DISCRETE-SIGNAL ANALYSIS AND DESIGN- P28

DISCRETE-SIGNAL ANALYSIS AND DESIGN- P28:Electronic circuit analysis and design projects often involve time-domainand frequency-domain characteristics that are difÞcult to work with usingthe traditional and laborious mathematical pencil-and-paper methods offormer eras. This is especially true of certain nonlinear circuits and sys-tems that engineering students and experimenters may not yet be com-fortable with.
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DISCRETE-SIGNAL ANALYSIS AND DESIGN- P28 THE POWER SPECTRUM 121 1 kHz f0 Upper Side Band 10 dBFigure 7-3 Single-sideband speech power spectrum, spectrum analyzerplot.(f 0 − 1) kHz and (f 0 + 4) kHz. This kind of display would be difÞcult toobtain using purely mathematical methods because the long-term spectralcomponents on adjacent channels caused by various mild system non-linearities combined with a very complicated complex signal would bedifÞcult, but not impossible, to model accurately. Another instrument, the vector network analyzer, displays dB ampli-tude and phase degrees or complex S -parameters in a polar or Smithchart pattern, which adds greatly to the versatility in RF circuit designand analysis applications. The important thing is that the signal is sam-pled in certain Þxed and known bandwidths, and further analyses of thetypes that we have been studying, such as Þltering, smoothing and win-dowing and others, both linear and nonlinear, can be performed on thedata after it has been transferred from the instrument. This processed spec-trum information can be transformed to the time or frequency domain forfurther evaluations.Wiener-Khintchine TheoremAnother way to get a two-sided power spectrum sequence is to carry outthe following procedures:122 DISCRETE-SIGNAL ANALYSIS AND DESIGN 1. From the x (n) time sequence, calculate the autocorrelation function C A (τ) using Eq. (6-12). Note that τ is the integer value (0 to N − 1) of shift of x (n) that is used to get C A (τ). 2. Perform the DFT on C A (τ) using Eq. (1-2) to get P (k ) [Carlson, 1986, Sec. 3]. Note that the shift of τ is carried out in steps of 1.0 over the range from 0 to N − 1 in Eq. (7-4). N −1 1 τ P (k) = F[CA (τ)] = CA (τ)exp − j 2π k (7-4) N N τ=0 This P (k ) spectrum is two-sided and can be converted to one-sided asexplained in Chapter 2 and earlier in this chapter. The Wiener-Khintchinetheorem is bi-directional and the two-sided autocorrelation C A (τ) can befound by performing the IDFT [Eq. (1-8)] on the two-sided P (k ): N −1 τ CA (τ) = F−1 [P (k)] = P (k) exp j 2π k (7-5) N k=0The FFT can be used to expedite the forward and reverse Fourier transfor-mations. This method is also useful for sequences that are unlimited (notperiodic) in the time domain, if the autocorrelation function is available.SYSTEM POWER TRANSFERThe autocorrelation and cross-correlation functions can be deÞned in termsof periodic repeating signals, in terms of Þnite nonrepeating signals, and interms of random signals that may be inÞnite and nonrepeating [Oppenheimand Schafer, 1999, Chap. 10]. We have said that for this introductory book we will assume that asequence of 0 to N − 1 of some reasonable length N contains enoughsigniÞcant information that all three types can be calculated to a sufÞcientdegree of accuracy using Eqs. (6-12) and (6-13). We will make N largeenough that circular correlation and circular convolution are not needed.We will continue to assume an inÞnitely repeating process. When a fairlylow value of noise contamination is present, we will perform averaging THE POWER SPECTRUM 123of many sequences to get an improved estimate of the correct values.We will also assume ergodic, wide-sense stationary processes that makeour assumptions reasonable. This means that expected (ensemble) valueand time average are “nearly” equal, especially for Gaussian noise. Wealso assume that windowing and anti-aliasing procedures as explained inChapters 3 and 4 have been applied to keep the 0 to N − 1 sequenceessentially “disconnected” from adjacent sequences. The Hanning andHamming windows are especially good for this. If a linear system, possibly a lossy and complex network, has the com-plex voltage or current input-to-output frequency response H (k ) and ifthe input power spectrum is P (k )in , the output power spectrum P (k )out ina 1.0 ohm resistor can be found using Eq. (7-6) P (k)out = [H (k)H (k)∗ ]P (k)in = |H (k)|2 P (k)in (7-6)where the asterisk(*) means complex conjugate. Because P (k )in andP (k )out are Fourier transforms of an autocorrelation, their values arereal and nonnegative and can be two-sided in frequency [Papoulis, 1965,p. 338]. This is an important fundamental idea in the design and analysisof linear systems. Equation (7-6) is related to the Fourier transform ofconvolution that we studied and veriÞed in Eqs. (5-6) to (5-10). Equation(7-6) for the power domain is easily deduced from that material by includ-ing the complex conjugate of H (k ). To repeat, P (k )in and P (k )out arereal-valued, equal to or greater than zero and two-sided in frequency.CROSS POWER SPECTRUMEquation (7-4) showed how to use the auto-correlation in Eq. (6-12) toÞnd the power spectrum of a single signal using the DFT. In a similarmanner, the cross-spectrum between two signals can be found from theDFT of the cross-correlation in Eq. (6-13). The cross-spectrum evaluatesthe commonality of the power in signals 1 and 2, and phase commonalityis included in the deÞnition. We will now use an example of a pair ofsinusoidal signals to illustrate some interesting ideas. Equation (7-7) compares the average power P 1 for the product of asi ...