Summary

This article explains the precise impulse-response constraints that make tapped-delay-line FIR filters exhibit perfect linear phase. Rick Lyons shows both the familiar real-coefficient symmetry/antisymmetry conditions and the less-known conjugate-reflection constraint for complex coefficients, and explains the implications for phase, group delay, and filter design.

Key Takeaways

• State the real-coefficient symmetry rules: h[n] = +h[N-1-n] or h[n] = -h[N-1-n] produce linear-phase FIRs with known phase offsets.
• Apply the complex-coefficient conjugate-reflection constraint (h[n] = e^{jφ} h*[N-1-n]) that generalizes linear phase to complex-valued FIRs.
• Compute group delay directly from filter length: Ï„ = (N-1)/2 for linear-phase FIRs and verify via phase unwrapping or numerical group-delay plots.
• Design linear-phase FIRs by enforcing symmetry (use windowing, Parks–McClellan, or frequency-sampling methods) and exploit coefficient folding to reduce computation.
• Detect and avoid pitfalls: recognize when near-symmetry, numerical round-off, or coefficient quantization breaks linear-phase behavior in practice.

Who Should Read This

DSP engineers and researchers (advanced undergrad/grad level or practicing engineers) designing or analyzing FIR filters for audio, communications, radar, or general signal-processing applications.

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