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Unformatted text preview: ECE 301 Detailed Solutions  Set 1: The Impulse Signal 1 11. Let a function f α be defined by the following piecewise form: f α = 2 ; α ; else Find the simplest math form for each of the following functions: (a) f t ; (b) f t ; (c) f t 4 ; (d) f 4 t . Also, plot each function on the t axis. DETAILED SOLUTION: (a) Find the simplest math form for f t : To find f t , make the substitution α = t in the defining equation for f α given in the problem statement: f t = f α ¬ ¬ ¬ α = t = 2 ; t ; else a Observe from equation a above that you have to make the substitution α = t everywhere α appears in the defining equation. A plot of f t as given by equation a is shown below. t 2 f(t) t = 0 (b) Find the simplest math form for f t : To find f t , make the substitution α = t in the defining equation for f α given in the problem state ment: f t = f α ¬ ¬ ¬ α = t = 2 ; t ; else b If we want to plot f t on the t axis, it is useful to rearrange the inequality in b so that positive t appears on one side of the inequality. Therefore, multiply both sides of the inequality t 0 by 1, and remember that multiplication by a negative number changes the sense of the inequality: t t c Substituting c into d then gives the desired math form f t = 2 ; t ; else d ECE 301 Detailed Solutions  Set 1: The Impulse Signal 2 11 (cont.) A plot of f t as given by equation d is shown below. t = 0 t f(t) 2 (c) Find the simplest math form for f t 4 : To find f t 4 , make the substitution α = t 4 in the defining equation for f α given in the problem statement: f t 4 = f α ¬ ¬ ¬ α = t 4 = 2 ; t 4 ; else e As in part (b) on the previous page, if we want to plot f t 4 on the t axis, it is useful to rearrange the inequality in e so that positive t appears on one side of the inequality. Therefore, add positive 4 to both sides of the inequality t 4 0, obtaining t 4 t 4 f Substituting f into e then gives the desired math form f t 4 = 2 ; t 4 ; else g A plot of f t 4 as given in equation g is shown below. t 2 f(t4) t = 0 t = 4 (d) Find the simplest math form for f 4 t : To find f 4 t , make the substitution α = 4 t in the defining equation for f α given in the problem statement: f 4 t = f α ¬ ¬ ¬ α = 4 t = 2 ; 4 t ; else h ECE 301 Detailed Solutions  Set 1: The Impulse Signal 3 11 (cont.) As done previously, if we want to plot f 4 t on the t axis, it is useful to rearrange the inequality in h so that positive t appears on one side of the inequality. Therefore, add t to both sides of the inequality 4 t 0, obtaining 4 t 4 t t 4 i Substituting i into h then gives the desired math form f 4 t = 2 ; t 4 ; else j A plot of f 4 t as given in equation j is shown below....
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 Spring '08
 Hoozturk
 simplest math form, impulse signal, ´bµ, ´t

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