Non-sinusoidal periodic functions are commonly used to describes signals which should rise and fall peri- odically. For example, a square waveform function could be used to describe a signal which periodically switches on and off. Square A square waveform (source: Wikipedia) Questions Consider the triangle sawtooth wave which can be defined for all a e R by the following statements: (i) f(z) = = for ze [0,=] (ii) f is even (mii) f is 2x-periodic 1. Plot the graph of f over the interval [-3x, 3x]. Use plotting software or carefully sketch by hand. When analyzing signals, we often prefer to work with a smooth function, rather than one that exhibits points of non-differentiability. 2. Plot the graph of the function g(x) = ] - } cos(x) with the graph of f over the interval [-3x, 3x]. Does g provide a good approximation to f? 3. To arrive at an improved approximation, consider the function h(s) = ] - acces(x) for some constant a. (Note, g is equal to h when a = -.) Find the value of a that results in the function h that best approximates f, where the error in approximation is given by the total squared error Ela) = [h(z) - f(x)] ax (Be sure to briefly explain what you are doing and why throughout each stage of your calculation.) 4. Plot the graph of this optimal h along with f over [-3x, 3x]