Random Process For a random process in the discrete time domain, we use to represent the probability distribution of n samples. If the random process is stationary, we have Hierarchy of probability density of random process
Markov Process Markov process is a special type of random process. For each Markov process, we have Intuitively, in a Markov process, given the current system state, the future system state is independent of the previous history.
Noise Noise is the negative factor impairing communication qualities. Without noise, we may transmit as much as we want without errors. In this chapter, we study the mechanisms, properties and descriptions of various types of noise. We follow the classical book: D. Middleton, An Introduction to Statistical Communication Theory, Peninsula Publishing, 1987
Three Types of Noise In this chapter, we consider three types of noises: Thermal noise Shot noise Impulse noise
Thermal Noise Thermal noise is the result of the random motion of the free electrons in a conductor with temperature T. The random movement results in a random current I(t). Two equivalent representations of a resistance at temperature T:
Spectrum of Thermal Current (detailed model) Using the theory of electrons (such as free path), we obtain the spectrum of thermal current When the wave length is 10^-6cm and T0=300K, the spectrum begins to depart from the uniform response when f is more than 10^13 rad/s.In the range of wireless signal, we can consider the thermal noise as ‘white’. The voltage spectrum is given by
An Alternative Derivation We can have another approach to derive the Nyquist equation:
Quiz Problem 1. Given the following band pass signal: write down the equivalent baseband signal in both time and frequency domains. Problem 2. Consider a two-path wireless channel with the following output: write down the frequency domain transfer function.
Generalization Nyquist’s result is mot limited to purely resistive elements in an equilibrium state, but can also be directly extended to general (passive) linear systems.
Noise Factor and figure The noise factor of a system is defined as The noise figure is defined as The noise factor is given by, where T_e and T_0 are the noise and physical temperatures. For a cascaded system, the noise factor is given by
Homework 3 Problem 1. If the temperature is 300K and the signal bandwidth is 1MHz, what is the value of noise power? Problem 2. Consider a series of devices with gains G1, G2, …, Gn and noise temperature T1, T2, …, Tn. What is the expression of the noise temperature of these concatenated devices? Problem 3. What is the expectation and variance of Poisson distribution? Deadline: Sept. 23, 2013.
Distribution of Thermal Noise We can assume that the thermal noise is Gaussian distributed: Usually we also assume that the thermal noise is white, i.e., the noise is independent for different time slots. In this case, we say that the communication channel is additive white Gaussian noise (AWGN).
White Noise When the noise spectrum is flat, we call it white noise. The spectral density is given by
Filtered (Colored) Noise When passed through a LTI filter with transfer function H(f), we have Example: noise passed through RC network
Noise Equivalent Bandwidth Average noise power: Noise equivalent bandwidth: The filtered noise is What about the RC circuit?
Bandpass Noise Bandpass noise results when white noise passes through a bandpass filter.
SNR The predetection signal-to-noise ratio is given by We also define a system parameter (W is the low pass filter bandwidth)
Quadrature Components The bandpass noise can be written as The power spectral densities are identical lowpass functions related to G_n(f):
Envelope and Phase The envelope of bandpass noise is a Rayleigh random variable The phase distribution is uniform over [0,2π]
Impulse Noise The noise inherent in transmitting and receiving systems is for the most part due to thermal effects in both the passive and active elements of the system. Additional noise may enter a communication link through the medium of propagation. One common source is interference, which has a noticeable different statistical character.
A General Model We assume that the noise process X(t;a) is the resultant of multiple events in the time interval (t,t+T). We have
Poisson Noise In this model, the process X(t,a) is assumed to be the result of the linear superposition of independent impulses.
Temperature-limited Shot Noise Shot noise is the name given to the noise that arises in vacuum tubes and crystals because of the random emission and motion of electrons in these active elements. Noise of this type appears as a randomly fluctuating component of the output current and along with thermal noise is an important factor inhibiting the performance of transmitting and receiving systems.
Expression of Distribution Consider the current of a temperature limited diode. The current waves can be written as The first order approximation is given by