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CHAPTER 2 MATHEMATICAL REPRESENTATION OF NOISE

CHAPTER 2 MATHEMATICAL REPRESENTATION OF NOISE. Noise – Introduction Noise – Unwanted Signals that tend to disturb the Transmission and Processing of Signals in Communication System and over which we have incomplete control.

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CHAPTER 2 MATHEMATICAL REPRESENTATION OF NOISE

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  1. CHAPTER 2 MATHEMATICAL REPRESENTATION OF NOISE

  2. Noise – Introduction • Noise – Unwanted Signals that tend to disturb the Transmission and Processing of Signals in Communication System and over which we have incomplete control. • Noise is a general term which is used to describe an unwanted signal which affects a wanted signal. • These unwanted signals arise from a variety of sources. Analog Communication - NOISE

  3. Why study noise? • It sets the lower limit for the detectable signals. • It sets the upper limit for system gains. • Develop mathematical models to take the effects of noise into account when analyzing electrical circuits/systems. • Find ways to reduce noise.

  4. Sources of Noise • Sources of noise may be: • External • Internal • Naturally occurring external noise sources include: • Atmosphere disturbance (e.g. electric storms, lighting, ionospheric effect etc), so called ‘Sky Noise’ • Cosmic noise which includes noise from galaxy, solar noise • ‘Hot spot’ due to oxygen and water vapour resonance in the earth’s atmosphere.

  5. Sources of Noise • Noise performance by external sources is shown below.

  6. Sources of Noise • Internal Noise is an important type of noise that arises from the SPONTANEOUS FLUCTUATIONS of Current or Voltage in Electrical Circuits internally. • This type of noise is the basic limiting factor of employing more complex Electrical Circuits in Communication System. • Noise is basically due to the discrete nature of electrical charges. • Most Common Internal Noises are: • Shot Noise • Thermal Noise

  7. Shot Noise • Shot Noise arises in Electronic Components like Diodes and Transistors. • Due to the discrete nature of Current flow In these components. • Shot noise is due to the random arrivals of electron packets at the potential barrier of forward biased P/N junctions. • It is always associated the a dc current flow in diodes and BJTs. • It is frequency independent (white noise) well into the GHz region.

  8. Example: A Photodiodecircuit. • Photodiode emits electrons from the cathode when light falls on it. • The circuit generates a current pulse when an electron is emitted. • The electrons are emitted at Random times, Ʈk where -∞ < k < ∞ and assume this random emission have been gone for a long time. • Thus the Total Current flowing through the Photodiode may be modeled as the sum of these Current Pulses. • This process X(t) is Stationary and is called SHOT NOISE

  9. Shot noise modeling: • The noise amplitude is represented by the rms value: • The rms noise current for a diode current of 1 mA is about 20 pA/Hz1/2. • The amplitude distribution is Gaussian with m = ID and s = in. • A parallel current source (in) can be added to a diode to account for the shot noise.

  10. Thermal Noise • Thermal Noise is the name given to the Electrical Noise arising from the Random motion of electrons in a conductor. • It is also called Jonson Noise or Nyquist Noise. • It is directly proportional to absolute temperature. • Let VTN is the Thermal Noise Voltage appearing across the two terminals of a resistor.Let the applied voltage have a bandwidth (or frequency), ∆f.

  11. Then the Mean Square value of VTN is given by: Where k = Boltzmann’s constant = 1.38 x 10-23 Joules per oK T = absolute temperature in oK R = resistance in ohms

  12. Jonson Noise or Nyquist Noise

  13. Thermal Noise Modeling: • We can model a noisy resistor using the Thevenin and Norton Equivalent Circuit as shown below: • The number of electrons inside a resistor is very large and their random motions inside the resistors are statistically independent. • The Central Limiting Theorem indicates that thermal Noise is a Gaussian Distribution with Zero mean.

  14. Low Frequency or Flicker Noise • Active devices, integrated circuit, diodes, transistors etc also exhibits a low frequency noise, which is frequency dependent (i.e. non uniform) known as flicker noise . • It is also called ‘one – over – f’ noise or 1/f noise because of its low-frequency variation. • Its origin is believed to be attributable to contaminants and defects in the crystal structure in semiconductors, and in the oxide coating on the cathode of vacuum tube devices • Flicker Noise is found in many natural phenomena such as nuclear radiation, electron flow through a conductor, or even in the environment. • The noise power is proportional to the bias current, and, unlike Thermal and Shot Noise, Flicker Noise decreases with frequency.

  15. An exact mathematical model does not exist for flicker noise because it is so device-specific. • However, the inverse proportionality with frequency is almost exactly 1/f for low frequencies, whereas for frequencies above a few kilohertz, the noise power is weak but essentially flat. • Flicker Noise is essentially random, but because its frequency spectrum is not flat, it is not a white noise. • It is often referred to as pink noise because most of the power is concentrated at the lower end of the frequency spectrum. • Flicker Noise is more prominent in FETs (smaller the channel length, greater the Flicker Noise), and in bulky carbon resistors. • The objection to carbon resistors mentioned earlier for critical low noise applications is due to their tendency to produce flicker noise when carrying a direct current. • In this connection, metal film resistors are a better choice for low frequency, low noise applications.

  16. Flicker noise modeling: • The noise amplitude is represented by the rms value: • The constant K1 is device dependent and must be determined experimentally. • The amplitude distribution is non-Gaussian. • It is often the dominating noise factor in the low-frequency region. • It can be described in more details with fractal theory.

  17. White Noise • The Noise Analysis of Communication System is done on the basis of an idealized form of noise called WHITE NOISE. • Its power spectral density is independent on operating frequency. • White – White light contain equal amount of all frequencies in visible spectrum.

  18. White Noise • Power spectral densityis given by: Where k:Boltzmann’s constant=1.38x10-23 joules/K, Te is the Equivalent noise temperature of the receiver. The 1/2 here emphasizes that the spectrum extends to both positive and negative frequencies.

  19. Power Spectral Density of White Noise • A random process W(t) is called white noise if it has a flat power spectral density, i.e., SW(f) is a constant c for all f.

  20. Ideal Low Pass Filtered White Noise • Let • w(t) = White Gaussian Noise applied to the LPF • B = Bandwidth of LPF • n(t) = noise appearing at the output of LPF • SN(f) = Power Spectral Density of n(t) • RN(Ʈ) = Auto Correlation function of n(t)

  21. Ideal Low Pass filtered White Noise

  22. Noise Parameters Signal to Noise ratio Noise Figure or Noise Factor Effective Noise temperature

  23. Signal to Noise Ratio (SNR) where: PS is the signal power in watts PN is the noise power in watts • Hartley-Shannon Theorem (also called Shannon’s Limit) states that the maximum data rate for a communications channel is determined by a channel’s bandwidth and SNR. • A SNR of zero dB means that noise power equals the signal power.

  24. Noise Figure / Factor (NF or F or Fn) • Electrical noise is defined as electrical energy of random amplitude, phase, and frequency. • It is present in the output of every radio receiver. • The noise is generated primarily within the input stages of the receiver system itself. • Noise generated at the input and amplified by the receiver's full gain greatly exceeds the noise generated further along the receiver chain.

  25. Noise Figure / Factor (NF or F or Fn) • The noise performance of a receiver is described by a figure of merit called the noise figure (NF). Where, G = Antenna Gain

  26. Effective Noise Temperature • The Equivalent noise temperature is defined as “the temperature at which a noisy resistor has to be maintained such that, by connecting the resistor to the input of a noiseless version of the system, it produces the same available noise power at the output of the system as that produced by all the sources of noise in actual system”. It depends only on system parameters Where, T = environmental temperature (Kelvin) N = noise power (watts) K = Boltzmann’s constant (1.38 10 -23 J/K) B = total noise frequency (hertz) Te= equivalent noise temperature F = noise factor (unitless)

  27. Narrow band noise • Preprocessing of received signals • Preprocessing done by a Narrowband Filter • Narrowband Filter – Bandwidth large enough to pass the modulated signal. • Noise also pass through this filter. • The noise appearing at the output of this NB filter is called NARROWBAND NOISE.

  28. Narrow band noise • Fig (a) – spectral components of NB Noise concentrates about +fc • Fig (b) – shows that a sample function n(t) of such process appears somewhat similar to a sinusoidal wave of frequency fc

  29. Narrow band noise • We need a mathematical representation to analyze the effect of this NB Noise. • There are 2 specific representation of NB Noise (depending on the application)

  30. Representation of narrowband noise in terms of In phase and Quadrature Components • Let n(t) is the Narrowband Noise with Bandwidth 2B centered at fc • We can represent n(t) in canonical (standard) form as: • We can extract nI(t) (In Phase Component) and nQ(t) (Quadrature Component) from n(t).

  31. Extraction of nI(t) and nQ(t) from n(t) • Each LPF have bandwidth ‘B’ • This is known as NARROWBAND NOISE ANALYSER

  32. Generation of n(t) from nI(t) and nQ(t) • This is known as NARROWBAND NOISE SYNTHESISER

  33. Important properties of nI(t) and nQ(t)

  34. Power spectral density (PSD) of a Random process • By definition, the power spectral density SX(t) and autocorrelation function RX(Ʈ)of an ergodic random process X(t) form a Fourier transform pairwith Ʈ and f as the variables of interest. • The power of an ergodic random process X(t) is equal to the total area under the graph of power spectral density. • The power spectral density is that characteristic of a random process which is easy to measure and which is used in communication engineering to characterize noise.

  35. Gaussian Process • Any random process X(t) is said to be Gaussian process ,if every linear functional of X(t) is a Gaussian random variable. • Any random variable Y is said to be a Gaussian random variable if its PDF has the form, Where, • A plot of this PDF i.e. Gaussian distribution of random variable is ,

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