Q2Digital Signal Processing

The given impulse response h(n) = {1,2,3,4,3,2,1} for n=0,...,6 (M=7 taps) is symmetric about its center: h(0)=h(6)=1, h(1)=h(5)=2, h(2)=h(4)=3, h(3)=4 (center, unpaired since M is odd). This satisfies h(n) = h(M-1-n) = h(6-n), confirming this is a Type I linear-phase FIR filter (odd length, symmetric coefficients), which guarantees exactly linear phase response.

Standard (non-exploiting) Direct Form realization would require 7 separate multipliers (one per tap) and 6 adders, with the output computed as:

Symmetric (linear-phase-optimized) Direct Form realization: because h(k) = h(6-k), pairs of input samples that are multiplied by the same coefficient value can first be added together, and the sum multiplied once by the shared coefficient, rather than performing two separate multiplications. Regrouping the sum using this symmetry:

This symmetric structure requires only 4 multipliers (instead of 7) — one for each distinct coefficient value {1, 2, 3, 4} — while still requiring the full 6-sample delay line (7 taps, so 6 unit-delay elements, z^-1 through z^-6) to hold all the required input samples. The structure is realized by tapping the delay line at symmetric pairs of positions (n and n-6, n-1 and n-5, n-2 and n-4), summing each symmetric pair using an adder before it reaches its shared multiplier, and finally summing all four scaled partial results together to form y(n). This exploitation of coefficient symmetry, possible specifically because linear-phase FIR filters by definition have symmetric or antisymmetric coefficients, reduces the multiplier count from M to approximately M/2 (exactly (M+1)/2 for odd M as here, since the middle tap is unpaired), which is a substantial hardware saving of nearly 43% (from 7 down to 4 multipliers) for this specific 7-tap filter, and is one of the primary practical motivations for preferring linear-phase FIR filter designs in resource-constrained DSP hardware implementations.

Verification of the exact linear-phase property: the frequency response of this symmetric filter can be written as H(ejω) = e^(-j3ω)·A(ω), where A(ω) is a real-valued amplitude function (since the coefficients are symmetric about the center tap at n=3), confirming that the phase response is exactly -3ω, i.e., perfectly linear in ω with a constant group delay of exactly 3 samples across all frequencies. This exact linearity of phase (as opposed to only approximately linear phase, as would result from an arbitrary, non-symmetric IIR or FIR design) is the defining practical benefit of this class of filters: every frequency component of the input signal experiences the identical 3-sample delay when passing through the filter, so the filter introduces no phase distortion (no relative smearing between different frequency components of a wideband signal), which is essential in applications such as audio processing, image processing, and data transmission where preserving the waveform shape (not just the frequency-magnitude content) of the signal matters.

General classification note: since M=7 is odd and the coefficients are symmetric (not antisymmetric), this filter belongs to the 'Type I' linear-phase FIR category in the standard four-type classification (Type I: odd length, symmetric; Type II: even length, symmetric; Type III: odd length, antisymmetric; Type IV: even length, antisymmetric), each of which has different inherent constraints on the filter's frequency response at ω=0 and ω=π (for instance, Type III and IV filters necessarily have a zero at ω=0, making them unsuitable for standard lowpass designs but well suited for differentiators and Hilbert transformers), a classification that is important context whenever selecting the appropriate FIR filter length and symmetry type for a given filter design specification.