Transfer Function

The transfer function of a linear time-invariant system is the ratio of the Laplace transform of the output to the Laplace transform of the input, with all initial conditions set to zero. Filter designers, control engineers, and communication system analysts all express system behavior through transfer functions because they directly reveal frequency response, stability, and gain.

Core Concept

The transfer function H(s) is defined only for systems that are linear and time-invariant. Setting all initial conditions to zero isolates the system's own dynamics from any energy stored before the input was applied. This makes H(s) a pure property of the system structure, not of any particular initial state.

The poles of H(s) are the values of s where the denominator is zero. They determine the natural frequencies of the system. If any pole has a positive real part, the system is unstable: its natural response grows without bound. Poles in the left half-plane correspond to decaying exponentials. A pole at the origin corresponds to integration.

The zeros of H(s) are the values of s where the numerator is zero. They are the input frequencies that are completely blocked by the system. Placing a zero at s = j*omega_0 means the system produces no output at angular frequency omega_0. Filter design often involves choosing zeros to suppress specific interfering tones.

Key Formula

H(s) = Y(s)/X(s) with zero initial conditions. For a system described by a differential equation a_n*y^(n) + ... + a_0*y = b_m*x^(m) + ... + b_0*x, the transfer function is H(s) = (b_m*s^m + ... + b_0)/(a_n*s^n + ... + a_0). The order of the system equals the degree of the denominator polynomial. The DC gain of the system is H(0) = b_0/a_0. The frequency response is obtained by substituting s = j*omega.

Given: d^2y/dt^2 + 5*dy/dt + 6*y = 2*dx/dt + x
Find H(s) and identify poles and zeros.

Step 1: Take Laplace transform (zero initial conditions)
  s^2*Y(s) + 5s*Y(s) + 6*Y(s) = 2s*X(s) + X(s)

Step 2: Factor
  Y(s)[s^2 + 5s + 6] = X(s)[2s + 1]

Step 3: Form transfer function
  H(s) = Y(s)/X(s) = (2s + 1) / (s^2 + 5s + 6)

Step 4: Factor denominator
  s^2 + 5s + 6 = (s+2)(s+3)

  H(s) = (2s + 1) / ((s+2)(s+3))

Step 5: Find poles and zeros
  Zero: 2s+1 = 0  =>  s = -1/2
  Poles: s = -2 and s = -3

Step 6: Stability check
  Both poles have negative real parts => system is BIBO stable

DC Gain: H(0) = 1 / (2*3) = 1/6

Exam Tip: BIBO stability of a causal LTI system requires ALL poles of H(s) to be in the open left half-plane, meaning Re(s) < 0 for every pole. A pole on the imaginary axis makes the system marginally stable, not stable. For GATE, when asked to determine stability from H(s), factor the denominator completely and check every pole location individually.

Properties Summary

• Definition: H(s) = Y(s)/X(s) with all initial conditions zero
• Poles are denominator roots; govern natural response and stability
• Zeros are numerator roots; frequencies where output is zero for sinusoidal input
• System order = degree of denominator of H(s)
• DC gain = H(s) evaluated at s=0 = b_0/a_0
• Frequency response: substitute s = j*omega to get magnitude and phase
• Impulse response h(t) = L^{-1}{H(s)}; system output for delta input

Quick Revision

• H(s) = Y(s)/X(s) with zero initial conditions; purely a system property
• Derived by taking Laplace transform of the governing differential equation
• Poles in left half-plane: stable; right half-plane: unstable; imaginary axis: marginally stable
• Zeros block specific frequency components from reaching the output
• DC gain is H(0); high-frequency gain is the leading coefficient ratio
• Frequency response found by substituting s = j*omega
• Cascading two systems: H_total(s) = H1(s) * H2(s)
• Exam trap: not setting initial conditions to zero before forming H(s); any non-zero initial condition produces extra source terms that make the ratio Y(s)/X(s) not equal to H(s)