Z80 technical manual by Joseph J. Carr PDF

By Joseph J. Carr

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7: If x(t) is an odd function, that is, x(nT ) = − x(− nT ) = − x (( N − n)T ), then the am coefficcients are all zero, cm = − c− m is imaginary, and the Fourier series is a sine series. Assume N is even. 2, show that, since xn = x− n , the terms in the sum for n = 0 and n = N/2 must both be zero. b. Show that if x(t) is odd, xn = − xN − n for any value of n. c. Using the condition given in part (b), separate the nonzero terms in the sum into two sums that cancel, thus proving the theorem. 0 and period of 500.

On the complex plane, e jθ becomes a unit vector at angle θ measured counterclockwise from the positive real axis. 6. That is, the vector at 3 o’clock is the term for n = 0, the vector at 12 o’clock is for n = 2, and so on. 21). 21) with N = 8 on the complex plane. 2 as a reference for the remainder of the text. Trigonometric functions, especially sine and cosine functions, appear in different combinations in all kinds of harmonic analysis—Fourier series, Fourier transforms, and so on. The identities that give sine and cosine functions in terms of exponentials are important, because they allow us to find sums of sines and cosines using the geometric series as in the example above.

11). 13). • Make a continuous plot of the least-squares parabola, fˆ (t), in the range t = [0, N − 1]. Use 1000 values to produce a smooth plot. • Finally, on the same graph, plot the elements of f as discrete points. Label the plot using functions xlabel, ylabel, and title, and observe whether the least-squares fit seems to be correct. a. f = [1, 3, 2] b. f = [1, 3, 4, 2] c. 0] d. 1 using a vector f = [−1, 3], that is, a vector with N < M. a. 10). b. Show from part (a) why the three linear equations are not independent, and why therefore, a unique solution for c does not exist.

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Z80 technical manual by Joseph J. Carr


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