Cascade and Parallel Systems

Cascade and parallel connections are the two basic ways to combine LTI systems into a larger processing chain. Understanding how overall transfer functions and impulse responses compose under these connections lets you design equalizers, multi-stage amplifiers, and filter banks from simpler building blocks.

Core Concept

In a cascade connection, the output of the first system feeds directly into the input of the second. Because multiplication in the frequency domain corresponds to convolution in the time domain, the overall impulse response is h1(t) convolved with h2(t), and the overall transfer function is H1(s) * H2(s). The order of cascading does not change the overall input-output relationship due to the commutativity of convolution.

In a parallel connection, the same input drives both systems simultaneously, and their outputs are summed. The overall impulse response is simply h1(t) + h2(t), and the overall transfer function is H1(s) + H2(s). This structure appears in bank-of-filters designs where different frequency bands are processed separately and then recombined.

Stability of combinations is straightforward. A cascade is BIBO stable only if every subsystem is individually stable. A parallel combination is stable if each branch is stable, because an unstable branch drives the sum to infinity. Checking poles of the combined transfer function confirms this: cascade poles are the union of both pole sets; parallel poles are the poles of the combined rational function after addition.

Key Formula

Cascade: H(s) = H1(s) * H2(s), equivalently h(t) = h1(t) * h2(t) (convolution).

Parallel: H(s) = H1(s) + H2(s), equivalently h(t) = h1(t) + h2(t).

Mixed: a combination of both uses algebraic manipulation of transfer function blocks following standard block diagram reduction rules.

Problem: Find overall H(s) for H1(s) = 1/(s+1) in cascade with H2(s) = s/(s+3).

Step 1 — Cascade rule:
  H(s) = H1(s) * H2(s)
       = [1/(s+1)] * [s/(s+3)]
       = s / [(s+1)(s+3)]

Step 2 — Poles and zeros:
  Zero at s = 0
  Poles at s = -1 and s = -3

Step 3 — Stability check:
  Both poles in left-half plane -> BIBO stable cascade.

Final Answer: H(s) = s / [(s+1)(s+3)]
Poles: -1, -3. Zero: 0. System is causal and stable.

Exam Tip: When two systems share a common factor in numerator and denominator after cascade, there is a pole-zero cancellation. GATE and VTU examiners sometimes place an unstable pole that cancels with a zero; the resulting H(s) appears stable, but the internal state between the two subsystems can grow without bound. Always check individual subsystem stability, not just the overall transfer function.

Properties Summary

• Cascade in s-domain: H(s) = H1(s) * H2(s); poles combine as union of both sets.
• Cascade in time-domain: h(t) = h1(t) * h2(t) (convolution of impulse responses).
• Parallel in s-domain: H(s) = H1(s) + H2(s); add over common denominator.
• Parallel in time-domain: h(t) = h1(t) + h2(t) (sum of impulse responses).
• Commutativity of cascade: H1*H2 = H2*H1; swapping order gives same I/O relation.
• Stability of cascade: stable only if all individual subsystems are stable.
• Block diagram reduction: cascade = multiply, parallel = add, feedback requires Mason's rule or the standard feedback formula.

Quick Revision

• Cascade: H = H1 * H2 (multiply transfer functions, convolve impulse responses).
• Parallel: H = H1 + H2 (add transfer functions, add impulse responses).
• Cascade poles are the union of H1 poles and H2 poles.
• Swapping cascade order does not change overall transfer function.
• Unstable subsystem in cascade makes the whole cascade unstable.
• Pole-zero cancellation in cascade can hide internal instability.
• Exam trap: students add H1 and H2 for cascade; cascade is multiplication, parallel is addition.