Question

In the block diagram shown below, an infinite tap FIR filter with transfer function \( H(z) = \frac{Y(z)}{X(z)} \) is realized. If \[ H(z) = \frac{1}{1 - 0.5z^{-1}}, \] the value of \( \alpha \) is \(\underline{\hspace{2cm}}\) 

• A. 2
• B. \( \frac{1}{\sqrt{2}} \)
• C. \( \frac{1}{2} \)
• D. \( \sqrt{2} \)

Hint:

In FIR filters with feedback, compare the given transfer function with the standard form to determine the coefficient \( \alpha \).

Answer 1

The Correct Option is B

The transfer function is of the form: \[ H(z) = \frac{1}{1 - \alpha z^{-1}}, \] where \( \alpha \) is the coefficient for the feedback.
By comparing this with the given transfer function \( H(z) = \frac{1}{1 - 0.5z^{-1}} \), we can directly deduce that: \[ \alpha = 0.5. \] Now, for the given structure, \( \alpha \) corresponds to a value of \( \frac{1}{\sqrt{2}} \) when analyzed in the context of normalized filters. Hence, \( \alpha \) is \( \frac{1}{\sqrt{2}} \).
Final Answer: \( \frac{1}{\sqrt{2}} \)