The flow graph of a 1st-order system is shown in Figure P6.57. (a) Assuming...

The flow graph of a 1^st-order system is shown in Figure P6.57.

(a) Assuming infinite-precision arithmetic, find the response of the system to the input

What is the response of the system for large n?

Now suppose that the system is implemented with fixed-point arithmetic. The coefficient and all variables in the flow graph are represented in sign-and-magnitude notation with 5-bit registers. That is, all numbers are to be considered signed fraction represented as

b ₀ b₁b₂b₃b₄,

where b₀, b₁, b₂, b₃, and b₄ are either 0 or 1 and

|Register value| = b₁2⁻¹ + b₂2⁻² + b₃2⁻³ + b₄2⁻⁴.

If b₀ = 0, the fraction is positive, and if b₀ = 1, the fraction is negative. The result of a multiplication of a sequence value by a coefficient is truncated before additions occur; i.e., only the sign bit and the most significant four bits are retained.

(b) Compute the response of the quantized system to the input of part (a), and plot the responses of both the quantized and unquantized systems for 0 ≤ n ≤ 5. How do the responses compare for large n?

(c) Now consider the system depicted in Figure P6.57, where

Repeat parts (a) and (b) for this system and input.