allpole

All-pole refers to a class of digital filters in which the transfer function has poles but no finite zeros. In the usual z-domain notation, an all-pole filter is described by H(z) = B(z)/A(z) with A(z) = 1 − a1 z^−1 − ... − ap z^−1 and B(z) being a constant. When B(z) is constant (often normalized to 1), the filter is written as H(z) = 1/A(z). This means the filter’s frequency response is shaped entirely by its poles.

In the time domain, an all-pole filter corresponds to an autoregressive (AR) process. If the input is an excitation signal e[n], the output follows y[n] = a1 y[n−1] + ... + ap y[n−p] + e[n]. The system is stable when all poles of A(z) lie inside the unit circle.

Applications are common in digital signal processing, especially in speech, acoustics, and audio coding. All-pole models are used to capture the spectral envelope created by resonances, such as the vocal tract in speech production. Linear predictive coding (LPC) employs all-pole synthesis filters to model speech signals efficiently, enabling compression and synthesis. All-pole filters are also used in resynthesis, filter design, and parametric equalization where a smooth, resonant spectrum is desired.

Parameter estimation typically relies on autocorrelation methods and the Levinson-Durbin algorithm to solve the Yule-Walker equations for the AR coefficients. Model order p is chosen to balance accurate spectral representation with computational efficiency. Limitations include the inability to represent zeros directly, which can be important for shaping notches or dips in the spectrum; in such cases a pole-zero model may be more appropriate.