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The frequency response H(jω) = |H(jω)|e^(jφ(ω)) shows how a system responds to sinusoidal inputs at different frequencies. Magnitude response reveals gain at each frequency; phase response shows time delay or lead. Together they fully characterize the steady-state behavior of linear systems.
From your study of transfer functions and poles and zeros, you know that a linear time-invariant (LTI) system is fully described by its transfer function H(s). You also know the representation of signals as magnitude and phase spectra. The frequency response connects these two ideas: it is what you get when you evaluate H(s) along the imaginary axis, replacing s with jω. The result H(jω) is a complex function of real frequency ω, and its magnitude and phase tell you exactly how any sinusoidal input is transformed by the system.
Why evaluate on the imaginary axis? Any stable LTI system driven by a sinusoidal input A·cos(ωt) at steady state produces an output A·|H(jω)|·cos(ωt + φ(ω)). The input frequency is preserved — LTI systems cannot create new frequencies — but the amplitude is scaled by magnitude response |H(jω)| and the phase is shifted by phase response φ(ω) = ∠H(jω). This is the foundation of filter design: a lowpass filter has |H(jω)| ≈ 1 at low frequencies (signals pass through unchanged) and |H(jω)| ≈ 0 at high frequencies (signals are attenuated). The phase response tells you how much each frequency component is delayed relative to the input.
Computing H(jω) from H(s). Given a transfer function H(s) = N(s)/D(s) with numerator and denominator polynomials, substitute s = jω to get H(jω) = N(jω)/D(jω). The magnitude is |H(jω)| = |N(jω)|/|D(jω)| and the phase is φ = ∠N(jω) − ∠D(jω). For factored forms H(s) = K·∏(s−z_i)/∏(s−p_j), the magnitude is the product of distances from ω on the jω-axis to each zero divided by distances to each pole, scaled by K. The phase is the sum of angles from ω to each zero minus the sum of angles to each pole. This geometric interpretation directly connects the pole-zero plot to the frequency response: poles near the imaginary axis create peaks in |H(jω)| at the corresponding frequency (resonances), while zeros on or near the axis create dips or notches.
Phase response and group delay. Phase response φ(ω) is not just a nuisance — it matters for signal fidelity. If φ(ω) is linear in ω (φ = −τω), then every frequency component is delayed by the same time τ, and the output waveform is a perfect replica of the input, just shifted. This is linear phase. Non-linear phase distorts waveforms: different frequency components arrive at different times, smearing pulses or creating dispersion. Group delay τ_g(ω) = −dφ/dω is a frequency-resolved measure of delay: constant group delay = linear phase = no dispersion. Audio amplifiers and communications receivers must have near-linear phase over their operating band; control systems care about phase at specific frequencies because phase shift determines stability margins.
Bode plots as a practical tool. The magnitude response plotted in decibels (dB = 20·log₁₀|H(jω)|) on a log-frequency axis is a Bode magnitude plot. In this representation, products become sums: |H| = |K|·∏|s−z_i|/∏|s−p_j| becomes a sum of log-magnitude contributions, one per pole and zero. Each real pole at s = −a contributes a −20 dB/decade rolloff above ω = a; each real zero contributes +20 dB/decade. Complex conjugate pole pairs near the imaginary axis create resonance peaks whose height depends on the quality factor Q (or equivalently the damping ratio ζ). These asymptotic approximations — the Bode straight-line method — let you sketch magnitude and phase responses by inspection from the pole-zero locations, building powerful intuition for filter behavior before any numerical computation.