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 ( )(/ ) ( ) [( ) ( ) ]/ / // / *