Q3Digital Signal Processing

(a) Minimum Phase System

A minimum phase system is a causal, stable LTI system whose z-transform H(z) has ALL of its zeros, in addition to all of its poles, located strictly inside the unit circle in the z-plane (recall that for any causal, stable system, all poles must already lie inside the unit circle; the minimum-phase distinction additionally requires all zeros to lie inside the unit circle as well, a further, separate condition not automatically satisfied by every stable system).

Significance - minimum group delay/phase lag: among all possible causal, stable LTI systems sharing an identical magnitude response |H(e^jw)|, the specific system whose zeros (as well as poles) all lie inside the unit circle exhibits the smallest possible phase lag (and correspondingly, the smallest group delay) at every frequency, hence the name 'minimum phase.' Any other system sharing the same magnitude response but having one or more zeros located outside the unit circle (a 'non-minimum-phase' or 'mixed-phase' system) necessarily has a larger phase lag at some or all frequencies.

Minimum-phase/all-pass decomposition: any non-minimum-phase causal stable system Hmixed(z) can always be decomposed as the cascade (product) of a minimum-phase system Hmin(z) and an all-pass system Hap(z) (discussed in part (b) below): Hmixed(z) = Hmin(z) * Hap(z), where Hmin(z) is obtained from Hmixed(z) by reflecting any zeros located outside the unit circle to their conjugate-reciprocal (mirror) locations inside the unit circle, which leaves the overall magnitude response |Hmixed(e^jw)| unchanged (since this reflection process preserves magnitude on the unit circle, a standard z-transform property) while producing a genuinely minimum-phase system. This decomposition is of significant practical importance in system identification, channel equalization, and speech processing applications, since a minimum-phase system additionally has the useful property that its inverse system, 1/Hmin(z), is itself also causal and stable (a property that does NOT hold for a non-minimum-phase system's inverse, since the original non-minimum-phase zeros outside the unit circle would become poles outside the unit circle in the inverted system, causing instability).

(b) All-Pass System

An all-pass system is an LTI system whose magnitude response is exactly constant (equal to 1, i.e., unity gain) at every frequency, |H(e^jw)|=1 for all w, while its phase response varies with frequency in a generally non-trivial, non-linear way (unless specifically designed otherwise).

Structure achieving unity magnitude: a first-order all-pass system has the general form:

where a is a complex constant with |a|<1 (required for the single pole at z=a to lie inside the unit circle, ensuring stability), and a denotes its complex conjugate. This structure places a zero at z=1/a (the conjugate-reciprocal, or 'mirror,' location relative to the pole at z=a, reflected through the unit circle) - this specific pole-zero mirroring arrangement is precisely what guarantees that the magnitude response evaluated on the unit circle remains constant at exactly 1 for all frequencies, a standard, easily-verified z-transform property whenever poles and zeros occur in this conjugate-reciprocal mirrored pairing about the unit circle.

Applications: all-pass systems are used primarily for phase equalization/correction - since they leave a signal's magnitude spectrum completely unaffected but can be designed to introduce any desired additional phase shift as a function of frequency, an all-pass system can be cascaded with another system exhibiting undesirable phase distortion (such as a non-minimum-phase channel or filter) to compensate for and correct that phase distortion, restoring a more desirable overall (such as approximately linear) phase response for the combined cascade, without altering the overall magnitude response that has already been correctly established by the other system in the cascade - a technique widely used in audio processing, communication channel equalization, and crossover network design in loudspeaker systems.

The minimum-phase property of a system has a direct and practically important consequence for system invertibility: because a minimum-phase system, by definition, has all of its poles and zeros located strictly inside the unit circle, its inverse system 1/H(z) also has all poles strictly inside the unit circle (since the inverse system's poles are located exactly where the original system's zeros were), meaning the inverse of a minimum-phase system is itself both stable and causal - this is a highly desirable property in applications such as channel equalization or deconvolution, where recovering the original input signal from the observed output requires implementing the inverse system, and only a minimum-phase forward system guarantees that this inverse can be realized as a stable, causal digital filter.

Among all causal, stable LTI systems sharing the same magnitude response |H(e^jw)|, the minimum-phase system is additionally distinguished by two further important extremal properties: it has minimum group delay (the phase response accumulates the least possible delay, at every frequency, compared to any other system with the identical magnitude response) and minimum energy delay (for any given output-energy budget, more of the system's total impulse-response energy is concentrated toward the beginning, n=0, of the impulse response than for any alternative system sharing the same magnitude response) - these properties collectively explain why minimum-phase filter designs are generally preferred whenever transient response speed and energy concentration near t=0 are important design considerations, such as in certain audio and control-system filtering applications.

The all-pass system, by contrast, is characterized by a magnitude response that is identically equal to unity (or some fixed constant gain) at every frequency, |Hap(e^jw)|=1 for all w, while its phase response varies with frequency in a generally nonlinear fashion - an all-pass system is constructed so that every pole inside the unit circle at location z=a is paired with a corresponding zero at the conjugate-reciprocal location z=1/a* (outside the unit circle for a stable, causal all-pass system), a pole-zero pairing pattern that mathematically guarantees the unit-magnitude response property at all frequencies. Because it does not alter a signal's magnitude spectrum at all, the all-pass system's sole practical function is to modify the phase (and hence group-delay) characteristics of a signal, making all-pass sections a standard building block for phase equalization (correcting unwanted phase nonlinearity introduced by some other component in a signal processing chain) without disturbing the overall magnitude response that has already been carefully designed elsewhere in the system.

It is a fundamental and useful theorem of digital signal processing that any stable, causal, rational system function H(z) can be uniquely decomposed as the product of a minimum-phase system Hmin(z) and an all-pass system Hap(z), H(z) = Hmin(z)*Hap(z) - this minimum-phase/all-pass decomposition is achieved by reflecting each zero of H(z) that lies outside the unit circle to its conjugate-reciprocal location inside the unit circle (forming Hmin(z)), and compensating for this reflection by cascading an all-pass section whose zero-pole pair restores the original zero location while preserving the overall magnitude response |H(e^jw)| unchanged, providing a systematic way to extract the best-possible (minimum-phase, and hence minimum-delay) causal, stable approximation to any given non-minimum-phase system's magnitude characteristics.