tailieunhanh - DISCRETE-SIGNAL ANALYSIS AND DESIGN- P19

DISCRETE-SIGNAL ANALYSIS AND DESIGN- P19: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. | 76 DISCRETE-SIGNAL ANALYSIS AND DESIGN WINDOWING REFERENCES Harris F. J. 1978 On the use of windows for harmonic analysis with the Fourier transform Proc. IEEE Jan. Oppenheim A. V. and R. W. Schafer 1975 Digital Signal Processing McGraw-Hill New York. 5 Multiplication and Convolution Multiplication and convolution are very important operations in discrete sequence operations in the time domain and the frequency domain. We will find that there is an interesting and elegant relationship between multiplication and convolution that is useful in problem solving. MULTIPLICATION For the kinds of discrete time x n or frequencies X k of interest in this book there are two types of multiplication. The n and k values are integers from 0 to N 1. The X k values to be multiplied are phasors that have amplitude frequency and phase attributes and the x n values have amplitude and time attributes. The Mathcad program sorts it all out. Each sequence is assumed by the software to be one realization of an infinite steady-state repetition with all of the significant information available in a single two-sided n or k sequence as explained previously and mentioned here again for emphasis. Discrete-Signal Analysis and Design By William E. Sabin Copyright 2008 John Wiley Sons Inc. 77 78 DISCRETE-SIGNAL ANALYSIS AND DESIGN Sequence Multiplication One type of multiplication is the distributed sequence multiplication seen in Eq. 4-2 and repeated here z n x n y n 0 n N 1 5-1 Each element of z n is the product of each element of x n and the corresponding element of y n . Frequently x n is a weighting factor for the y n value. For example x n can be a window function that modifies a signal waveform y n . Chapter 4 showed some examples that will not be repeated here. The values x n and y n may in turn be functions of one or more parameters of n at each value of n which is grunt work for the computer. We are often interested in the sum z n over the range 0 to N 1 as the sum of the product of each