universal definitions that distinguish noise from interference. Most often the term interference refers to unwanted signals entering the passband of the desired system from other systems that intentionally radiate electromagnetic waves.

An intentional radiator is any radio system that uses electromagnetic waves to perform its function. The term noise often refers to unwanted signals arising from natural phenomena or unintentional radiation by man-made systems. Examples of natural phenomena that produce noise are atmospheric disturbances, extraterrestrial radiation, and the random motions of electrons. Examples of man-made unintentional radiators that produce noise are power generators, automobile ignition systems, electronic instruments, and microwave ovens. The distinction between noise and interference will become more evident when we discuss interference management in cellular systems.

The effects of interference and certain types of noise can often be mitigated and sometimes eliminated by appropriate engineering techniques and/or the establishment of rules to restrict both intentional and unintentional radiation. One particular type of electrical noise, however, is ubiquitous insofar as it arises in the very components used to implement a system. This noise, called thermal noise , arises from the thermal motion of electrons in a conducting medium and is present in any circuit consisting of resistive elements such as wires, semiconductors, and, of course, resistors. It is, therefore, present in any system that uses these components. The presence of thermal noise limits the sensitivity of all electronic wireless systems. Sensitivity is a measure of a system’s ability to reliably detect a signal.

In addition to assuming that waves propagate in free space, our development of the range equation tacitly assumed ideal transmit and receive antennas and a lossless connection between the receive antenna and an ideal receiver. Under these optimal conditions there is no minimum limit to a receiver’s ability to detect a signal, and therefore the operating range of such a system is limitless. Real antennas, receivers, and interconnecting circuits, however, all have resistive (lossy) elements and electronic components. These elements and components introduce thermal noise throughout a receiver and in particular in its front end, that is, the first stages immediately following the receiving antenna. When the information-carrying signal is comparable to the noise level, the information may become corrupted or may not even be detectable or distinguishable from the noise. The maximum range is, therefore, constrained by the need to maintain the information-carrying signal at some level relative to the noise level at the input to the receiver. This noise level is often called the noise floor.

Characterizing Noise Sources

Taking into account the thermodynamic nature of the phenomenon, the noise generated by a resistor has a normalized power spectrum  given by

Noise having a power spectrum that is constant at all frequencies is called “white” noise, by analogy with white light, which has a constant power spectrum at all wavelengths. Equation (2.32) shows that thermal noise can be modeled as white noise for all radio frequencies of current practical interest. We often write the power spectrum of white noise as

when we do not want to imply that the noise was necessarily generated by a resistor. Figure 2.9 shows a signal x (t ) with Fourier transform X (ƒ  ) passing through a filter with frequency response H (ƒ ). The output signal y (t) has Fourier transform Y ( ƒ ). We know that

Taking the magnitude squared of both sides of Equation (2.34) gives a relation between the energy spectrum of the input signal and the energy spectrum of the output signal:

Now noise does not have an energy spectrum, but it can be shown that an equation similar to Equation (2.35) applies to noise power spectra. Thus, if noise x (t ) having a power spectrum Sx ( ƒ  ) is passed through the filter, then the power spectrum Sy ( ƒ ) of the output noise y (t ) is given by

Equation (2.37) gives the average power of a white noise signal measured using an instrument of bandwidth B. If the noise is actually thermal noise generated by a resistor, then Equation (2.32) gives us

We need to be more precise about the meaning of the term average power,  as several notions of average power will be used in the sequel. If x (t ) is a signal, the average power is given by

This is actually the mean square value of x (t ) or the average power that a voltage or current x (t ) would deliver to a 1-ohm resistor. The average power Px  is sometimes referred to as the “normalized” power in x (t ) and is sometimes written x² (t )  to emphasize the mean-square concept. We note that the square root of the mean-square value is the RMS value, so

J. B. Johnson of the Bell Telephone Laboratories was the first to study and model thermal noise (also known as “Johnson” noise) in the late 1920s. In his model, thermal noise arising from a resistance of value R is represented as an ideal voltage source in series with a noiseless resistance of value R  as shown in Figure 2.12. The mean-square open-circuit voltage of the ideal voltage source is given by Equation (2.39).

Next:  Thermal Noise and Receiver Analysis - Continued

Introduction to Wireless Systems by Bruce A. Black, Philip S. DiPiazza, Bruce A. Ferguson, David R. Voltmer, Frederick C. Berry, Published Jun 7, 2011 by Prentice Hall, is reprinted with permission by Pearson Publishing.