Z-Plane (Pole-Zero Plot)
Magnitude Response |H(e^jw)|
Impulse Response h[n]
Understanding the Z-Transform
What is the Z-Transform?
The Z-Transform converts a discrete-time signal x[n] into a complex frequency-domain representation X(z). It is defined as X(z) = sum(x[n] * z^(-n)) for n from -infinity to +infinity. The region of convergence (ROC) is the set of z values for which the series converges. For causal systems, the ROC is the exterior of a circle centered at the origin whose radius equals the magnitude of the outermost pole.
Poles, Zeros, and the Z-Plane
The transfer function H(z) of a linear time-invariant (LTI) system can be expressed as a ratio of polynomials in z. The roots of the numerator are zeros (where H(z) = 0), and the roots of the denominator are poles (where H(z) diverges). On the z-plane, poles are marked with x and zeros with o. Their placement determines the system's frequency response, stability, and filter characteristics. Poles near the unit circle create peaks in the frequency response, while zeros near the unit circle create notches.
Unit Circle and Frequency Response
Evaluating H(z) on the unit circle (z = e^(jw)) yields the Discrete-Time Fourier Transform (DTFT), which is the frequency response of the system. The angle w (omega) ranges from 0 to 2*pi and corresponds to normalized frequency. As w increases from 0, we traverse the unit circle counterclockwise. The point z = 1 (w = 0) represents DC, and z = -1 (w = pi) represents the Nyquist frequency. The magnitude |H(e^jw)| shows how the system amplifies or attenuates different frequencies.
Stability Criterion (BIBO)
A system is Bounded-Input Bounded-Output (BIBO) stable if and only if all poles of its transfer function lie strictly inside the unit circle (|p| < 1 for all poles). If any pole is on or outside the unit circle, the system is unstable and the impulse response grows without bound. The stability indicator in the stats panel checks this criterion in real-time as you move poles on the z-plane. Conjugate pairs ensure real-valued impulse responses.
Transfer Function
H(z) = K * Product(z - z_i) / Product(z - p_j)
Where K is the gain, z_i are the zeros, and p_j are the poles. The frequency response is obtained by substituting z = e^(jw) and evaluating over w from 0 to 2*pi. The magnitude at each frequency is |H(e^jw)| = K * product(|e^jw - z_i|) / product(|e^jw - p_j|).