Electrical Engineering 20N - Spring 1999 - Midterm 2

Electrical Engineering 20N - Spring 1999 - Midterm 2 - 1 30...

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1. 30 points Consider an LTI system with impulse response h given by n Ints ,h ( n )= δ ( n )+ δ ( n - 1) , where δ is the Kronecker delta function. Let H denote the frequency response of this system. Suppose that two copies of this system are connected in cascade, meaning that the output of the ﬁrst one feeds the input of the second. (a) Find the impulse response g of the cascade system. (b) Give the frequency response G of the cascade system in terms of H . (c) Find H and G as functions of ω , frequency in radians/sample. Solution: (a) If the input to the ﬁrst system is δ , the Kronecker delta function, then the output will be h , the impulse response. But h just consists of two delta functions, one of them delayed. By linearity and time invariance, the output will therefore be y = h 1 h ,or n Ints ,y ( n δ ( n δ ( n - 1) + δ ( n - 1) + δ ( n - 2) or y ( n δ ( n )+2 δ ( n - 1) + δ ( n - 2) . (b) G = H 2 (c) ω Reals ,H ( ω X n = -∞ h ( n ) e - jωn =1+ e - , using the sifting property of the delta function. G ( ω )=(1+ e - ) 2 . 2. 30 points Consider an LTI system L with impulse response h given by n Ints ( n δ ( n +2)+ δ ( n - 2) , where δ is the Kronecker delta function.

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Electrical Engineering 20N - Spring 1999 - Midterm 2 - 1 30...

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