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DISCRETE-SIGNAL ANALYSIS AND DESIGN- P29

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DISCRETE-SIGNAL ANALYSIS AND DESIGN- P29: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. | 126 Figure 7-4 Phase noise on a test signal sine wave. THE POWER SPECTRUM 127 At the upper left is a noise-free discrete sine wave Vi n at frequency f 1.0 amplitude 1.0 in 128 positions of n . A discrete cosine wave V2 n amplitude 1.0 at the same frequency has some phase noise added 0.1- rnd 1 0.5 . The rnd 1 function creates a random number from 0 to 1 at each position of n . The value 0.5 is subtracted so the random number is then between 0.5 and 0.5. The index of the phase modulation is 0.1. a The plot of the noise-free sine wave. b The plot of the cosine wave with the noise just barely visible. c We now multiply the sine wave and the cosine wave. This multiplication produces the baseband phase noise output VT n and a sine wave of amplitude 0.5 at twice the frequency of the two input waves. We subtract this unwanted wave so that only the phase noise is visible in part c . This is equivalent to a lowpass filter that rejects the times 2 frequency. Note the vertical scale in the graph of part c that shows the phase noise greatly amplified. d We next use the DFT to get the noise spectrum VT k in dB format. At this point we also perform two 3-point smoothing operations on VT k first to get VT 1 k and then to get VT2 k . This operation smoothes the spectrum of VT k so that VT k in the graph in part e is smoothed to VT2 k in the graph in part f . This is postdetection filtering that is used in spectrum analyzers and many other applications to get a smoother appearance and reduce noise peaks it improves readability of the noise shelf value. e Also in part d we perform lowpass filtering 20 log 1 k2 a Butterworth lowpass filter to get VN k . This result is also smoothed two times and the comparison of VN k and VN2 k is seen in the graphs of parts e and f . f Note that in parts e and f the upper level of the phase noise plot VT k and VT2 k is 53 dB below the 0-dB reference level of the test signal V 2 at frequency k 1. This is called the relative noise shelf for the noisy