ECE_313_Fall22_HW3_soln

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ECE313 - Linear Systems and Signals - Prof. Jon Tamir Homework #3 Solutions 1. (30 pts) Determine (with justification) whether the following systems are stable, time invariant and/or linear: (a) y ( t ) = x ( t − 1) + 4 x (3 − t ) . (b) y [ n ] = nx [ n ] . • Solution: a) Linear, stable, time variant: Time-invariance: Let x 1 ( t ) = x ( t − t 0 ) be the shifted input, and y 1 ( t ) be the output of the shifted input, then y 1 ( t ) = x 1 ( t − 1) + 4 x 1 (3 − t ) = x ( t − 1 − t 0 ) + 4 x (3 − t − t 0 )) Now let y 2 ( t ) = y ( t − t 0 ) be the shifted output, then: y 2 ( t ) = y ( t − t 0 ) = x ( t − t 0 − 1) + 4 x (3 − ( t − t 0 )) = x ( t − t 0 − 1) + 4 x (3 − t + t 0 ) Hence, y 2 ( t ) ̸ = y 1 ( t ). b) Linear, Non-stable, time variant Time-invariance: Let x 1 [ n ] = x [ n − n 0 ] be the shifted input, and y 1 [ n ] be the output of the shifted input, then y 1 [ n ] = nx 1 [ n ] = nx [ n − n o ] Now let y 2 [ n ] = y [ n − n 0 ] be the shifted output, then: y 2 [ n ] = y [ n − n 0 ] = ( n − n 0 ) x [ n − n 0 ] Hence, y 2 [ n ] ̸ = y 1 [ n ]. Stability: Because of the n multiplied by x [ n ]. 2. (30 points) Systems Engineering. For the system given in Figure 1, answer the following: 1

Figure 1: System for Question 2 (a) (12 points) Find the equation that defines the given system. Solution: x 1 ( t ) = Ax ( t − t 0 ) x 2 ( t ) = x 1 ( t ) x ( t ) = Ax ( t ) x ( t − t 0 ) x 3 ( t ) = Bx ( t ) y ( t ) = x 2 ( t ) − x 3 ( t ) y ( t ) = Ax ( t ) x ( t − t 0 ) − Bx ( t ) (b) (6 points) Suppose t 0 ̸ = 0. For what values (or range of values) of A and B is the system memoryless? Your answers can include =, ≤ , < , ≥ , > , −∞ , + ∞ , etc. Solution: for memoryless: y ( t ) only depends on current value of x ( t ) ⇒ Need A = 0. hence, A = 0 , −∞ < B < ∞ (c) (6 points) Suppose A ̸ = 0. For what values of B and t 0 is the system causal? Your answer can include =, ≤ , < , ≥ , > , −∞ , + ∞ , etc. Solution: Causal: y ( t ) depends on current or past values of x ( t ). Need t 0 ≥ 0. Hence, −∞ < B < ∞ , t 0 ≥ 0 . (d) (6 points) For what values of A , B , and t 0 is the system linear? Your answer can include =, ≤ , < , ≥ , > , −∞ , + ∞ , etc. Solution: Linearity: cannot have ( x ( t )) 2 or x ( t ) x ( t − t 0 ) ⇒ A = 0. A = 0 , −∞ < B < ∞ , −∞ < t 0 < ∞ 3. (15 pts) Compute the convolution between 2

and in the following two ways: (a) direct calculation, by which you consider the input as a series of delta functions, and sum up the resulting (shifted, scaled) responses. You can express y [ n ] mathematically as a signal or graphically. • Solution : (a) Direct calculation: We can rewrite the given signal using kronecker delta’s. x 1 [ n ] = δ [ n + 1] + 2 δ [ n ] + 3 δ [ n − 1] x 2 [ n ] = − δ [ n ] + δ [ n − 1] 3

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