Notes

Outline

Noise: An Introduction Adapted from a presentation in: Transmission Systems for Communications, Bell Telephone Laboratories, 1970, Chapter 7

Noise: An Introduction What is noise? Waveforms with incomplete information Analysis: how? What can we determine? Example: sine waves of unknown phase Energy Spectral Density Probability distribution function: P(v) Probability density function: p(v) Averages Common probability density functions Gaussian Exponential Noise in the real-world Noise Measurement Energy and Power Spectral densities

Background Material Probability Discrete Continuous The Frequency Domain Fourier Series Fourier Transform

Noise Definition Any undesired signal that interferes with the reproduction of a desired signal Categories Deterministic: predictable, often periodic, noise often generated by machines Random: unpredictable noise, generated by a “stochastic” process in nature or by machines

Random Noise Unpredictable “Distribution” of values Frequency spectrum: distribution of energy (as a function of frequency) We cannot know the details of the waveform only its “average” behavior

Noise analysis Introduction: a sine wave of unknown phase Single-frequency interference n(t) = A sin(wnt + f) A and wn are known, but f is not known We cannot know its value at time “t”

Energy Spectral Density Here the “Energy Spectral Density” is just the magnitude squared of the Fourier transform of n(t) since all of the energy is concentrated at wn and each half of the energy is at ± w since the Fourier transform is based on the complex exponential not sine and cosine.

Probability Distribution The “distribution” of the ‘noise” values Consider the probability that at any time t the voltage is less than or equal to a particular value “v”  The probabilities at some values are easy P(-A) = 0 P(A) = 1 P(0) = ½ The actual equation is: P(vn) = ½ + (1/p)arcsin(vn/A)

Probability Distribution continued The actual equation is: P(vn) = ½ + (1/p)arcsin(v/A) Note that the noise spends more time near the extremes and less time near zero.  Think of a pendulum: It stops at the extremes and is moving slowly near them It move fastest at the bottom and therefore spends less time there. Another useful function is the derivative of P(vn): the “Probability Density Function”, p(vn)   (note the lower case p)

Probability Density Function The area under a portion of this curve is the probability that the voltage lies in that region. This PDF is zero for |vn| > A

Averages Time Average of signals “Ensemble” Average Assemble a large number of examples of the noise signal. (the set of all examples is the “ensemble”) At any particular time (t0) average the set of values of  vn(t0) to get the “Expected Value” of vn When the time and ensemble averages give the same value (they usually do), the noise process is said to be “Ergodic”

Averages (2) Now calculate the ensemble average of our sinusoidal “noise” Which is obviously zero (odd symmetry, balance point, etc. as it should since this noise the has no DC component.)

Averages (3) E[vn] is also known as the “First Moment” of p(vn) We can also calculate other important moments of p(vn).  The “Second Central Moment” or “Variance” (s2) is: Which for our sinusoidal noise is:

Averages (4) Integrating this requires “Integration by parts

Averages (5) Continuing Which corresponds to the power of our sine wave noise Note: s (without the “squared”) is called the “Standard Deviation” of the noise and corresponds to the RMS value of the noise

Common Probability Density Functions: The Gaussian Distribution Central Limit Theorem The probability density function for a random variable that is the result of adding the effects of many small contributors tends to be Gaussian as the number of contributors gets large.

Common Probability Density Functions: The Exponential Distribution Occurs naturally in discrete “Poison Processes” Time between occurrences Telephone calls Packets

Common Noise Signals Thermal Noise Shot Noise 1/f Noise Impulse Noise

Thermal Noise From the Brownian motion of electrons in a resistive material. pn(f) = kT is the power spectrum where: k = 1.3805 * 10-23 (Boltzmann’s constant) and T is the absolute temperature (°Kelvin) This is a “white” noise (“flat” spectrum) From a color analogy White light has all colors at equal energy The probability distribution is Gaussian

Thermal Noise (2) A more accurate model (Quantum Theory) Which corrects for the high frequency roll off (above 4000 GHz at room temperature) The power in the noise is simply Pn = k*T*BW Watts  or Pn = -174 + 10*log10(BW) in dBm (decibels relative to a milliwatt) Note: dB = 10*log10 (P/Pref ) = 20*log10 (V/Vref )

Shot Noise From the irregular flow of electrons Irms = 2*q*I*f  where: q = 1.6 * 10-19 the charge on an electron This noise is proportional to the signal level (not temperature) It is also white (flat spectrum) and Gaussian

1/f Noise Generated by: irregularities in semiconductor doping contact noise Models many naturally occurring signals “speech” Textured silhouettes (Mountains, clouds, rocky walls, forests, etc.) pn(f) =A / f a  (0.8 < a < 1.5)

Impulse Noise Random energy spikes, clicks and pops Common sources Lightning Vehicle ignition systems This is a white noise, but NOT Gaussian Adding multiple sources - more impulse noise An exception to the “Central Limit Theorem”

Noise Measurement The Human Ear Average Performance The Cochlea Hearing Loss Noise Level A-Weighted C-Weighted

Hearing Performance (an average, good, ear) Frequency response is a function of sound level 0 dB here is the threshold of hearing Higher intensities yield flatter response

The Cochlea A fluid-filled spiral vibration sensor Spatial filter: Low frequencies travel the full length High frequencies only affect the near end Cillia: hairs put out signals when moved Hearing damage occurs when these are injured Those at the near end are easily damaged (high frequency hearing loss)

Noise Intensity Levels: The A- Weighted Filter Corresponds to the sensitivity of the ear at the threshold of hearing; used to specify OSHA safety levels (dBA)

An A-Weighting Filter Below is an active filter that will accurately perform A-Weighting for sound measurements Thanks to: Rod Elliott at http://sound.westhost.com/project17.htm

Noise Intensity Levels: The C- Weighted Filter Corresponds to the sensitivity of the ear at normal listening levels; used to specify noise in telephone systems (dBC)

Energy Spectral Density (ESD)

Energy Spectral Density (ESD) and Linear Systems Therefore the ESD of the output of a linear system is obtained by multiplying the ESD of the input by |H(w)|2

Power Spectral Density (PSD) Functions that exist for all time have an infinite energy so we define power as:

Power Spectral Density (PSD-2) As before, the function in the integral is a density.  This time it’s the PSD Both the ESD and PSD functions are real and even functions