DISCRETE-SIGNAL ANALYSIS AND DESIGN- P28:Electronic circuit analysis and design projects often involve time-domain and frequency-domain characteristics that are difÞcult to work with using the traditional and laborious mathematical pencil-and-paper methods of former eras. This is especially true of certain nonlinear circuits and sys- tems that engineering students and experimenters may not yet be com- fortable with. | THE POWER SPECTRUM 121 Figure 7-3 Single-sideband speech power spectrum spectrum analyzer plot. f o 1 kHz and f 0 4 kHz. This kind of display would be difficult to obtain using purely mathematical methods because the long-term spectral components on adjacent channels caused by various mild system nonlinearities combined with a very complicated complex signal would be difficult but not impossible to model accurately. Another instrument the vector network analyzer displays dB amplitude and phase degrees or complex S -parameters in a polar or Smith chart pattern which adds greatly to the versatility in RF circuit design and analysis applications. The important thing is that the signal is sampled in certain fixed and known bandwidths and further analyses of the types that we have been studying such as filtering smoothing and windowing and others both linear and nonlinear can be performed on the data after it has been transferred from the instrument. This processed spectrum information can be transformed to the time or frequency domain for further evaluations. Wiener-Khintchine Theorem Another way to get a two-sided power spectrum sequence is to carry out the following procedures 122 DISCRETE-SIGNAL ANALYSIS AND DESIGN 1. From the x n time sequence calculate the autocorrelation function Ca t using Eq. 6-12 . Note that t is the integer value 0 to N 1 of shift of x n that is used to get Ca t . 2. Perform the DFT on Ca t using Eq. 1-2 to get P k Carlson 1986 Sec. 3 . Note that the shift of t is carried out in steps of over the range from 0 to N 1 in Eq. 7-4 . 4 N 1 Z _ 1 A I T P k F Ca t n 12 cA T expi j2n M 7-4 T 0 This P k spectrum is two-sided and can be converted to one-sided as explained in Chapter 2 and earlier in this chapter. The Wiener-Khintchine theorem is bi-directional and the two-sided autocorrelation Ca t can be found by performing the IDFT Eq. 1-8 on the two-sided P k N 1 Z X Ca t F 1 P k P k expi j2nNM 7-5 k 0 The FFT can be used to expedite the forward
