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Linear Systems and Signal Analysis

# Linear Systems and Signal Analysis - Part 1 Mathematics...

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Part 1: Mathematics Revisit Straight Line L is a straight line on x-y plane P1: (0,b) P2: (a,0) The slope of L: b is called the y-intercept a is called the x-intercept y x a b L a b a b m = = 0 0

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Equation for a straight line Two point form: where (x 1 ,y 1 ) and (x 2 , y 2 ) are any two points on L Point-slope form: where (x 1 ,y 1 ) is a point on L and m is the slope of L Slope-intercept form: where m and b are the slope and the y-intercept of L Intercept form: where a, b are the x- and y-intercepts, respectively. 1 = + b y a x m x x y y = 1 1 b mx y + = 1 2 1 2 1 1 x x y y x x y y = Question Express the following function f(x): f(x) x 1 2 2 3 4 Hint: Separate into several segments
Transformation Consider the previous f(x), draw the following functions: y=f(2x) y=2f(x) y=f(x-4) y=f(-x) y x y x y x y x Trigonometric Functions z Sine, Cosine, Tangent,.. z Consider a unit circle: z sin θ = b z cos θ = a z z Important Equalities: a b = = θ θ θ cos sin tan a b θ 1 ( ) ( ) ( ) ( ) [ ] ( ) ( ) [ ] y x y x y x y x y x y x y x y x y x y x y x y x x x + =

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Linear Systems and Signal Analysis - Part 1 Mathematics...

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