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Noise and Distortion in Microwave System

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Noise and Distortion in Microwave System Noise and Distortion in Microwave System Prof. Tzong-Lin Wu EMC Laboratory Department of Electrical Engineering National Taiwan University 2016/3/16 MW & RF Design / Prof. T. -L. Wu 1 Introduction Noise is a random process from many sources: thermal, atmosphere, inter-stellar radiation and man-made radiation. 2016/3/16 MW & RF Design / Prof. T. -L. Wu 2 3.1 Review of Random Processes Probability and Random Variables 2016/3/16 MW & RF Design / Prof. T. -L. Wu 3 3.1 Review of Random Processes Cumulative Distribution Function The cumulative distribution function (CDF), Fx(x), of the random variable X is defined as the probability that X is less than or equal to a particular value, x. Properties: Monotonic (non decreasing) function 2016/3/16 MW & RF Design / Prof. T. -L. Wu 4 3.1 Review of Random Processes Probability Density Function The probability density function (PDF)of a random variable X is defined as the derivative of the CDF: 2016/3/16 MW & RF Design / Prof. T. -L. Wu 5 3.1 Review of Random Processes Probability Density Function PDF and CDF with 2 random variables Joint CDF associated with random variables X and Y is defined as the joint PDF is calculated as The probability of x and y both occurring in given ranges is found from 2016/3/16 MW & RF Design / Prof. T. -L. Wu 6 3.1 Review of Random Processes Probability Density Function The individual probability density functions for X and Y can be recovered from the joint PDF by integration over one of the variables: Statistically independent 2016/3/16 MW & RF Design / Prof. T. -L. Wu 7 3.1 Review of Random Processes Some Important Probability Density Functions Uniform distribution Gaussian distribution where m is the mean of the distribution, and is the variance. Rayleigh distribution 2016/3/16 MW & RF Design / Prof. T. -L. Wu 8 3.1 Review of Random Processes Expected Values The expected value is also sometimes called the mean, or average value. For discrete random variables Continuous random variables: Taking the expected value of a random variable is a linear operation Expected value of a function of a random variable Discrete case Continuous random variable 2016/3/16 MW & RF Design / Prof. T. -L. Wu 9 Higher order statistical average nth moment of the random variable The variance, Expected value of joint PDF Independent random variables: 2016/3/16 MW & RF Design / Prof. T. -L. Wu 10 3.1 Review of Random Processes Autocorrelation Autocorrelation: how fast a signal changes with time Defined for a complex deterministic signal, x(t) For stationary random processes, such as noise Power spectral density (PSD) 2016/3/16 MW & RF Design / Prof. T. -L. Wu 11 3.1 Review of Random Processes Autocorrelation Power delivered to 1 load is For a noise voltage, the power spectral density represents the noise power density in the spectral (frequency) domain, assuming a 1ohm load resistor. 2016/3/16 MW & RF Design / Prof. T. -L. Wu 12 2016/3/16 MW & RF Design / Prof. T. -L. Wu 13 2016/3/16 MW & RF Design / Prof. T. -L. Wu 14 2016/3/16 MW & RF Design / Prof. T. -L. Wu 15 3.2 Thermal Noise • Random motion of charge carrier. • Also called Nyquist, or Johnson noise • Most important source of noise in microwave band. • Other noise: shot noise, flicker, plasma, quantum Shot noise, due to the random motion of charge carriers in electron tubes and solid-state devices; flicker noise, also occurring in solid-sate devices and vacuum tubes; plasma noise, caused by random motions of charged particles in an ionized gas or sparking electrical contacts; Quantum noise, resulting from the quantized nature of charge carriers and photons. 2016/3/16 MW & RF Design / Prof. T. -L. Wu 16 3.2 Thermal Noise Noise Voltage and Power Rayleigh Jeans approximation: The electrons in the resistor are in random motion, with a kinetic energy that is proportional to the temperature, T. These random motions produce small random voltage fluctuations across the terminals of the resistor. The mean value of this voltage is zero, but its nonzero rms value in a narrow frequency bandwidth B is given by 2016/3/16 MW & RF Design / Prof. T. -L. Wu 17 3.2 Thermal Noise Noise Voltage and Power The noisy resistor can be modeled using a Thevenin equivalent circuit as an ideal (noiseless) resistor with a voltage generator to represent the noise voltage. The available noise power is defined as the maximum power that can be delivered from the noise source to a load resistor. Note that the noise power decreases as the system bandwidth decrease. This implies that systems with smaller bandwidths collect less noise power. Noise power decreases as temperature decreases, which implies that internally generated noise effects can be reduced by cooling a system to low temperatures. Since thermal noise power is independent of frequency, it is referred to as white noise 2016/3/16 MW & RF Design / Prof. T. -L. Wu 18 3.2 Thermal Noise Noise Voltage and Power Noise spectrum is flat from DC~1000GHz Two-sided power spectral density Autocorrelation of a white noise PDF of white noise: Gaussian with zero mean 2016/3/16 MW & RF Design / Prof. T. -L. Wu 19 Since the variance of the sum of two independent random variables is the sum of the individual variances (see Problem 3.5) The variance is equivalent to power delivered to a 1 Q load, the noise powers generated by two independent noise sources add in a common load. This is in contrast to the case of deterministic sources, where voltages add. 2016/3/16 MW & RF Design / Prof. T. -L. Wu 20 Ex 3.2 2016/3/16 MW & RF Design / Prof. T. -L. Wu 21 3.3 NOISE IN LINEAR SYSTEMS In a wireless radio receiver, both desired signals and undesired noise pass through various stages, such as RF amplifiers, filters, and mixers. These functions generally alter the statistical properties of the noise, and so it is useful to study these effects by considering the general case of transmission of noise through a linear system. Autocorrelation and Power Spectral Density in Linear Systems 2016/3/16 MW & RF Design / Prof. T. -L. Wu 22 3.3 Noise in Linear Systems Autocorrelation and Power Spectral Density in Linear Systems This result shows that the autocorrelation of the output is given by the double convolution of the autocorrelation of the input with the impulse response: 2016/3/16 MW & RF Design / Prof. T. -L. Wu 23 Autocorrelation and Power Spectral Density in Linear Systems the equivalent result in terms of power spectral density 2016/3/16 MW & RF Design / Prof. T. -L. Wu 24 3.3 Noise in Linear Systems Gaussian White Noise through an Ideal Low-pass Filter Filter notation The output noise power is proportional to the filter bandwidth. 2016/3/16 MW & RF Design / Prof. T. -L. Wu 25 3.3 Noise in Linear Systems Gaussian White Noise through an Integrator 2016/3/16 MW & RF Design / Prof. T. -L. Wu 26 3.3 Noise in Linear Systems Mixing of Noise LO signal with random phase: This result shows that mixing reduces the average noise power by half. In this case, the 2 factor of one-half is due to the ensemble averaging of the cos w0t term over the range of random phase. 2016/3/16 MW & RF Design / Prof. T. -L. Wu 27 3.3 Noise in Linear Systems Mixing of Noise LO signal with deterministic phase: The variance of the output signal is now a function of time, and represents the instantaneous power of the output signal. To find the time-average output power, we must take the time average of the variance We see that the same average output power is obtained whether the averaging is over the ensemble phase variation, or over time. 2016/3/16 MW & RF Design / Prof. T. -L. Wu 28 2016/3/16 MW & RF Design / Prof. T. -L. Wu 29 3.3 Noise in Linear Systems Narrowband Representation of Noise Gaussian white noise passing through a bandpass filter with a center frequency w0 and bandwidth Δw. x(t) and y(t) are random processes, but are slowly varying due to the narrow bandwidth of the filter. How to prove the valid of this form? 2016/3/16 MW & RF Design / Prof. T. -L. Wu 30 3.3 Noise in Linear Systems Narrowband Representation of Noise After low-pass filtering with a low-pass cutoff frequency of , only the x(t) and y(t) terms will remain in the above results. 2016/3/16 MW & RF Design / Prof. T. -L. Wu 31 We now find the statistics of x(t) and y(t), and show that these Gaussian random processes have zero mean, the same variance as n(t), and are uncorrelated. 2016/3/16 MW & RF Design / Prof. T. -L. Wu 32 3.3 Noise in Linear Systems Narrowband Representation of Noise Comparing (3.48) to (3.47) shows that Meaning? 2016/3/16 MW & RF Design / Prof. T. -L. Wu 33 3.3 Noise in Linear Systems Narrowband Representation of Noise Ans: 2016/3/16 MW & RF Design / Prof. T. -L. Wu 34 3.4 Basic Threshold Detection 2016/3/16 MW & RF Design / Prof. T. -L. Wu 35 Probability of error When a binary "1" is transmitted, a detection error will occur if the received signal and noise is less than the threshold level at the sampling time. For a threshold of vo/2, the probability of this event is Gaussian probability density function set Lower limit Define: complementary error function: 2016/3/16 MW & RF Design / Prof.
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