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DISCRETE-SIGNAL ANALYSIS AND DESIGN- P3: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. | viii CONTENTS Frequency and Time Scaling Number of Samples Complex Frequency-Domain Sequences x n Versus Time and X k Versus Frequency 2 Sine Cosine and 0 27 One-Sided Sequences Combinations of Two-Sided Phasors Time and Spectrum Transformations Transforming Two-Sided Phasor Sequences into One-Sided Sine Cosine 0 Example 2-1 Nonlinear Amplifier Distortion and Square Law Modulator Example 2-2 Analysis of the Ramp Function 3 Spectral Leakage and Aliasing 43 Spectral Leakage. Noninteger Values of Time x n and Frequency X k Example 3-1 Frequency Scaling to Reduce Leakage Aliasing in the Frequency Domain Example 3-2 Analysis of Frequency-Domain Aliasing Aliasing in the Time Domain 4 Smoothing and Windowing 61 Smoothing the Rectangular Window Without Noise and with Noise Smoothed Sequences Near the Beginning and End Rectangular Window Hamming Window Hanning Hann Window Relative Merits of the Three Windows Scaling the Windows 5 Multiplication and Convolution 77 Sequence Multiplication Polynomial Multiplication CONTENTS ix Convolution Discrete Convolution Basic Equation Relating Convolution to Polynomial Multiplication Fold and Slide Concept Circular Discrete Convolution Try to Avoid Sequence Time and Phase Shift DFT and IDFT of Discrete Convolution Fig. 5-6. Compare Convolution and Multiplication Deconvolution 6 Probability and Correlation 95 Properties of a Discrete Sequence Expected Value of x n Include Some Additive Noise Envelope Detection of Noisy Sequence Average Power of Noiseless Sequence Power of Noisy Sequence Sequence Averaging Variance Gaussian Normal Distribution Cumulative Distribution Correlation and Covariance Autocorrelation Cross-Correlation Autocovariance Cross-Covariance Correlation Coefficient 7 The Power Spectrum 113 Finding the Power Spectrum Two-Sided Phasor Spectrum One-Sided Power Spectrum Example 7-1 The Use of Eq. 7-2 Random Gaussian Noise Spectrum Measuring the Power Spectrum Spectrum Analyzer Example Wiener-Khintchine Theorem x CONTENTS .
