Nguyên tắc phân tích tín hiệu ngẫu nhiên và thiết kế tiếng ồn thấp P9

Principles of Random Signal Analysis and Low Noise Design: The Power Spectral Density and Its Applications. Roy M. Howard Copyright ¶ 2002 John Wiley & Sons, Inc. ISBN: 0-471-22617-3 9 Principles of Low Noise Electronic Design 9.1 INTRODUCTION This chapter details noise models and signal theory, such that the effect of noise in linear electronic systems can be ascertained. The results are directly applicable to nonlinear systems that can be approximated around an operating point by an affine function. An introductory section is included at the start of the chapter to provide an insight into the nature of Gaussian white noise—the most common form of noise encountered in electronics. This is followed by a description of the standard types of noise encountered in electronics and noise models for standard electronic components. The central result of the chapter is a system-atic explanation of the theory underpinning the standard method of character-izing noise in electronic systems, namely, through an input equivalent noise source or sources. Further, the noise equivalent bandwidth of a system is defined. This method of characterizing a system, simplifies noise analysis— especially when a signal to noise ratio characterization is required. Finally, the input equivalent noise of a passive network is discussed which is a generaliz-ation of Nyquist’s theorem. General references for noise in electronics include Ambrozy (1982), Buckingham (1983), Engberg (1995), Fish (1993), Leach (1994), Motchenbacher (1993), and van der Ziel (1986). 9.1.1 Notation and Assumptions When dealing with noise processes in linear time invariant systems, an infinite timescale is often assumed so power spectral densities, consistent with previous notation, should be written in the form G( f). However, for notational 256 INTRODUCTION 257 RS + VS Vo − Figure 9.1 Schematic diagram of signal source and amplifier. convenience, the subscript is removed and power spectral densities are written as G( f ). Further, the systems are assumed to be such that the fundamental results, as given by Theorems 8.1 and 8.6, are valid. 9.1.2 The Effect of Noise In electronic devices, noise is a consequence of charge movement at an atomic level which is random in character. This random behaviour leads, at a macro level, to unwanted variations in signals. To illustrate this, consider a signal V , from a signal source, assumed to be sinusoidal and with a resistance R , which is amplified by a low noise amplifier as illustrated in Figure 9.1. The equivalent noise signal at the amplifier input for the case of a 1k source resistance, and where the noise from this resistance dominates other sources of noise, is shown in Figure 9.2. A sample rate of 2.048 kSamples/sec has been used, and 200 samples are displayed. The specific details of the amplifier are described in Howard (1999b). In particular, the amplifier bandwidth is 30kHz. Amplitude (Volts) 4 · 10 6 2 · 10 6 0 −2 · 10 6 −4 · 10 6 0.02 0.04 0.06 0.08 0.1 Time (Sec) Figure 9.2 Time record of equivalent noise at amplifier input. 258 PRINCIPLES OF LOW NOISE ELECTRONIC DESIGN Amplitude (Volts) 0.000015 0.00001 5 · 10 6 0 −5 · 10 6 −0.00001 −0.000015 0.02 0.04 0.06 0.08 0.1 Time (Sec) Figure 9.3 Sinusoid of 100Hz whose amplitude is consistent with a signal-to-noiseratio of 10. In Figure 9.3 a 100Hz sine wave is displayed, whose amplitude is consistent with a signal-to-noise ratio of 10 assuming the noise waveform of Figure 9.2. The addition of this 100Hz sinusoid, and the noise signal of Figure 9.2, is shown in Figure 9.4 to illustrate the effect of noise corrupting the integrity of a signal. For completeness, in Figure 9.5, the power spectral density of the noise referenced to the amplifier input is shown. In this figure, the power spectral Amplitude (Volts) 0.000015 0.00001 5 · 10 6 0 −5 · 10 6 −0.00001 −0.000015 0.02 0.04 0.06 0.08 0.1 Time (Sec) Figure 9.4 100Hz sinusoidalsignal plus noise due to the source resistance and amplifier. The signal-to-noise ratio is 10. GAUSSIAN WHITE NOISE 259 Figure 9.5 Power spectral density of amplifier noise referenced to the amplifier input. density has a 1/f form at low frequencies, and at higher frequencies is constant. For frequencies greater than 10Hz, the thermal noise from the resistor dominates the overall noise. 9.2 GAUSSIAN WHITE NOISE Gaussian white noise, by which is meant noise whose amplitude distribution at a set time has a Gaussian density function and whose power spectral density is flat, that is, white, is the most common type of noise encountered in electronics. The following section gives a description of a model which gives rise to such noise. Since the model is consistent with many physical noise processes it provides insight into why Gaussian white noise is ubiquitous. 9.2.1 A Model for Gaussian White Noise In many instances, a measured noise waveform is a consequence of the weighted sum of waveforms from a large number of independent random processes. For example, the observed randomly varying voltage across a resistor is due to the independent random thermal motion of many electrons. In such cases, the observed waveform z, can be modelled according to z(t) w z (t) z E (9.1) where w is the weighting factor for the ith waveform z, which is from the ith 260 PRINCIPLES OF LOW NOISE ELECTRONIC DESIGN Figure 9.6 One waveform from a binary digital random process on the interval [0,8D]. ensemble E defining the ith random process Z . Here, z is one waveform from a random process Z which is defined as the weighted summation of the random processes Z ,...,Z . Consider the case, where all the random processes Z , ..., Z are identical, but independent, signalling random processes and are defined, on the interval [0, ND], by the ensemble Ez(, . . . ,, t) (t (k 1)D) 1, 1 P[1]0.5 (9.2) where the pulse function is defined according to (t) 1 0tD elsewhere ( f ) Dsinc(fD)e (9.3) All waveforms in the ensemble have equal probability, and are binary digital information signals. One waveform from the ensemble is illustrated in Figure 9.6. One outcome of the random process Z, as defined by Eq. (9.1), has the form illustrated in Figure 9.7 for the case of equal weightings, w 1, D1, and M500. The following subsections show, as the number of waveforms M, increases, that the amplitude density function approaches that of a Gaussian function, and that over a restricted frequency range the power spectral density is flat or ‘‘white’’. 9.2.2 Gaussian Amplitude Distribution The following, details the reasons why, as the number of waveforms, M, comprising the random process increases, the amplitude density function approaches that of a Gaussian function. The waveform defined by the sum of M equally weighted independent binary digital waveforms, as per Eq. (9.1), has the following properties: (1) the amplitudes of the waveform during the intervals [iD, (i 1)D), and [jD,( j1)D), are independent for ij; (2) the amplitude A, in any inter-val [iD, (i 1)D] is, for the case where M is even, from the set ... - tailieumienphi.vn