Homework ECE320 1 Cornell University
Spring 2006 T.L.Fine
Please hand in this assignment at the end of lecture on Tuesday, 7 February. Use only your assigned three-digit code and not your name. 1. (a) What is the domain, codomain, image, and graph(f ) of the usual quadratic polynomial x2 - x + 2? (b) If a function f has the graph {(1, a), (2, b), (-1, a), (c, 0)} what are its domain and codomain? Can you evaluate f (0)? (c) Describe the domain and codomain for a color image created on your computer screen. (d) Provide an example of a non-constant function f having a domain of {a, , &, !} and a codomain of {1, 2, b}. What is the image of your function? 2. (a) Given the logistic function f specified by graph(f ) = {(x, y) : x [0, 1], y = 3x(1 - x)}, plot its graph. (b) Plot the graph of the composition h = f (f ) of the logistic with itself. 3. Is the "function" that is 0 on the irrational numbers and 1 on the rational numbers a function? If so, what are its domain, codomain, and graph? Can you plot this function? 4. If we define f (x) = xx for domain (0, ), then what is the composition h function = f (f )? Evaluate it for h(2). Does h(2) differ from 2(2
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5. Show that the exponential function ez , for complex-valued z is a periodic function with period j2. (Hint: recall the Euler formula ej = cos() + j sin().) 1
6. (ECE220 revisited!) Consider the information-bearing signal f (t) = cos(1 t) encoded as an AM signal g(t) = (1 + f (t)) cos(0 t) for 0 >> 1 . Determine the Fourier transform G() of g. 7. For the same information-bearing signal f of Problem 6, consider its encoding as an FM signal g(t) = cos(0 t + sin(1 t)). Take 1 = .2 and 0 = 5. Choose a discrete set of times 0 : 0.1 : 1023. (a) Using Matlab, plot the first 2000 values of g for each of the values of the modulation index = .2, .8, 2. (c) Plot the magnitude (abs) of the discrete Fourier transform dft (fft) for the three values of modulation index. (d) Replot the magnitude of the discrete Fourier transform dft only for values in the range of 8000:end and do so for the three values of modulation index. (e) What do you observe about the dft bandwidth of the FM signal as increases?
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