DSP 101
M7 · L4
Module 7 — FIR Filter Design
Linear Phase Property
Why coefficient symmetry is the key to distortion-free filtering — and how it spawns four distinct types of FIR filters, each with unique frequency-response constraints.
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DSP 101
M7 · L4
Definition
Equal Delay for All Frequencies
Linear Phase Response
H(e^{j\omega}) = A(\omega)\, e^{-j\omega(N-1)/2}
A(ω) is real — all phase information is in the exponential. Every frequency component is delayed by exactly (N−1)/2 samples, so the waveform shape is perfectly preserved at the output.
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DSP 101
M7 · L4
Group Delay
Constant Delay = Zero Distortion
Group Delay
\tau(\omega) = -\frac{d\phi(\omega)}{d\omega} = \frac{N-1}{2}
- Constant τ(ω) — every frequency waits the same time to exit the filter
- Variable τ(ω) in IIR filters — some frequencies arrive earlier, smearing the waveform
- For an N-tap FIR: group delay = (N−1)/2 samples at every frequency
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DSP 101
M7 · L4
Symmetry Condition
The Structural Guarantee
Even / Odd Symmetry
h[n] = h[N\!-\!1\!-\!n] \;\text{(even)} \qquad h[n] = -h[N\!-\!1\!-\!n] \;\text{(odd)}
Why It Works
Symmetry forces all roots of H(z) to appear in conjugate-reciprocal pairs — the algebraic reason the phase is exactly linear.
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DSP 101
M7 · L4
Four Filter Types
Symmetry × Length = Four Types
I
Sym, Odd N
II
Sym, Even N
III
Anti, Odd N
IV
Anti, Even N
Type I is the most versatile — no forced zeros, works for all filter shapes. The windowed-sinc method produces Type I by default.
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DSP 101
M7 · L4
Forced Zeros
Not a Choice — a Constraint
- Type II: H(e^jπ) = 0 always — cannot implement HP or BS filters
- Type III: H(e^j0) = H(e^jπ) = 0 — blocked at DC and Nyquist
- Type IV: H(e^j0) = 0 — cannot implement LP filters
- These zeros are mathematical consequences of length and symmetry — they cannot be removed by changing coefficients
Design Rule
HP filter → use Type I or IV. LP filter → use Type I or II. Hilbert transformer → use Type III or IV.
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DSP 101
M7 · L4
Applications
When Does Phase Matter?
- Audio: preserves transient character and stereo image in crossovers and EQ
- Digital communications: prevents ISI in matched filter receivers
- Biomedical: preserves ECG/EEG waveform morphology (P, QRS, T waves)
- Image processing: prevents edge shift in symmetric 2D FIR kernels
- Not needed: power spectral estimation, envelope detection, energy measurements — use IIR for efficiency
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DSP 101
M7 · L4
Key Takeaways
What You Learned
- Linear phase: φ(ω) = −α ω — all frequencies delayed equally, waveform preserved
- Group delay τ = (N−1)/2 is constant — the quantitative measure of zero phase distortion
- Symmetry h[n] = ±h[N−1−n] is necessary and sufficient for linear phase in FIR
- Four types from symmetry × length: Type I is the universal workhorse
- Types II, III, IV have forced zeros — wrong type = degraded response near DC or Nyquist
- Use FIR when waveform fidelity matters; use IIR when efficiency matters
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