FIR Structures

Once the FIR filter coefficients are determined, they must be implemented in hardware or software using a specific filter structure. The choice of structure affects computational complexity, memory requirement, numerical precision, and whether the linear-phase property is exploited efficiently. FIR structures are directly examined in GATE problems on signal flow graphs and complexity estimation.

Core Concept Explanation

An FIR filter of order M has M+1 coefficients h[0], h[1], ..., h[M]. The output is the convolution sum y[n] = sum over k from 0 to M of h[k]·x[n-k]. Any implementation of this equation is called a filter structure, and different structures rearrange the computations in different ways without changing the input-output relationship.

The direct form structure implements this convolution sum literally. It uses a chain of M unit delay elements (z-1 blocks), M+1 multipliers (one for each coefficient), and M adders to produce the final output. This is also called a tapped delay line or transversal filter. For a filter of length N (meaning N coefficients, order N-1), the direct form requires N multiplications and N-1 additions per output sample.

The cascade form structure factorizes the transfer function H(z) into a product of second-order sections: H(z) = product of H_k(z) where each H_k(z) = (b_k0 + b_k1·z-1 + b_k2·z-2). Each second-order section is implemented independently as a small FIR filter, and the sections are connected in series. For filters with complex conjugate zeros, cascading second-order sections keeps all coefficients real, which is numerically preferable to direct form for long filters.

Linear Phase FIR Structure

When an FIR filter has a symmetric impulse response, meaning h[n] = h[N-1-n], the filter has exactly linear phase. This symmetry can be exploited structurally to reduce the number of multiplications by approximately half. Instead of multiplying each coefficient independently, pairs of symmetric coefficients h[k] and h[N-1-k] are combined: their corresponding delayed input samples are added first, and then only one multiplication is performed. For a filter of odd length N, the number of multiplications reduces to ceil(N/2).

There are four types of linear phase FIR filters based on the length and symmetry. Type I has odd length and even symmetry, and it can implement any frequency selective filter. Type II has even length and even symmetry, and always has a zero at ω = π, making it unsuitable for high-pass filters. Type III has odd length and odd symmetry (antisymmetric), with zeros at ω = 0 and ω = π. Type IV has even length and odd symmetry, with a zero at ω = 0. These constraints are critical for GATE filter design questions.

Mathematical Expression

For a Type I linear phase FIR filter of odd length N = 2M+1, the transfer function exploiting symmetry is: H(z) = z^(-M) · [h[M] + sum from k=1 to M of h[M-k]·(z^k + z^(-k))]. This shows the filter as a linear phase delay of M samples multiplied by a zero-phase filter. The computational savings are direct: instead of 2M+1 multiplications, only M+1 multiplications are needed after exploiting symmetry.

Numerical Example

Given an FIR filter length, determine the number of multiplications needed in direct form and in the linear-phase-exploiting structure, and find the savings.

Given:
FIR filter with N = 31 coefficients (Type I, odd length, symmetric)

Why this formula applies:
Direct form uses one multiplier per coefficient.
Linear phase form pairs symmetric coefficients: multiplications = ceil(N/2)

Formula:
Direct form multiplications = N
Linear phase form multiplications = ceil(N/2)
Savings = N - ceil(N/2)

Substitution:
N = 31
Direct form = 31 multiplications
Linear phase = ceil(31/2) = ceil(15.5) = 16 multiplications

Calculation:
Savings = 31 - 16 = 15 multiplications per output sample
Savings percentage = 15/31 × 100 ≈ 48.4%

Final Answer:
Linear phase structure saves 15 multiplications, approximately 48% reduction in multiply operations.

Exam Tip: GATE commonly asks about Type II FIR filters. Always remember Type II has even length with symmetric coefficients and always has H(e^jπ) = 0 (zero at ω=π). This makes it unsuitable for high-pass or band-stop filter design. A very common trap is selecting Type II for a high-pass specification.

Cascade vs Direct Form Comparison

• Direct form uses a single tapped delay line with N multiplications. It is simple to implement but prone to coefficient sensitivity issues for long high-order filters.
• Cascade form breaks the filter into second-order sections. Each section has only 3 coefficients, which minimizes roundoff error accumulation and is preferred for fixed-point implementations.
• Linear phase structure exploits h[n] = h[N-1-n] symmetry to reduce multiply operations by nearly 50 percent. This directly reduces hardware area and power consumption in VLSI implementations.
• All four FIR structure types produce the same frequency response. The differences are only in computational efficiency and numerical behavior, not in filter specifications.
• For real-time DSP processors, the direct form is often implemented using a circular buffer and multiply-accumulate (MAC) instructions, making it efficient despite requiring N multiplications.

Quick Revision

• Direct form FIR: N multiplications, N-1 additions, N-1 delay elements for N coefficients.
• Linear phase Type I (odd N, symmetric): ceil(N/2) multiplications. Nearly 50% savings.
• Type I: odd N, even symmetry. Type II: even N, even symmetry (zero at ω=π, no HPF). Type III: odd N, antisymmetric. Type IV: even N, antisymmetric.
• Cascade form uses second-order sections: better numerical stability for long filters.
• Exam trap: Type II cannot implement high-pass or band-stop filters due to mandatory zero at ω=π.
• All FIR structures have the same input-output transfer function. Structure choice affects only implementation cost and precision.
• Group delay of a linear phase FIR = (N-1)/2 samples, constant for all frequencies.