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Question 1 (9 points) Given a recursive system depicted in the figure below. x (n) y(n) + -3/4 Z 1 (a) 1p Write down the difference equation describing the system. (b) 1p What is the impulse response of the system? Suppose that the system is implemented with fixed-point arithmetic based on three bits for magnitude and a sign bit (SM representation). Suppose, further, that the quantization that takes place after multiplication rounds the resulting product, such that the maximum quantization error is 2-b (c) 1p For this quantizer, make a table with 16 rows and 3 columns, to associate the 4-bit SM representation to the corresponding numerical value. As the first column, show the input range of values that leads to the corresponding quantized output. (d) 2p Assume an input sequence 0.5 for n = 0 x[n] = otherwise. Compute the first 5 samples of the quantized output signal, yq[0], ,yq[4]. What is the range of amplitudes of the dead band of the quantized system? From which sample on does the system reach its steady-state output sequence? (e) 1p Is this steady-state output sequence a constant value or an oscillation? Why? (f) 1p How can you change the system design (without changing its functionality) in order to decrease the dead band? (g) 2p Let's consider now an arbitrary input and an additive noise model for the quantization error in the above system. What is the variance of quantization error and the variance of the output noise? Give the numerical values.

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