Solving for the constant no, we obtain n,= Pww( B)/2B, which we put into Eq (73. 20)to get the spectral density as a function of temperature and resistance using Nyquist's result above Swu(w)=n,= Pww(B)/4B= 4kTR2T B/4TB= 2kTR watts/(rad/s)(73.26) Some Examples The parasitic capacitance in the terminals of a resistor may cause a roll-off of about 20 dB/octave in actual resistors [Brown, 1983, P. 139]. At 290K (room temperature), we have 2kT=2X 1.38 X 10-2X 290=0.8 X 10-20 W/Hz due to each ohm [see Ott, 1988. For R= 1 MQ2(1022), Swwdw)=0.8x 10-4. Over a band of10°Hz, we have Pwn(B)=Swww)B=0.8×10-×10=0.8×106w=0.8 uw by Eqs.(73.24)and (73. 26). In practice, parasitic capacitance causes thermal noise to be bandlimited(pink noise). Now consider Fig. 73. 6(b)and let the temperature be 300k,R=10632, C=I pf (1 picofarad = 10-12 farads), and assume L is OH. By Eq (73.26), the thermal noise power is (w)=2kTR=2×1.38×10-23×300×106=828×10-17W/Hz The power across a bandwidth B=10% is Pww(B)= Sww(w)B=8280 X 10-l2W, so the rms voltage is wms [Pw(B)]=91μV. Now let Y(t) be the output voltage across the capacitor. The transfer function can be seen to be H(w) I(w)(1/fwOH{r(w)R+(1/jiO=(1/mO/[R+1/jnO=l/1+jwRCl(wherew)istherouriertransfor of the current). The output psdf [see Eq(73. 22)is Srr(w)=H(w)/(w)=(1/[1+w2R"C21)Sww(w) Integrating Snx(w)=(1/[1+w2R-CD)Swwfw)over all radian frequencies w=2f [see Eq.(73. 21)], we obtain the antiderivative(828 X 10-7)(1/RC)atan(RCw)/2. Upon substituting the limits w= too, this becomes 828 X 1017[m2+m/21/2RC=414×10-(1/2RC)=207×1017×106=2070×10- W/Hz. Then o2=Y()= y(-∞,∞)=2070×10-w,soYm(t)=y=[P(∞,)2=45.5μV. The half-power(cut- off) radian frequency is w=1/RC=10rad/s, orf= w/2T=159.2 kHz. Approximating Snx(w) by the rectangular spectrum Srrw)=no-10< w<10 rad/s(0 elsewhere), we have that RnT)=(w/sinc(wr), which has the first zeros at:|=T, that is =1/(2f)[see Fig. 73. 4(b)]. We approximate the autocorrelation by Rrr(t)=0 for 521/2) Measuring Thermal noise In Fig. 73.7, the thermal noise from a noisy resistor R is to be measured, where R, is the measurement load. The incremental noise power in R over an incremental frequency band of width dfis Pwu(df)=4kTRdf w by Eg(73. 24). Pn(df) is the integral of Sn(w)over df by Egs.(73. 21), where Sn(w)=H(w)PSwn(w), by Eq (73. 22). In this case, the transfer function H(w)is nonreactive and does not depend upon the radian frequency we can factor it out of the integral). Thus, Rr(df)= H()I(2KTR)df=(R/(R+Ri)14KTRdf To maximize the power measured, let RL=R. The incremental available power measured is then Prdf) 4kTR-df/(4R2)=kTdf [see Ott, 1988, P. 201; Gardner, 1990, P. 288; or Peebles, 1987, P. 227]. Thus, we have the result that incremental available power over bandwidth df depends only on the temperature T. Prrdf=kTf (output power over df) Albert Einstein used statistical mechanics in 1906 to postulate that the mean kinetic energy per degree freedom of a particle,(1/2)mE[v(o)) is equal to(1/2)kT, where m is the mass of the particle, yt)is
© 2000 by CRC Press LLC Solving for the constant no , we obtain no = PWW(B)/2B, which we put into Eq. (73.20) to get the spectral density as a function of temperature and resistance using Nyquist’s result above. SWW(w) = no = PWW(B)/4pB = 4kTR2pB/4pB = 2kTR watts/(rad/s) (73.26) Some Examples The parasitic capacitance in the terminals of a resistor may cause a roll-off of about 20 dB/octave in actual resistors [Brown, 1983, p. 139]. At 290K (room temperature), we have 2kT = 2 3 1.38 3 10–23 3 290 = 0.8 3 10–20 W/Hz due to each ohm [see Ott, 1988]. For R = 1 MW (106 W), SWW(w) = 0.8 3 10–14. Over a band of 108 Hz, we have PW W(B) = SW W(w)B = 0.8 3 10–14 3 108 = 0.8 3 10–6 W = 0.8 mW by Eqs. (73.24) and (73.26). In practice, parasitic capacitance causes thermal noise to be bandlimited (pink noise). Now consider Fig. 73.6(b) and let the temperature be 300K, R = 106 W, C = 1 pf (1 picofarad = 10–12 farads), and assume L is 0H. By Eq. (73.26), the thermal noise power is SWW(w) = 2kTR = 2 3 1.38 3 10–23 3 300 3 106 = 828 3 10–17 W/Hz The power across a bandwidth B = 106 is PWW(B) = SWW(w)B = 8280 3 10–12 W, so the rms voltage is Wrms = [PWW(B)]1/2 = 91 mV. Now let Y(t) be the output voltage across the capacitor. The transfer function can be seen to be H(w) = {I(w)(1/jwC)}/{I(w)[R + (1/jwC)]} = (1/jwC)/[R + 1/jwC] = 1/[1 + jwRC] (where I(w) is the Fourier transform of the current). The output psdf [see Eq. (73.22)] is SYY(w) = *H(w)* 2SWW(w) = (1/[1 + w2R2C2 ])SWW(w) Integrating SYY(w) = (1/[1 + w2 R2C2 ])SWW(w) over all radian frequencies w = 2pf [see Eq. (73.21)], we obtain the antiderivative (828 3 10–17)(1/RC)atan(RCw)/2p. Upon substituting the limits w = ±`, this becomes 828 3 10–17[p/2 + p/2]/2pRC = 414 3 10–17(1/2RC) = 207 3 10–17 3 106 = 2070 3 10–12 W/Hz. Then sY 2 = E[Y(t)2 ] = PYY(–`,`) = 2070 3 10–12 W, so Yrms(t) = sY = [PYY(–`,`)]1/2 = 45.5 mV. The half-power (cut-off) radian frequency is wc = 1/RC = 106 rad/s, or fc = wc /2p = 159.2 kHz.Approximating SYY(w) by the rectangular spectrum SYY(w) = no, –106 < w < 106 rad/s (0 elsewhere), we have that RYY(t) = (wc /p)sinc(wct), which has the first zeros at uwctu = p, that is utu = 1/(2fc) [see Fig. 73.4(b)].We approximate the autocorrelation by RYY(t) = 0 for usu ³ 1/2fc . Measuring Thermal Noise In Fig. 73.7, the thermal noise from a noisy resistor R is to be measured, where RL is the measurement load. The incremental noise power in R over an incremental frequency band of width df is PWW(d f ) = 4kTRdf W, by Eq. (73.24). PYY(d f ) is the integral of SYY(w) over df by Eqs. (73.21), where SYY(w) = *H(w)* 2 SWW(w), by Eq. (73.22). In this case, the transfer function H(w) is nonreactive and does not depend upon the radian frequency (we can factor it out of the integral). Thus, To maximize the power measured, let RL = R. The incremental available power measured is then PYY(d f ) = 4kTR2 df /(4R2 ) = kTdf [see Ott, 1988, p. 201; Gardner, 1990, p. 288; or Peebles, 1987, p. 227]. Thus, we have the result that incremental available power over bandwidth df depends only on the temperature T. PY Y(df) = kTdf (output power over df) (73.27) Albert Einstein used statistical mechanics in 1906 to postulate that the mean kinetic energy per degree of freedom of a particle, (1/2)mE[v2 (t)], is equal to (1/2)kT, where m is the mass of the particle, v(t) is its P df H f kTR df R R R kTRdf YY L L df df ( ) = ( ) ( ) = { + ) }( ) -Ú * *2 2 2 /( 4
RL=Load FIGURE 73.7 Measuring thermal noise voltage instantaneous velocity in a single dimension, k is Boltzmanns constant, and T is the temperature in kelvin. A shunt capacitor Cis charged by the thermal noise in the resistor [ see Fig. 73.6(b), where L is taken to be zero] The average potential energy stored is(1/2)CE[ W(()2). Equating this to 1/2kT and solving, we obtain the mean ElW(t)2]=kT/C (73.28) For example, let T= 300K and C=50 pf, and recall that k= 1.38 X 10-23J/K Then E[W(0)]=kT/C=82.8 X 10-13, so that the input rms voltage is IE[W(t112=9.09 uV. Effective noise and Antenna noise Let two series resistors R, and R2 have respective temperatures of T and t. The total noise power over an ncremental frequency band df is Pota(df)=Pi(df)+ P2(df)=4kTRdf+ 4ktRdf=4k(TR,+ tR)df. By putting TE=(T1R1+T2R2)(R1+R2) (73.29) we can write PTotal(df)=4kTE(R,+ R2)df. TE is called the effective noise temperature [see Gardner, 1990, P. 289; or Peebles, 1987, P. 228]. An antenna receives noise from various sources of electromagnetic radiation, such as radio transmissions and harmonics, switching equipment(such as computers, electrical motor controllers), thermal(blackbody) radiation of the atmosphere and other matter, solar radiation, stellar radiation, and galaxial radiation(the ambient noise of the universe). To account for noise at the antenna output, we model the noise with an equivalent thermal noise using an effective noise temperature TE. The incremental available power (output)over an incremental frequency band dfis Prr(df)=kTe df from Eq (73. 27). TE is often called antenna temperature, denoted by T. Although it varies with the frequency band, it is usually virtually constant over a small bandwidth noise Factor and noise ratio In reference to Fig. 73. 8(a), we define the noise factor F=(noise power output of actual device)/(noise power output of ideal device), where(noise power output of ideal device)=(power output due to thermal noise source) The noise source is taken to be a noisy resistor R at a temperature T, and all output noise measurements must be taken over a resistive load R, (reactance is ignored). Letting Pww(B)=4ktRB be the open circuit thermal oise power of the source resistor over a frequency bandwidth B, and noting that the gain of the device is G, the output power due to the resistive noise source becomes G Pww(B)=4kTRBG/RL. Now let Y(o be the output voltage measured at the output across Ri. Then the noise factor is e 2000 by CRC Press LLC
© 2000 by CRC Press LLC instantaneous velocity in a single dimension, k is Boltzmann’s constant, and T is the temperature in kelvin. A shunt capacitor C is charged by the thermal noise in the resistor [see Fig. 73.6(b), where L is taken to be zero]. The average potential energy stored is (1/2)CE[W(t)2 ]. Equating this to 1/2kT and solving, we obtain the mean square power E[W(t)2] = kT/C (73.28) For example, let T = 300K and C = 50 pf, and recall that k = 1.38 3 10–23 J/K. Then E[W(t)2 ] = kT/C = 82.8 3 10–12, so that the input rms voltage is {E[W(t)2 ]}1/2 = 9.09 mV. Effective Noise and Antenna Noise Let two series resistors R1 and R2 have respective temperatures of T1 and T2 . The total noise power over an incremental frequency band df is PTotal(df ) = P11 (df ) + P22(df ) = 4kT1R1df + 4kT2R2df = 4k(T1R1 + T2R2 )df. By putting TE = (T1R1 + T2R2 )/(R1 + R2 ) (73.29) we can write PTotal(df ) = 4kTE (R1 + R2 )df. TE is called the effective noise temperature [see Gardner, 1990, p. 289; or Peebles, 1987, p. 228]. An antenna receives noise from various sources of electromagnetic radiation, such as radio transmissions and harmonics, switching equipment (such as computers, electrical motor controllers), thermal (blackbody) radiation of the atmosphere and other matter, solar radiation, stellar radiation, and galaxial radiation (the ambient noise of the universe). To account for noise at the antenna output, we model the noise with an equivalent thermal noise using an effective noise temperature TE . The incremental available power (output) over an incremental frequency band df is PYY (df) = kTE df, from Eq. (73.27). TE is often called antenna temperature, denoted by TA . Although it varies with the frequency band, it is usually virtually constant over a small bandwidth. Noise Factor and Noise Ratio In reference to Fig. 73.8(a), we define the noise factor F = (noise power output of actual device)/(noise power output of ideal device), where (noise power output of ideal device) = (power output due to thermal noise source). The noise source is taken to be a noisy resistor R at a temperature T, and all output noise measurements must be taken over a resistive load RL (reactance is ignored). Letting PWW (B) = 4kTRB be the open circuit thermal noise power of the source resistor over a frequency bandwidth B, and noting that the gain of the device is G, the output power due to the resistive noise source becomes G2 PWW (B) = 4kTRBG2 /RL . Now let Y(t) be the output voltage measured at the output across RL . Then the noise factor is FIGURE 73.7 Measuring thermal noise voltage
G= Gain kT. RI 凡L T=290K RaT。+T6 W-4kRBn+下 FIGURE 73.8 Equivalent input noise and noise factor. F=(Py(B)/R2)(G2Pw(B)/R1)=(Py(B)/(4kTRBG2) (73.30) Fis seen to be independent of R,, but not R. To compare two noise factors, the same source must be used In the ideal noiseless case, F=l, but as the noise level in the device increases, Fincreases. Because this is a power ratio, we may take the logarithm, called the noise ratio, which NE=10 logo(F=10 logo(Pyx( B))-10 logo(4kTRBG2 (73.31) a The noise power output Pr(B)of an actual device is a superposition of the amplified source thermal noise Pww(B)and the device noise, i. e, Pry (B)=G2Pww(B)+(device noise). The output noise across R, can be measured by putting a single frequency(in the passband) source generator S(n)as input. First, S(n) is turned off, and the output rms voltage Y(o) is measured and the output power Prw (B)is recorded. This is the sum of the thermal available power and the device noise. Next, S(n)is turned on and adjusted until the output power doubles, i.e., until the output power Pr(w(B)+ Pxs)(B)=2Px w(B). This Pss(B) is recorded. Solving for Pns(B)=Prw(B), we substitute this in F= Pr(w(B)/(G Pww(B))to obtain F=Pns (B)/(G2. PwuB))=(GPss(B))/(G4kTRB)=Pss( B)/4kTRB (73.32) a better way is to input white noise w(t) in place of S(t)(a noise diode may be used). The disadvantages of noise factors are(1)when the device has low noise relative to thermal noise the noise factor has value close to 1;(2)a low resistance causes high values; and (3)increasing the source resistance decreases the noise factor while increasing the total noise in the circuit [Ott, 1988, p. 216]. Thus, accuracy is not good. For cascaded devices, the noise factors can be conveniently computed[see Buckingham, 1985, P. 67; or Ott, 1988, p. 228] Equivalent Input Noise shot noise(see below) and other noise can be modeled by equivalent thermal noise that would be generated in an input resistor by increased temperature. Recall that the(maximum)incremental available power(output) in a frequency bandwidth dfis Pwr(f= kTdffrom Eq (73. 27). Figure 73. 8(b) presents the situation. Let the resistor be the noise source at temperature T. with thermal noise W(n). Then ELW(n2]=4kT Rdf, by Eq (73.24 yquist's result). Let the open circuit output noise power at Ri be ely(r2. The incremental available noise power Prr(df) at the output(Ri= R)can be considered to be due to the resistor R having a higher temperature and an ideal (noiseless)device, usually an amplifier. We must find a temperature T at which a pseudothermal e 2000 by CRC Press LLC
© 2000 by CRC Press LLC F = (PYY(B)/RL)/(G 2PWW(B)/RL) = (PYY(B))/(4kTRBG 2) (73.30) F is seen to be independent of RL , but not R. To compare two noise factors, the same source must be used. In the ideal noiseless case, F = 1, but as the noise level in the device increases, F increases. Because this is a power ratio, we may take the logarithm, called the noise ratio, which is NF = 10 log10(F) = 10 log10(PYY(B)) – 10 log10(4kTRBG2) (73.31) The noise power output PYY (B) of an actual device is a superposition of the amplified source thermal noise G2 PWW (B) and the device noise, i.e., PYY (B) = G2PWW (B) + (device noise). The output noise across RL can be measured by putting a single frequency (in the passband) source generator S(t) as input. First, S(t) is turned off, and the output rms voltage Y(t) is measured and the output power PY(W)(B) is recorded. This is the sum of the thermal available power and the device noise. Next, S(t) is turned on and adjusted until the output power doubles, i.e., until the output power PY(W)(B) + PY(S)(B) = 2PY(W)(B). This PSS (B) is recorded. Solving for PY(S)(B) = PY(W)(B), we substitute this in F = PY (W)(B)/(G2PWW (B)) to obtain F = PY( S)(B)/(G2 · PW W(B)) = (G2PSS(B))/(G24kTRB) = PSS(B)/4kTRB (73.32) A better way is to input white noise W(t) in place of S(t) (a noise diode may be used). The disadvantages of noise factors are (1) when the device has low noise relative to thermal noise, the noise factor has value close to 1; (2) a low resistance causes high values; and (3) increasing the source resistance decreases the noise factor while increasing the total noise in the circuit [Ott, 1988, p. 216]. Thus, accuracy is not good. For cascaded devices, the noise factors can be conveniently computed [see Buckingham, 1985, p. 67; or Ott, 1988, p. 228]. Equivalent Input Noise Shot noise (see below) and other noise can be modeled by equivalent thermal noise that would be generated in an input resistor by increased temperature. Recall that the (maximum) incremental available power (output) in a frequency bandwidth df is PWW(df) = kTdf from Eq. (73.27). Figure 73.8(b) presents the situation. Let the resistor be the noise source at temperature To with thermal noise W(t). Then E[W(t)2 ] = 4kToRdf, by Eq. (73.24) (Nyquist’s result). Let the open circuit output noise power at RL be E[Y(t)2 ]. The incremental available noise power PYY(df) at the output (RL = R) can be considered to be due to the resistor R having a higher temperature and an ideal (noiseless) device, usually an amplifier. We must find a temperature Te at which a pseudothermal FIGURE 73.8 Equivalent input noise and noise factor
noise power E[W (02]=4kT Rdf yields the extra"input"noise power. Let V(n)=w(r)+ w(r). Then Pw(df 4kTRdf+ 4kT Rdf=4k(T+ T)RdfW, from Eq (73.24). T is called the equivalent input noise temperature. It related to the noise factor F by T:=290(F-1). In cascaded amplifiers with gains G,, G2,... and equivalent the total equivalent input noise temperature is Te(Total)=Tel T2/G1+ T3/GG+ (73.33) [see Gardner, 1990, P. 289] Other electrical noise Thermal noise and shot noise(which can be modeled by thermal noise with equivalent input noise)are the main noise sources. Other noises are discussed in the following paragraphs. Shot noise In a conductor under an external emf, there is an average flow of electrons, holes, photons, etc. In addition this induced net flow and thermal noise, there is another effect. The potential differs across the boundaries of metallic grains and particles of impurities, and when the kinetic energy of electrons exceeds electrons jump across the barrier. This summed random flow is known as shot noise[see Gardner, 1990, P. 239 Ott, 1988, P. 208. The shot effect was analyzed by Schottky in 1918 as Ih=(2qLd b)n, where q=1.6X 10-19 oulombs per electron, Idc average dc current in amperes, and B=noise bandwidth(Hz) Partition noise Partition noise is caused by a parting of the flow of electrons to different electrodes into streams of randomly varying density. Suppose that electrons from some source S flow to destination electrodes A and B. Let n(A) and n( B) be the average numbers of electrons per second that go to nodes A and B respectively, so that n(S) n(A)+n(B)is the average total number of electrons emitted per second. It is a success when an electron goes to A, and the probability of success on a single trial is P, where P=n(A)/n(S),1-p=n(B)/n(S) (73.34) The current to the respective destinations is I(A)=n(A)a, I(B)=n(B)a, where q is the charge of an electron, that I(A)I(S)=pand I(B)/I(S=l-P. Using the binomial model, the average numbers of successes are E[n(A)]=n(S)p and E[n(B)]=n(S)(1-P). The variance is Var(n(A))=n(S)p(1-P)=Var(n(B))(from the binomial formula for variance). Therefore, substitution yields Var(I(A))=q[n(S)p(I-p)]= gn(S)I(A)I(B)/[I(A)+I(B) 73.35) Partition noise applies to pentodes, where the source is the cathode, A is the anode(success), and B is the grid. For transistors, the source is the emitter, A is the collector, and B represents recombination in the bas In photo devices, a photoelectron is absorbed, and either an electron is emitted (a success)or not. Even a partially silvered mirror can be considered to be a partitioner: the passing of a photon is a success and reflection is a failure. While the binomial model applies to partitions with destinations A and B, multinomial models are analogous for more than two destinations. Flicker, Contact, and Burst Noise B. Johnson first noticed in 1925 that noise across thermionic gates exceeded the expected shot noise at lower frequencies. It is most noticeable up to about 2 kHz. The psdf of the extra noise, called flicker noise, is S(刀)=P/axf,f>0 (73.36) where I is the dc current flowing through the device and fis the positive frequency. Empirical values of a are about 1 to 1.6 for different sources. These sources vary but include the irregularity of the size of macro regions e 2000 by CRC Press LLC
© 2000 by CRC Press LLC noise power E[We (t)2 ] = 4kTeRdf yields the extra “input” noise power. Let V(t) = W(t) + We(t). Then PVV (df) = 4kToRdf + 4kTeRdf = 4k(To + Te)Rdf W, from Eq. (73.24). Te is called the equivalent input noise temperature. It is related to the noise factor F by Te = 290(F – 1). In cascaded amplifiers with gains G1, G2 , . . . and equivalent input noise temperatures Te1 , Te2 , . . ., the total equivalent input noise temperature is Te(Total) = Te1 + Te2 /G1 + Te3 /G1G2 + . . . (73.33) [see Gardner, 1990, p. 289]. Other Electrical Noise Thermal noise and shot noise (which can be modeled by thermal noise with equivalent input noise) are the main noise sources. Other noises are discussed in the following paragraphs. Shot Noise In a conductor under an external emf, there is an average flow of electrons, holes, photons, etc. In addition to this induced net flow and thermal noise, there is another effect. The potential differs across the boundaries of metallic grains and particles of impurities, and when the kinetic energy of electrons exceeds this potential, electrons jump across the barrier. This summed random flow is known as shot noise [see Gardner, 1990, p. 239; Ott, 1988, p. 208]. The shot effect was analyzed by Schottky in 1918 as Ish = (2qIdcB)1/2, where q = 1.6 3 10–19 coulombs per electron, Idc = average dc current in amperes, and B = noise bandwidth (Hz). Partition Noise Partition noise is caused by a parting of the flow of electrons to different electrodes into streams of randomly varying density. Suppose that electrons from some source S flow to destination electrodes A and B. Let n(A) and n(B) be the average numbers of electrons per second that go to nodes A and B respectively, so that n(S) = n(A) + n(B) is the average total number of electrons emitted per second. It is a success when an electron goes to A, and the probability of success on a single trial is p, where p = n(A)/n(S), 1 – p = n(B)/n(S) (73.34) The current to the respective destinations is I(A) = n(A)q, I(B) = n(B)q, where q is the charge of an electron, so that I(A)/I(S) = p and I(B)/I(S) = 1 – p. Using the binomial model, the average numbers of successes are E[n(A)] = n(S)p and E[n(B)] = n(S)(1 – p). The variance is Var(n(A)) = n(S)p(1 – p) = Var(n(B)) (from the binomial formula for variance). Therefore, substitution yields Var(I(A)) = q2[n(S)p(1 – p)] = q2n(S){I(A)I(B)/[I(A) + I(B)]} (73.35) Partition noise applies to pentodes, where the source is the cathode, A is the anode (success), and B is the grid. For transistors, the source is the emitter, A is the collector, and B represents recombination in the base. In photo devices, a photoelectron is absorbed, and either an electron is emitted (a success) or not. Even a partially silvered mirror can be considered to be a partitioner: the passing of a photon is a success and reflection is a failure. While the binomial model applies to partitions with destinations A and B, multinomial models are analogous for more than two destinations. Flicker, Contact, and Burst Noise J.B. Johnson first noticed in 1925 that noise across thermionic gates exceeded the expected shot noise at lower frequencies. It is most noticeable up to about 2 kHz. The psdf of the extra noise, called flicker noise, is S(f) = I2/af, f > 0 (73.36) where I is the dc current flowing through the device and f is the positive frequency. Empirical values of a are about 1 to 1.6 for different sources. These sources vary but include the irregularity of the size of macro regions
of the cathode surface, impurities in the conducting channel, and generation and recombination noise in transistors. In the early days of transistors, this generation-recombination was of great concern because the materials were not of high purity. Flicker noise occurs in thin layers of metallic or semiconducting material, olid state devices, carbon resistors, and vacuum tubes [see Buckingham, 1985, p. 143]. It includes contact nois because it is caused by fluctuating conductivity due to imperfect contact between two surfaces, especially switches and relays. Flicker noise may be high at low frequencies Burst noise is also called popcorn noise: audio amplifiers sound like popcorn popping in a frying pan background(thermal noise). Its characteristic is 1/fn(usually n=2), so its power density falls off rapidly, where f is frequency. It may be problematic at low frequencies. The cause is manufacturing defects in the junction of transistors(usually a metallic impurity) Barkhousen noise is due to the variations in size and orientation of small regions of ferromagnetic material and is especially noticeable in the steeply rising region of the hysteresis loop. There is also secondary emission, photo and collision ionization, etc. Measurement and Quantization Noise Measurement error The measurement X, of a signal X(r) at any t results in a measured value x, = x that contains error, and so is not equal to the true value X,=xr. The probability is higher that the magnitude of e=(x-xr) is closer to zero. The bell-shaped Gaussian probability density f(e)=[1/(2o2"exp(-e/2o) fits the error well. This noise process is stationary over time. The expected value is He=0, the mean-square error is o2, and the rms error is Oe. Its instantaneous power at time t is o2. To see this, the error signal e(t)=(x-xr) has instantaneous power per Q2 of P=e(ti(t=aDlet/r=e(t) (73.37) where r=1 Q2 and i(t) is the current. The average power is the summed instantaneous power over a period of time T, divided by the time, taken in the limit as T-o,i.e Pve=lim(/T)e2(t)dt This average power can be determined by sampling on known signal values and then computing the sample variance(assuming ergodicity: see Gardner[1990, p. 163]). The error and signal are probabilistically independent (unless the error depends on the values of X). The signal-to-noise power ratio is computed by S/N= Psignal/Pave Quantization noise Quantization noise is due to the digitization of an exact signal value v=Mt) captured at sampling time t by an A/D converter. The binary representation is bm-1bm-2.. b,b(an n-bit word). The n-bit digitization has 2" different values possible, from 0 to 2"1. Let the voltage range be R. The resolution is dv=R/2". Any voltage dv. Thi e are distributed over the interval [o, dv] in an equally likely fashion that implies the uniform distribution on [0, dv]. The expected value of e=e, =e(r) at any time is ue=dw2, and the variance is u2=dv/12(the variance of a uniform distribution on an interval [a, b) is o=(b-a)/12). Thus the noise is ws and the power of quantization noise is o2=(e-dv/2)(1/dv)de 73.38) (e-dv/2)/3dv o=[(dv)+(dv)]/24dv dv2/12 e 2000 by CRC Press LLC
© 2000 by CRC Press LLC of the cathode surface, impurities in the conducting channel, and generation and recombination noise in transistors. In the early days of transistors, this generation-recombination was of great concern because the materials were not of high purity. Flicker noise occurs in thin layers of metallic or semiconducting material, solid state devices, carbon resistors, and vacuum tubes [see Buckingham, 1985, p. 143]. It includes contact noise because it is caused by fluctuating conductivity due to imperfect contact between two surfaces, especially in switches and relays. Flicker noise may be high at low frequencies. Burst noise is also called popcorn noise: audio amplifiers sound like popcorn popping in a frying pan background (thermal noise). Its characteristic is 1/f n (usually n = 2), so its power density falls off rapidly, where f is frequency. It may be problematic at low frequencies. The cause is manufacturing defects in the junction of transistors (usually a metallic impurity). Barkhousen and Other Noise Barkhousen noise is due to the variations in size and orientation of small regions of ferromagnetic material and is especially noticeable in the steeply rising region of the hysteresis loop. There is also secondary emission, photo and collision ionization, etc. Measurement and Quantization Noise Measurement Error The measurement Xt of a signal X(t) at any t results in a measured value Xt = x that contains error, and so is not equal to the true value Xt = xT . The probability is higher that the magnitude of e = (x – x T) is closer to zero. The bell-shaped Gaussian probability density f(e) = [1/(2ps2 ]1/2exp(–e2 /2ps) fits the error well. This noise process is stationary over time. The expected value is me = 0, the mean-square error is se 2 , and the rms error is se . Its instantaneous power at time t is se 2 . To see this, the error signal e(t) = (x – xT) has instantaneous power per W of Pi = e(t)i(t) = e(t)[e(t)/R] = e2(t) (73.37) where R = 1 W and i(t) is the current. The average power is the summed instantaneous power over a period of time T, divided by the time, taken in the limit as T Æ `, i.e., This average power can be determined by sampling on known signal values and then computing the sample variance (assuming ergodicity: see Gardner [1990, p. 163]). The error and signal are probabilistically independent (unless the error depends on the values of X). The signal-to-noise power ratio is computed by S/N = Psignal /Pave . Quantization Noise Quantization noise is due to the digitization of an exact signal value vt = v(t) captured at sampling time t by an A/D converter. The binary representation is bn–1bn–2 . . . b1b0 (an n-bit word). The n-bit digitization has 2n different values possible, from 0 to 2n –1. Let the voltage range be R. The resolution is dv = R/2n . Any voltage vt is coded into the nearest lower binary value xb , where the error e = xt – xb satisfies 0 £ e £ dv. Thus, the errors e are distributed over the interval [0, dv] in an equally likely fashion that implies the uniform distribution on [0, dv]. The expected value of e = et = e(t) at any time is me = dv/2, and the variance is me 2 = dv2/12 (the variance of a uniform distribution on an interval [a,b] is s = (b – a)2 /12). Thus the noise is ws and the power of quantization noise is (73.38) P T e t dt T T ave = Æ• Ú lim( / ) ( )1 2 0 se dv dv e dv dv de e dv dv dv dv dv dv 2 2 0 3 0 33 2 2 1 2 3 24 12 = - =- = + = Ú ( )(/ ) ( ) [( ) ( ) ]/ / // / *