Z-transform and digital filters · Definition

Discrete transfer function

signal processingz-transform and digital filters

Before you start you need to know

Transfer functions describe a discrete system through a rational function in \(z\). This representation simplifies stability analysis, frequency response, and filter design.

Definition

The discrete transfer function \(H(z)\) is the Z-transform of the impulse response \(h[n]\) of a discrete-time LTI system, with \(H(z)=Y(z)/X(z)\) under zero initial conditions.

\[H(z)=\sum_{n=-\infty}^{\infty} h[n]\,z^{-n}\]

If the system is linear and time-invariant, the output is the convolution \(y[n]=h[n]*x[n]\), and applying the Z-transform turns it into the product \(Y(z)=H(z)X(z)\). The poles of \(H(z)\) determine stability: the system is BIBO stable if all poles lie inside the unit circle \(|z|<1\), as Oppenheim and Schafer explain. The zeros of \(H(z)\) indicate frequencies the system attenuates.

For a first-order filter \(y[n]=0.5\,y[n-1]+x[n]\), taking the Z-transform with zero initial conditions gives \(Y(z)(1-0.5z^{-1})=X(z)\), so \(H(z)=\frac{1}{1-0.5z^{-1}}\), whose pole is at \(z=0.5\), so the system is stable. The impulse response is \(h[n]=0.5^n u[n]\), confirming the link between both representations.

A typical error is confusing the transfer function with the time-domain impulse response, or writing \(H(z)\) without specifying the region of convergence. Another common mistake is ignoring initial conditions, which invalidate the relation \(Y(z)=H(z)X(z)\). Always check that the region of convergence includes the unit circle when analyzing the frequency response.

Think about it

What would happen to stability if a pole of \(H(z)\) lay exactly on the unit circle \(|z|=1\)?

It would not be BIBO stable: \(h[n]\) is not absolutely summable, and a bounded input such as the unit step can produce an output that grows without bound.

Exercises 10

  1. What does \(H(z)\) represent for a discrete LTI system?
  2. If \(Y(z)=H(z)X(z)\), what time-domain operation corresponds to the product in \(z\)?
  3. What is the name of the region in the \(z\)-plane where the transform defining \(H(z)\) converges?
  4. For \(H(z)=\frac{1}{1-0.5z^{-1}}\), where is the pole?
  5. What condition guarantees BIBO stability of a discrete system with a rational transfer function?
  6. For \(y[n]=0.8\,y[n-1]+x[n]\) with zero initial conditions, what is \(H(z)\)?
  7. If \(h[n]=\delta[n]+0.5\,\delta[n-1]\), what is \(H(z)\)?
  8. A system has \(H(z)=\frac{z}{z-0.3}\). What is its causal impulse response \(h[n]\)?
  9. For \(H(z)=\frac{1}{1-1.2z^{-1}}\) with region of convergence \(|z|>1.2\), is the causal system stable?
  10. A system satisfies \(y[n]=0.6\,y[n-1]-0.08\,y[n-2]+x[n]\) with zero initial conditions. What is \(H(z)\)?

References

  • Alan V. Oppenheim, Alan S. Willsky, Signals and Systems, 2nd edition, Pearson, 1996.
  • Alan V. Oppenheim, Ronald W. Schafer, Discrete-Time Signal Processing, 3rd edition, Pearson, 2010.