Q4Digital Signal Processing

General Method for Finding Particular Solutions

The particular solution of a linear constant-coefficient difference equation is found by assuming a solution of the same functional form as the forcing (input) function x(n), substituting this assumed form into the difference equation, and solving for the undetermined coefficient(s) that make the equation hold for all n - if the assumed form happens to coincide with one of the equation's own homogeneous (characteristic-root) solutions, the assumed particular-solution form must be modified (typically by multiplying by an additional factor of n) to avoid degeneracy, exactly analogous to the corresponding technique used for solving linear constant-coefficient differential equations.

(a) y(n) + 3y(n-1) = x(n)

Homogeneous solution: setting x(n)=0, the equation becomes y(n)+3y(n-1)=0, giving the characteristic equation r+3=0 (obtained by assuming y(n)=r^n and dividing through by r^(n-1)), so the single characteristic root is r=-3. The homogeneous solution is therefore yh(n) = C*(-3)^n, where C is a constant determined by initial conditions. Since |r|=3>1, this root lies outside the unit circle, meaning the homogeneous solution grows without bound as n increases - this system is unstable in the absence of external forcing to counteract this growth.

Particular solution: assuming a unit-step input x(n)=A*u(n) (A a constant amplitude, a standard test input), and since the constant/step forcing function does not coincide with the characteristic root r=-3, assume a particular solution of the same constant form, yp(n)=K for n>=0. Substituting into the difference equation: K + 3K = A, giving 4K=A, so K=A/4. The particular solution is therefore yp(n) = A/4 for n>=0.

Total solution: y(n) = yh(n) + yp(n) = C*(-3)^n + A/4, with the constant C determined from the system's specified initial condition (such as y(-1) or y(0)).

(b) y(n) + 3y(n-1) - 4y(n-2) = x(n)

Homogeneous solution: setting x(n)=0 gives the characteristic equation r^2+3r-4=0 (assuming y(n)=r^n and dividing through by r^(n-2)). Using the quadratic formula:

giving roots r1 = (-3+5)/2 = 1 and r2 = (-3-5)/2 = -4 (the equation factors neatly as (r-1)(r+4)=0, confirming these roots). The homogeneous solution is therefore yh(n) = C1(1)^n + C2(-4)^n = C1 + C2*(-4)^n.

Particular solution: for a unit-step input x(n)=Au(n), the naive assumption of a constant particular solution yp(n)=K would coincide with the characteristic root r=1 (since a constant is the same as C1(1)^n), making this assumed form degenerate (it would satisfy the homogeneous equation and could not, by itself, balance the forcing term). To resolve this root-coincidence case, the particular solution must instead be assumed in the modified form yp(n) = Kn (multiplying the naive constant guess by an additional factor of n, exactly as done for repeated/coincident roots in differential equation particular-solution theory). Substituting yp(n)=Kn into the difference equation: Kn + 3K(n-1) - 4K(n-2) = A, expanding: Kn + 3Kn - 3K - 4Kn + 8K = A, simplifying the n-coefficients: (K+3K-4K)n + (-3K+8K) = A, giving 0*n + 5K = A (the n-terms cancel exactly, as expected since this modified form was specifically chosen to satisfy the homogeneous part appropriately), so 5K=A, K=A/5.

Total solution: y(n) = yh(n) + yp(n) = C1 + C2(-4)^n + (A/5)n, with C1 and C2 determined from the system's two required initial conditions (since this is a second-order difference equation).

Stability Analysis via Characteristic Root Magnitude

For any linear constant-coefficient difference equation, the overall system's stability is determined entirely by the magnitudes of its characteristic roots (equivalently, the poles of the corresponding transfer function): the homogeneous solution's contribution to the total response is a linear combination of terms of the form C*r^n for each characteristic root r, and each such term decays toward zero as n increases if and only if |r|<1 - if any root satisfies |r|>1, the corresponding term grows without bound, rendering the overall system BIBO-unstable, while a root with |r|=1 produces a homogeneous term that neither grows nor decays (marginal stability, oscillatory or constant persistence).

Applying this criterion to the two systems above: in part (a), the single characteristic root r=-3 has magnitude |r|=3, which exceeds 1, so this first-order system is unstable - any non-zero initial condition or any imperfect cancellation causes the homogeneous response C(-3)^n to grow in magnitude without bound as n increases, alternating in sign each sample due to the negative root. In part (b), the two characteristic roots are r1=1 (magnitude exactly 1, marginally stable/persistent, neither growing nor decaying) and r2=-4 (magnitude 4, unstable, growing and alternating in sign) - so this second-order system is likewise unstable overall, dominated in the long run by the exponentially growing (-4)^n term, and additionally exhibits the special root-at-unity behavior that necessitated the modified particular-solution form Kn derived above.

General Solution Method: Comparison to Linear ODEs

The overall method used above - finding the homogeneous solution from the characteristic equation's roots, then finding a particular solution matched to the forcing function's form (with the modified n-multiplied form used whenever the forcing function's natural form coincides with a homogeneous/characteristic-root solution), and finally summing the two to get the total solution with constants fixed by initial conditions - is structurally identical to the classical method of solving linear constant-coefficient ordinary differential equations (ODEs) in continuous-time system analysis. There, the characteristic equation is formed from the ODE's derivative-coefficient pattern (rather than from a difference equation's shift-coefficient pattern), homogeneous solutions take the exponential form Ce^(st) governed by the characteristic roots s (rather than Cr^n governed by characteristic roots r), and a particular solution is likewise assumed to match the forcing function's form, modified by an extra factor of t (rather than n) whenever a root-coincidence (resonance) case arises. This parallel is not a coincidence - it reflects that both problem classes belong to the same general theory of linear constant-coefficient systems, differing only in whether the underlying independent variable is continuous or discrete.

Forced response terminology: the particular solution yp(n) found in each part above is also commonly referred to as the forced response of the system, since it represents the specific response component directly attributable to (and persisting for as long as) the external forcing input x(n), as distinct from the homogeneous solution yh(n), which is correspondingly termed the natural response, representing the system's own inherent characteristic behavior (governed solely by its characteristic roots) that would persist even in the absence of continued external forcing, decaying or growing according to the stability properties discussed above.