Theory Communication

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Theory Communication

  1. 1. CHAPTER 4: NOISE Prepared by: DR NOORSALIZA BINTI ABDULLAH DEPARTMENT OF COMMUNICATION ENGINEERING FACULTY OF ELECTRICAL AND ELECTRONIC ENGINEERING
  2. 2. DEFINATION OF RANDOM VARIABLES A real random is mapping from the sample space Ω (or S) to the set of real numbers. A schematic diagram representing a random variable is given below Ω ω1 ω2 ω3 X (ω1 ) X (ω2 ) ω4 X (ω3 ) X (ω4 ) Figure 4.1 : Random variables as a mapping from Ω to R Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  3. 3. A random variable, usually written X, is a variable whose possible values are numerical outcomes of a random phenomenon, etc.; individuals values of the random variable X are X(ω). There are two types of random variables, which is Discrete Random Variables and Continuous Random Variables. Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  4. 4. Discrete Random Variables A sample space is discrete if the number of its elements are finite or countable infinite, i.e., a discrete random variable is one which may take on only a countable number of distinct values such as 0,1,2,3,4,........ Examples of discrete random variables include the number of children in a family, the Friday night attendance at a cinema, the number of patients in a doctor's surgery, the number of defective light bulbs in a box of ten. A non-discrete sample space happens when the sample space of the random experiment is infinite and uncountable. Example of non-discrete sample space is randomly chosen number from 0 to 1 (continuous random variables). Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  5. 5. Continuous Random Variables A continuous random variable is one which takes an infinite number of possible values. Continuous random variables are usually measurements. Examples include height, weight, the amount of sugar in an orange, the time required to run a mile. A continuous random variable is not defined at specific values. Instead, it is defined over an interval of values, and is represented by the area under a curve (in advanced mathematics, this is known as an integral). The probability of observing any single value is equal to 0, since the number of values which may be assumed by the random variable is infinite. Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  6. 6. Figure 4.2 : Random variables (a) continuous (b) discrete. Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  7. 7. Example 4.1 Which of the following random variables are discrete and which are continuous? c) X = Number of houses sold by real estate developer per week? X = Number of heads in ten tosses of a coin? X = Weight of a child at birth? d) X = Time required to run 100 yards? a) b) Answer: (a) Discrete (b) Discrete (c) Continuous (d) Continuous Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  8. 8. SIGNALS: DETERMINISTIC VS. STOCHASTIC DETERMINISTIC SIGNALS Most introductions to signals and systems deal strictly with deterministic signals as shown in Figure 4.3. Each value of these signals are fixed and can be determined by a mathematical expression, rule, or table. Because of this, future values of any deterministic signal can be calculated from past values. For this reason, these signals are relatively easy to analyze as they do not change, and we can make accurate assumptions about their past and future behavior. Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  9. 9. RANDOM SIGNALS Random signals cannot be characterized by a simple, welldefined mathematical equation and their future values cannot be predicted. Rather, we must use probability and statistics to analyze their behavior. Also, because of their randomness as shown in Figure 4.4, average values from a collection of signals are usually studied rather than analyzing one individual signal. Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  10. 10. Deterministic Signal Figure 4.3: An example of a deterministic signal, the sine wave. Random Signal Figure 4.4: We have taken the above sine wave and added random noise to it to come up with a noisy, or random, signal. These are the types of signals that we wish to learn how to deal with so that we can recover the original sine wave. Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  11. 11. RANDOM PROCESSES As mentioned before, in order to study random signals, we want to look at a collection of these signals rather than just one instance of that signal. This collection of signals is called a random process. Is an extension of random variables Also known as Stochastic Process Model Random Signal and Random Noise Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  12. 12. Outcome of a random experiment is a function An indexed set of random variables Typically the index is time in communications The difference between random variable and random process is that for a random variable, an outcome is the sample space mapped into a number, whereas for a random process it is mapped into a function of time. Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  13. 13. Figure 4.5: Example of random process represent the temperature of a city at 20 hours. Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  14. 14. POWER SPECTRAL DENSITY Random process is a collection of signals, and the spectral characteristics of these signals determine the spectral characteristic of the random process. Slow varying signals (of a random process) have power concentrated at low frequencies. Fast changing signals (of a random process) have power concentrated at high frequencies. Power spectral density determines the power distribution (or power spectrum) of the random process. PSD of a random process X(t) is denoted by SX(f), denotes the strength of power in the random process as a function of frequency. Units for PSD is Watts/Hz. Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  15. 15. RELATIONSHIP OF RANDOM PROCESS AND NOISE Unwanted electric signals come from variety of sources, generally classified as human interference or naturally occurring noise. Human interference comes from other communication systems and the effects of many unwanted signals can be reduced or eliminated completely. However there always remain inescapable random signals, that present a fundamental limit to systems performance. Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  16. 16. THERMAL NOISE Thermal noise is the noise resulting from the random motion of electrons in a conducting medium. Thermal noise arises from both the photodetector and the load resistor. Amplifier noise also contributes to thermal noise. A reduction in thermal noise is possible by increasing the value of the load resistor. However, increasing the value of the load resistor to reduce thermal noise reduces the receiver bandwidth. Figure 4.6 Fluctuating voltage produced by random movements of mobile electrons. Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  17. 17. GAUSSIAN PROCESS Gaussian process is important in communication systems. The main reason is that thermal noise in electrical devices produced by movement of electrons due to thermal agitation is closely modeled by a Gaussian process. Due to the movements of electrons, sum of small currents of a very large number of sources was introduced. Since majority sources are independent, hence the total current is sum of large number of random variables. Therefore the total currents has Gaussian distribution. Figure 4.7 Histogram of some noise voltage measurements Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  18. 18. Definition A random process X(t) is a Gaussion process if for all n and all (t1,t2,…,tn) the random variable {X(ti)}ni=1 have a jointly Gaussian density function. Gaussian or Normal Random Variables ( x − m )2 − where m = mean 1 2σ 2 (4.1) f X ( x) = e σ = standard deviation 2πσ σ2 = variance A Gaussian random variable with mean m and variance σ2 is denoted by Ν(m, σ2) Assuming X is a standard normal random variable, we defined the function Q(x) as P(X > x). The Q function is given by relation 2 Q( x) = P( X > x) = ∫ ∞ x 1 − t2 e dt 2π (4.2) Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  19. 19. The Q function represent the area under the tail of a standard random variable. It is well tabulated and used in analyzing the performance of communication system. Q(x) satisfy the following relations: (4.3a) Q(-x) = 1 – Q(x) (4.3b) Q(0) = ½ Q(∞) = 0 (4.3c) Table 3.1 gives the value of this function for various value of x. For Ν(m, σ2) random variable, a simple change of variable in the integral that computes P(X > x) results in P(X > x) = Q[(x – m)/σ]. tail probability in Gaussian random variable. Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  20. 20. Figure 4.8: The Q-function as the area under the tail of a standard normal random variable. Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  21. 21. Table 4.1 Table of the Q function Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  22. 22. Example 4.2 X is a Gaussian random variable with mean 1 and variance 4. Find the probability X between 5 and 7. Ans. We have m = 1 and σ = √4 = 2. Thus, P( 5 < X < 7) = P (X > 5) – P(X > 7) = Q ((5 – 1)/2) – Q((7 – 1)/2) = Q(2) – Q(3) ≈ 0.0214 Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  23. 23. WHITE NOISE There are many ways to characterize different noise sources, one is to consider the spectral density, that is, the mean square fluctuation at any particular frequency and how that varies with frequency. In what follows, noise will be generated that has spectral densities that vary as powers of inverse frequency, more precisely, the power spectra P(f) is proportional to 1 / fβ for β ≥ 0. When β = 0 the noise is referred to white noise, when β = 2, it is referred to as Brownian noise, and when it is 1 it normally referred to simply as 1/f noise which occurs very often in processes found in nature. White process is a process in which all frequency component appear with equal power, i.e. power spectral density is constant for all frequencies. A process X(t) is called a white process if it has a flat spectral density,i.e., if SX(f) is constant for all f. Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  24. 24. White Noise, β = 0 β =0 β =2 Brownian noise white noise β =1 β =3 1/f noise Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  25. 25. White Noise Spectral density of white noise is a constant, N0/2 N0 SX ( f ) = 2 SX (f) N0 2 f 0 (3.4) White noise power spectrum - Where N0 = kT RXX ( ) Autocorrelation function: N0 2 ⎛N ⎞ RXX (τ ) = F −1 ⎜ 0 ⎟ ⎝ 2 ⎠ ∞ = N 0 j 2π ft ∫ 2 e df −∞ N = 0 δ (τ ) 2 k = Boltzmann’s constant = 1.38 x 0 (3.5) 10-23 White noise autocorrelation - Figure 4.9: White noise (a) power spectrum (b) autocorrelation Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  26. 26. Properties of Thermal Noise Thermal noise is a stationary process Thermal noise is a zero-mean process Thermal noise is a Gaussian process Thermal noise is a white noise with power spectral density SX(f)=kT/2=Sn(f)=N0/2. It is clear that power spectral density of thermal noise increase with increasing the ambient temperature, therefore, keeping electric circuit cool makes their noise level low. Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  27. 27. TYPE OF NOISE Noise can be divided into : 2 general categories Correlated noise – implies relationship between the signal and the noise, exist only when signal is present Uncorrelated noise – present at all time, whether there is signal or not. Under this category there are two broad categories which are:i) Internal noise ii) External noise Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  28. 28. UNCORRELATED NOISE Can be divided into 2 categories 1. External noise Generated outside the device or circuit Three primary sources are atmospheric, extraterrestrial and man made (a) Atmospheric Noise Naturally occurring electrical disturbance originate within Earth’s atmosphere Commonly called static electricity Most static electricity is naturally occurring electrical conditions, such as lighting In the form of impulse, spread energy through wide range of frequency Insignificant at frequency above 30 MHz Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  29. 29. (b) Extraterrestrial Noise Consists of electrical signals that originate from outside earth atmosphere, deep-space noise Divide further into two (i) Solar noise – generated directly from sun’s heat. There are 2 parts to solar noise:Quite condition when constant radiation intensity exist and high intensity Sporadic disturbance caused by sun spot activities and solar flare-ups which occur every 11 years (ii) Cosmic noise – continuously distributed throughout the galaxies, small noise intensity because the sources of galactic noise are located much further away from sun. It's also often called as black-body noise. Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  30. 30. (c) Man-made noise Source – spark-producing mechanism such as from commutators in electric motors, automobile ignition etc Impulsive in nature, contains wide range of frequency that propagate through space the same manner as radio waves Most intense in populated metropolitan and industrial areas and is therefore sometimes called industrial noise. Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  31. 31. (d) Impulse noise High amplitude peaks of short duration in the total noise spectrum. Consists of sudden burst of irregularly shaped pulses. More devastating on digital data, Produce from electromechanical switches, electric motor etc. (e) Interference External noise Signal from one source interfere with another signal. It occurs when harmonics or cross product frequencies from one source fall into the passband of the neighboring channel. Usually occurs in radio-frequency spectrum Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  32. 32. 2. Internal noise Generated within a device or circuit. 3 primary kinds, shot noise, transit-time noise and thermal noise (a) Shot noise Caused by random arrival of carriers (hole and electron) at the output element of an electronic device such as diode, field effect transistor or bipolar transistor. The currents carriers (ac and dc) are not moving in a continuous, steady flow, as the distance they travel varies because of their random paths of motion. Shot noise randomly varying and is superimposed onto any signal present. When amplified, shot noise sounds similar to metal pellets falling on a tin roof. Sometimes called transistor noise Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  33. 33. (b) Transit-time noise (Ttn) Any modification to a stream of carriers as they pass from the input to the output of a device produce irregular, random variation (emitter to the collector in transistor). Time it takes for a carrier to propagate through a device is an appreciable part of the time of one cycle of the signal , the noise become noticeable. Ttn is transistors is determined by carrier mobility, bias voltage, and transistor construction. Carriers traveling from emitter to collector suffer from emitter delay, base Ttn,and collector recombination-time and propagation time delays. If transmit delays are excessive at high frequencies, the device may add more noise than amplification of the signal. Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  34. 34. (c) Thermal noise Due to rapid and random movement of electrons within a conductor due to thermal agitation Present in all electronic components and communication system. Uniformly distributed across the entire electromagnetic frequency spectrum, often referred as white noise. Form of additive noise, meaning that it cannot be eliminated , and it increases in intensity with the number of devices and circuit length. Set as upper bound on the performance of communication system. Temperature dependent, random and continuous and occurs at all frequencies. Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  35. 35. Noise Spectral Density In communications, noise spectral density No is the noise power per unit of bandwidth; that is, it is the power spectral density of the noise. It has units of watts/hertz, which is equivalent to watt-seconds or joules. If the noise is white, i.e., constant with frequency, then the total noise power N in a bandwidth B is BNo. This is utilized in Signal-to-noise ratio calculations. The thermal noise density is given by No = kT, where k is Boltzmann's constant in joules per kelvin, and T is the receiver system noise temperature in kelvin. No is commonly used in link budgets as the denominator of the important figure-of-merit ratios Eb/No and Es/No. Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  36. 36. NOISE POWER Noise power is given as N0 df −B 2 = N0 B PN = ∫ B and can be written as PN = kTB [W] (3.6) (3.7) where PN = noise power, k = Boltzmann’s constant (1.38x10-23 J/K) B = bandwidth, T = absolute temperature (Kelvin)(17oC or 290K) Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  37. 37. NOISE VOLTAGE Figure 4.10 shows the equivalent circuit for a thermal noise source. Internal resistance RI in series with the rms noise voltage VN. For the worst condition, the load resistance R = RI , noise voltage dropped across R = half the noise source (VR=VN/2) and From equation 4.5 the noise power PN , developed across the load resistor = kTB Figure 4.10 : Noise source equivalent circuit The mathematical expression : PN (V = kTB = N / 2) V N2 = R 4R 2 (4.8a) V N2 = 4 RkTB VN = 4 RkTB (4.8b) Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  38. 38. OTHER NOISE SOURCES 1. 2. 3. There are 3 other noise mechanisms that contribute to internally generated noise in electronic devices. Generation-Recombination Noise - The result of free carriers being generated and recombining in semiconductor material. Can consider these generation and recombination events to be random. This noise process can be treated as shot noise process. Temperature-Fluctuation Noise – The result of the fluctuating heat exchange between a small body, such as transistor, and it’s environment due to the fluctuations in the radiation and heat-conduction processes. If a liquid or gas is flowing past the small body, fluctuation in heat convection also occurs. Flicker Noise – It is characterized by a spectral density that increases with decreasing frequency. The dependence on spectral density on frequency is often found to be proportional to the inverse first power of the frequency. Sometimes referred as one-over-f noise. Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  39. 39. Example 4.3 Calculate the thermal noise power available from any resistor at room temperature (290 K) for a bandwidth of 1 MHz. Calculate also the corresponding noise voltage, given that R = 50 Ω. Ans a) Thermal noise power b) Noise voltage N = kTB VN = = 1.38 × 10 − 23 × 290 × 1×10 6 = 4 × 50 × 4 × 10 = 0 . 895 μ V = 4 × 10 −15W 4 RkTB − 15 Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  40. 40. Example 4.4 For an electronic device operating at a temperature of 17 oC with a bandwidth of 10 kHz, determine a) Thermal noise power in watts and dBm b) rms noise voltage for a 100 Ω internal resistance and 100 Ω load resistance. Ans. N = 1.38 ×10 −23 × 290 × 10 ×103 b) V = 4 RkTB a) N = 4.002 × 10 −17 W ⎛ 4 × 10−17 ⎞ N (dBm ) = 10 log⎜ ⎟ ⎜ 1 × 10− 3 ⎟ ⎠ ⎝ = −134dBm = 4 × 100 × 4 ×10 −17 = 0.127 μV (rms ) Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  41. 41. Example 4.5 Two resistor of 20 kΩ and 50 kΩ are at room temperature (290 K). For a bandwidth of 100 kHz, calculate the thermal noise voltage generated by 1. each resistor 2. the two resistor in series 3. the two resistor in parallel Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  42. 42. Ans. a) VN 2 = 4 R2 kTB VN 1 = 4 R1kTB − 23 3 3 =    4 × 20 × 103 × 1.38 × 10 − 23 × 290 × 100 × 103    4 × 50 ×10 ×1.38 ×10 × 290 ×100 ×10 = =  8.95 ×10 −6 V    5.66 × 10 − 6V = b) RT= 20 × 10 + 50 × 10 = 70 × 10 Ω 3 V Ntotal = 3 3 4 RT kTB    4 × 70 × 10 3 × 1 .38 × 10 − 23 × 290 × 100 × 10 3 =    1 .06 × 10 − 5 V = ( 20 + 50 )10 3 c) RT= = 14 .28 kΩ 3 3 20 × 50 × 10 × 10 V Ntotal = 4 RT kTB    4 × 14 .29 k × 1.38 × 10 − 23 × 290 × 100 × 10 3 =   4.78 μV = Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  43. 43. CORRELATED NOISE 1. Mutually related to the signal, not present if there is no signal Produced by nonlinear amplification, and include nonlinear distortion such as harmonic and intermodulation distortion Harmonic Distortion (HD) Harmonic distortion – unwanted harmonics of a signal produced through nonlinear amplification (nonlinear mixing). Harmonics are integer multiples of the original signal. There are various degrees of harmonic distortion. 2nd order HT, ratio of the rms amplitude of the second harmonic to the rms amplitude of the fundamental. 3rd oder HT, ratio of the rms amplitude of the third harmonic to the rms amplitude of the fundamental. Total harmonic distortion (THD), ratio of the quadratic sum of the rms values of all the higher harmonics to the rms value of the fundamental. Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  44. 44. Figure 4.11(a) show the input and output frequency spectrums for a nonlinear device with a single input frequency f1. Mathematically, THD is %THD = vhigher vfundamental x100 Where, %THD = percent total harmonic distortion vhigher = quadratic sum of the rms 2 2 2 voltages, v2 + v3 + vn vfundamental = rms voltage of the fundamental frequency (4.9) Figure 4.11: Correlated noise: (4.10) (a) Harmonic distortion (b) Intermodulation distortion Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  45. 45. 2. Intermodulatin Distortion (ID) Intermodulation distortion is the generation of unwanted sum and difference frequency when two or more signal are amplified in a nonlinear device such as large signal amplifier. The sum and difference frequencies are called cross products. Figure 4.11(b) show the input and output frequency spectrums for a nonlinear device with two input frequencies (f1 and f2). Mathematically, the sum and difference frequencies are (4.11) Cross products =mf1 ± nf2 Where f1 and f2 = fundamental frequencies, f1 > f2 m and n = positive integers between one and infinity Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  46. 46. Example 4.6 Determine 2nd, 3rd and 12th harmonics for a 1 kHz repetitive wave. b) Percent 2nd order, 3rd order and total harmonic distortion for a fundamental frequency with an amplitude of 8 Vrms, a 2nd harmonic amplitude of 0.2 Vrms and a 3rd harmonic amplitude of 0.1 Vrms. a) Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  47. 47. Ans a) b) 2nd harmonic = 2×fundamental freq. = 2×1 kHz =2 kHz 3rd harmonic = 3×fundamental freq. = 3×1 kHz =3 kHz 12th harmonic = 12×fundamental freq. = 12×1 kHz =12 kHz % 2nd V2 0.2 × 100 = 2.5% order = × 100 = V1 8 % 3rd order = V3 0.1 × 100 = ×100 = 1.25% V1 8 0.2 2 + 0.12 × 100% = 2.795% % THD = 8 Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  48. 48. Example 4.7 For a nonlinear amplifier with two input frequencies, 3 kHz and 8 kHz, determine, a) First three harmonics present in the output for each input frequency. b) Cross product frequencies for values of m and n of 1 and 2. Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  49. 49. Ans f1 = 8 kHz, f2 = 3 kHz a) For freqin =3kHz 1st harmonic = original signal freq. = 3 kHz 2nd harmonic = 2× original signal freq. = 2×3 kHz =6 kHz 3rd harmonic = 3× original signal freq. = 3×3 kHz =9 kHz For freqin =8kHz 1st harmonic = original signal freq. = 8 kHz 2nd harmonic = 2× original signal freq. = 2×8 kHz =16 kHz 3rd harmonic = 3× original signal freq. = 3×8 kHz =24 kHz b) m 1 n 1 8±3 Cross Product 5kHz and 11kHz 1 2 2 2 1 2 8±6 16±3 16±6 2kHz and 14kHz 13kHz and 19kHz 10kHz and 22kHz Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  50. 50. Table 4.2 Electrical Noise Source Summary Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  51. 51. SIGNAL-TO-NOISE RATIO (SNR) Signal-to-noise power ratio (S/N) is the ratio of the signal power level to the noise power Mathematically, S P = S N PN where, (4.12) PS = signal power (watts) PN = noise power (watts) In dB S PS (dB) = 10log N PN (4.13) Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  52. 52. If the input and output resistances of the amplifier, receiver, or network being evaluated are equal ⎛V ⎞ ⎛V ⎞ S ( dB ) = 10 log ⎜ s 2 ⎟ = 10 log ⎜ s ⎟ N ⎝ Vn ⎠ ⎝ Vn ⎠ 2 ⎛ Vs ⎞ = 20 log ⎜ ⎟ ⎝ Vn ⎠ where 2 (4.14) Vs = signal voltage (volts) Vn = noise voltage (volts) Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  53. 53. Example 4.8 For an amplifier with an output signal power of 10 W and an output noise power of 0.01W, determine the S/N. Ans S/N = 10 = 1000    [unitless ] 0.01 S / N ( dB ) = 10 log 1000 = 30 [ dB ] Example 4.9 For an amplifier with an output signal voltage of 4 V, an output noise voltage of 0.005 V and an input and output resistance of 50 Ω, determine the S/N. Ans Vs S/N = VN 2 R = 2 42 = 640000   [unitless] 0.0052 S / N ( dB ) = 10 log 640000 = 58 [ dB ] R Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  54. 54. NOISE FACTOR (F) & NOISE FIGURE (NF) Noise factor and noise figure are figures of merit to indicate how much a signal deteriorate when it pass through a circuit or a series of circuits Noise factor F= Noise figure input signal-to-noise ratio [unitless] output signal-to-noise ratio input signal-to-noise ratio [dB] NF = 10log output signal-to-noise ratio = 10log F (4.15) (4.16) For perfect noiseless circuit, F = 1, NF = 0 dB Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  55. 55. For ideal noiseless amplifier with a power gain (AP), an input signal power level (Si) and an input noise power level (Ni) as shows in Figure 4.12 (a). The output signal level is simply APSi, and the output noise level is APNi. Ap Si Sout Si = = N out Ap N i N i [unitless] (4.17) Figure 4.12 (b) shows a nonideal amplifier that generates an internal noise Nd Ap Si Sout Si = = N out Ap N i + N d N i + N d Ap [unitless] (4.18) Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  56. 56. Figure 4.12 Noise Figure: (a) ideal, noiseless device (b) amplifier with internally generated noise Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  57. 57. When two or more amplifiers are cascaded as shown in Figure 4.13, the total noise factor is the accumulation of the individual noise factors. Friiss’ formula is used to calculate the total noise factor of several cascaded amplifiers. Mathematically, Friiss formula is Fn − 1 F2 − 1 F3 − 1 [unitless] (4.19) FT = F1 + A1 + A1 A2 + A1 A2 ..... An −1 Figure 4.13 Noise figure of cascaded amplifiers Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  58. 58. Where FT = total noise factor for n cascaded amplifiers F1, F2, F3…n = noise factor, amplifier 1,2,3…n A1, A2…. An = power gain, amplifier 1,2,…..n Notification remarks Change unit of all noise factors F and power gains A from [dB] to [unitless] before insert its into Friss formula equation Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  59. 59. Example 4.10 The input signal to a telecommunications receiver consists of 100 μW of signal power and 1 μW of noise power. The receiver contributes an additional 80 μW of noise, ND, and has a power gain of 20 dB. Compute the input SNR, the output SNR and the receiver’s noise figure. Ans. a) Input SNR = Si 100 × 10 -6 = = 100 [ unitless ] -6 Ni 1 × 10 Input SNR(dB) = 10 log 100 = 20 [ dB ] Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  60. 60. b) The output noise power = internal noise + amplified input noise N out = N D + Ap N i = 80 μW + (100 × 1× 10 −6 W )    1.8 × 10 − 4 [W ] = The output signal power = amplified input signal S out = Ap Si = 100 × 100 × 10 −6    × 10 − 2 [W ] =1 S out 1× 10 -2 Output SNR= = = 55.56[unitless ] -4 N out 1.8 × 10 Output SNR(dB) = 10 log 55 .56 = 17 .45[ dB ] Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  61. 61. c) Noise Figure NF = 10 log input SNR[unitless ] 100 = 10 log output  SNR[unitless ] 55.56 = 2 . 55 [ dB ] Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  62. 62. Example 4.11 For a non-ideal amplifier and the following parameters, determine Input signal power = 2 x 10-10 W Input noise power = 2 x 10-18 W Power Gain = 1,000,000 Internal Noise (Nd) = 6 x 10-12 W a. b. c. Input S/N ratio (dB) Output S/N ratio (dB) Noise factor and noise figure Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  63. 63. Ans a) Input SNR S i 2 × 10 -10 = = 1 × 10 8 [unitless ] N i 2 × 10 -18 10 Input SNR(dB) = log 100000000 = 80 [ dB ] b) The output noise power Nout = ND + Ap Ni = 6×10−12 + (1×106 × 2×10−18)     ×10−12[W ] =8 The output signal power S out = Ap Si = 1×106 × 2 ×10 −10     2 ×10 − 4 [W ] = 2 × 10 -4 = 74 [ dB ] Output SNR(dB) = 10 log - 12 8 × 10 Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  64. 64. c) Noise factor F = input SNR [ unitless ] 100000000 = = 4[ unitless ] output  SNR [unitless ] 25000000 Noise figure NF = 10 log 4 = 6.02[ dB ] Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  65. 65. Example 4.12 For three cascaded amplifier stages, each with noise figures of 3 dB and power gains of 10 dB, determine the total noise figure. Ans. Change all noise figure and power gain from [dB] unit to [unitless] 10 Power gain A = A = A = 10 10 = 10[unitless ] 1 2 3 3 10 Noise Factor F1 = F2 = F3 = 10 = 2[unitless ] Using Friss formula , F − 1 F3 − 1 + [unitless ] Total noise factor FT = F1 + 2 A1 A1 A2 2 −1 2 −1 = 2+ + 10 10 × 10 [unitless ] = 2.11        Total noise figure NFT = 10 log 2 . 11 = 3 . 24 [ dB ] Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  66. 66. EQUIVALENT NOISE TEMPERATURE (Te) The noise produced from thermal agitation is directly proportional to temperature, thermal noise can be expressed in degrees as well as watts or dBm. Mathematically, N T= KB (4.20) where T = environmental temperature (kelvin) N = noise power (watts) K = Boltzmann’s constant (1.38 x 10-23 J/K) B = bandwidth (hertz) Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  67. 67. Te is a hypothetical value that cannot be directly measured Convenient parameter often used . It’s also indicates reduction in the signal-to-noise ratio a signal undergoes as it propagates through a receiver. The lower the Te , the better the quality of a receiver. Typically values for Te , range from (20 K – 1000 K) for noisy receivers. Mathematically, (4.21) Te = T (F − 1) Where Te =equivalent noise temperature (kelvin) T = environmental temperature (290 K) F = noise factor (unitless) Conversely, F can be represented as a function of Te : Te F =1+ T (4.22) Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia
  68. 68. Example 4.13 Determine, a) Noise figure for an equivalent noise temperature of 75 K. b) Equivalent noise temperature for noise figure of 6 dB. Ans. a) Noise factor F = 1 + Te = 1 + 75 = 1 .258 [unitless ] 290 T Noise figure NF = 10 log 1 . 258 = 1[ dB ] b) Noise factor F = anti log( NF 6 ) = anti log( ) = 4[unitless] 10 10 Equivalent noise temperature Te = T ( F − 1) = 290 ( 4 − 1)    870[ K ] = Dept. Of Communication Engineering, Faculty Of Electrical And Electronics, Universiti Tun Hussein Onn Malaysia

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