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# HW4sol - (c No If we let y t = t R-∞ x τ dτ with x t =...

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ECEN 314: Signals and Systems Solutions to HW 4 Problem 1.35 We need a smallest N o such that m (2 π/N ) N o = 2 πk or N o = kN/m , where k is an in- teger. If N o has to be an integer, then N must be a multiple of m/k , and m/k must be an integer. This implies that m/k is a divisor of both m and N . Also, if we want the smallest possible N o , then m/k should be the greatest common divisor (gcd) of m and N . Therefore, N o = N/gcd ( m,N ) Problem 1.37 (a) From the deﬁnition of φ xy ( t ), we have φ xy ( t ) = Z -∞ x ( t + τ ) y ( τ ) = Z -∞ y ( - t + τ ) x ( τ ) = φ yx ( - t ) (b) From part (a), we see that φ xx ( t ) = φ xx ( - t ). This implies that φ xx ( t ) is even. Therefore, the odd part of φ xx ( t ) is zero. (c) φ xy ( t ) = Z -∞ x ( t + τ ) y ( τ ) = Z -∞ x ( t + τ ) y ( τ + T ) = Z -∞ x ( τ 0 - T + t ) x ( τ 0 ) 0 (Letting τ + T = τ 0 ) = φ xx ( t - T ) 1

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φ yy ( t ) = Z -∞ y ( t + τ ) y ( τ ) = Z -∞ x ( t + τ + T ) x ( τ + T ) = Z -∞ x ( τ 0 + t ) x ( τ 0 ) 0 (Letting τ + T = τ 0 ) = φ xx ( t ) Problem 1.40 (a) If the system is additive, then 0 = x ( t ) - x ( t ) y ( t ) - y ( t ) = 0 If the system is homogenous, then 0 = 0 .x ( t ) y ( t ) . 0 = 0 (b) y ( t ) = x 2 ( t ) is such a system.
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Unformatted text preview: (c) No. If we let y ( t ) = t R-∞ x ( τ ) dτ with x ( t ) = u ( t )-u ( t-1). Then x ( t ) = 0 for t > 1, but y ( t ) = 1 for t > 1. Problem 1.43 (a) We have x ( t ) S → y ( t ) Since S is time-invariant, x ( t-T ) S → y ( t-T ) Now if x ( t ) is periodic with period T, x ( t ) = x ( t-T ). Therefore, we may we conclude that y ( t ) = y ( t-T ). This implies that y ( t ) is periodic with period T . A similar argument can be made in discrete-time. (b) Consider the system y ( t ) = ∞ X k =-∞ x 2 ( t-2 k ) The above system is time-invariant. Consider the input x ( t ) = u ( t )-u ( t-1). Then it is easy to see that while the input is not periodic, the output is periodic. 2...
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HW4sol - (c No If we let y t = t R-∞ x τ dτ with x t =...

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