TAILIEUCHUNG - DISCRETE-SIGNAL ANALYSIS AND DESIGN- P7

DISCRETE-SIGNAL ANALYSIS AND DESIGN- P7: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. | 16 DISCRETE-SIGNAL ANALYSIS AND DESIGN part and an imaginary part the real parts add coherently and the imaginary parts add coherently and the power is complex real watts and imaginary vars . There is much more about this later. If Kx in Eq. 1-1 then cycles would be visible the spectrum would contain many frequencies and the final phase would change to 2n radians. The value of the phase angle in degrees for each complex X k is W arctan 180 degrees 1-3 Re X k- J n For an example of this type of sequence look ahead to Fig. 1-6. A later section of this chapter gives more details on complex frequency-domain sequences. At this point notice that the complex term exp mt is calculated by Mathcad using its powerful and efficient algorithms eliminating the need for an elaborate complex Taylor series expansion by the user at each value of n or . This is good common sense and does not derail us from our discrete time frequency objectives. At each k stop the sum is performed at 0 to N 1 values of time n for a total of N values. It may be possible to evaluate accurately enough the sum at each k value with a smaller number of time steps say N 2 or N 4. For simplicity and best accuracy N will be used for both k and n . Using Mathcad to find the spectrum without assigning discrete k values from 0 to N 1 a very large number of frequency values are evaluated and a continuous graph plot is created. We will do this from time to time and the summation X becomes more like an integral but this is not always a good idea for reasons to be seen later. Note also that in Eq. 1-2 the factor 1 N ahead of the sum and the minus sign in the exponent are used but are not used in Eq. 1-8 look ahead . This notation is common in engineering applications as described by Ronald Bracewell 1986 and is also an option in Mathcad functions FFT and IFFT . See also Oppenheim and Willsky et al. 1983 p. 321 . This agrees with the practical engineering emphasis of this book. It also agrees with our .

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