Digital Signal Processing Basics
Which of the following is the transfer function H(z) = Y(z)/X(z) of the system described by the difference equation y[n] = x[n] + 0.5x[n − 1] + 0.25y[n − 1]?
- A(1 + 0.5z⁻¹)/(1 + 0.25z⁻¹)
- B(1 − 0.25z⁻¹)/(1 + 0.5z⁻¹)
- C(1 + 0.5z⁻¹)/(1 − 0.25z⁻¹)
Show answer and explanation
Answer: C. (1 + 0.5z⁻¹)/(1 − 0.25z⁻¹)
A delay of one sample corresponds to multiplying by z⁻¹ in the z-transform, so transforming both sides gives Y(z) = X(z) + 0.5z⁻¹X(z) + 0.25z⁻¹Y(z). Collecting the Y(z) terms on the left gives (1 − 0.25z⁻¹)Y(z) = (1 + 0.5z⁻¹)X(z), so H(z) = (1 + 0.5z⁻¹)/(1 − 0.25z⁻¹). The numerator holds the coefficients on the input side and the denominator those of the output feedback; z = 0.25, which makes the denominator 0, is a pole, and z = −0.5, which makes the numerator 0, is a zero.
- A. This forgets to flip the sign of the coefficient when moving the feedback term to the left side.
- B. This is the reciprocal of the correct H(z) (numerator and denominator swapped).
- C. Correct. Moving the feedback term to the left side flips the sign of the denominator coefficient.