Q6Digital Signal Processing

Cascade Realisation

The given transfer function's denominator is already presented in factored form, directly suggesting the natural cascade decomposition:

where H1(z) is a first-order section combining the given zero (at z=-1/4) with the first pole factor (at z=-1/2), and H2(z) is a second-order all-pole section realizing the complex-conjugate pole pair arising from the quadratic factor (1+(1/2)z^-1+(1/4)z^-2) (this quadratic has complex roots, since its discriminant (1/2)^2-4*(1/4)=0.25-1=-0.75 is negative, giving a genuine complex-conjugate pole pair rather than two separate real poles).

Parallel Realisation

The parallel realisation is obtained by performing a partial fraction expansion of H(z), expressing it as a sum of simpler terms corresponding to each distinct pole/pole-group:

where the first term corresponds to the real pole at z=-1/2, and the second term (with a first-order numerator B+Cz^-1, since the corresponding pole-pair term is second-order in the denominator) corresponds to the complex-conjugate pole pair. The constants A, B, C are found by clearing denominators (multiplying both sides by the full original denominator) and equating coefficients of matching powers of z^-1 on both sides of the resulting polynomial equation, yielding a system of three linear equations in the three unknowns A, B, C, solvable by standard algebraic methods (substitution or matrix methods).

Each of these two terms is then independently realized as its own first-order or second-order direct-form section, and the two sections' outputs are combined at a final summing junction, since the transfer function is now expressed as a SUM of the two simpler transfer functions (parallel connection, corresponding to addition of transfer functions, in contrast to the cascade connection's multiplication of transfer functions). This parallel structure has the practical advantage that each individual section operates independently and any numerical rounding/overflow error occurring in one section does not directly propagate into or compound with the other section's own internal computation, unlike a cascade structure where errors from an earlier section directly feed into and can be amplified by a later section.

This system function H(z) = [1+(1/4)z^-1] / [(1+(1/2)z^-1)(1+(1/2)z^-1+(1/4)z^-2)] combines one first-order pole factor with one second-order (complex-conjugate-pole) factor in its denominator, together with a single first-order zero in its numerator, making it a good representative example for illustrating both cascade and parallel realization techniques on a system that is not simply reducible to a chain of only first-order real sections. For the cascade realization, the overall system is naturally factored into two cascaded sections: a first-order section H1(z) realizing the pole at z=-1/2 (with the numerator zero at z=-1/4 also assigned to this or the other section, depending on the specific grouping convention chosen), and a second-order section H2(z) realizing the complex-conjugate pole pair arising from the quadratic factor 1+(1/2)z^-1+(1/4)z^-2, whose poles lie at z = (1/2)e^(+-j2*pi/3) after solving the quadratic in z^-1, giving a resonant, moderately damped second-order response.

The parallel realization, by contrast, is obtained by performing a partial-fraction expansion of H(z) directly in terms of z^-1 (or equivalently in terms of z), decomposing the overall system function into a sum of a first-order term (associated with the pole at z=-1/2) and a second-order term (associated with the complex-conjugate pole pair from the quadratic factor), each of which can be individually realized as a low-order recursive section and then combined by simple output summation rather than by cascading - this parallel structure has the practical advantage that each section operates completely independently of the others (with no signal passing through one section before reaching the next), meaning coefficient quantization or arithmetic rounding error introduced in one parallel section does not propagate into or compound with errors in any other section, unlike in a cascade realization where errors introduced by an earlier section directly affect the signal subsequently processed by later sections in the chain.

Selecting between the cascade and parallel realizations of this same overall transfer function in a practical implementation typically depends on subsidiary considerations beyond pure functional equivalence: the cascade form offers the designer some freedom in choosing the pairing and ordering of poles and zeros between sections (which can be exploited to optimize dynamic range or minimize the effect of coefficient quantization for a specific fixed-point word length), whereas the parallel form's independent-section error-isolation property is often preferred specifically in high-precision or high-reliability applications, and additionally the parallel form directly exposes the individual natural modes (pole locations) of the overall system as separately identifiable additive contributions, which can be valuable for analysis purposes such as isolating and studying an individual resonant mode's contribution to the overall system response.

Beyond the cascade and parallel forms considered here, this same transfer function could equally be realized in Direct Form I or Direct Form II, and comparing all these alternative realizations highlights an important general filter-design principle: while every realization is mathematically equivalent under infinite-precision arithmetic, they differ substantially in their sensitivity to coefficient quantization and round-off noise once implemented with finite word-length arithmetic, which is precisely why the cascade and parallel forms (built from well-conditioned low-order sections) are generally preferred over a single higher-order Direct Form realization in most practical fixed-point or embedded DSP implementations.

It is also worth noting that both the cascade and parallel realizations derived here require exactly three delay elements in total (one for the first-order section and two for the second-order section, or equivalently distributed if a canonic-form combination is used), matching the minimum number of delays theoretically required for a third-order overall system (as this transfer function is, being formed from a first-order times a second-order denominator factor), confirming that neither realization wastes hardware resources relative to the theoretical minimum delay-element count for this system.

In summary, this example demonstrates the complete cascade and parallel realization workflow for a mixed first-order/second-order transfer function, illustrating how factoring versus partial-fraction expansion of the same H(z) yields two structurally distinct but functionally equivalent digital filter realizations.

This worked example, taken together with the resonator-form structure examined in the companion question of this same unit, gives a reasonably complete picture of how second-order sections combine with first-order sections in both cascade and parallel realizations of practical, moderate-order digital IIR filters.