FIR Characteristics

FIR filters are among the most widely used structures in digital signal processing. Their properties, particularly linear phase and unconditional stability, make them the preferred choice in applications where phase distortion must be controlled. Understanding these characteristics is central to filter design and GATE problem solving.

Core Concept: What Makes FIR Different

A finite impulse response (FIR) filter is one in which the impulse response h[n] has a finite number of nonzero samples. The output y[n] is a weighted sum of the current and past N input samples only, with no feedback from past outputs. This no-feedback condition is the defining structural feature. The transfer function H(z) = sum of h[k]*z^(-k) for k=0 to N-1, which is a polynomial in z^(-1) with no denominator other than z^(-(N-1)).

Linear Phase Property

The most important characteristic of FIR filters is their ability to achieve exact linear phase. Linear phase means the phase response angle(H(e^jw)) = -alpha*w for some constant alpha, which implies all frequency components are delayed by the same amount (group delay = alpha = constant). This preserves the shape of the signal and prevents phase distortion.

Linear phase is achieved when the impulse response is symmetric: h[n] = h[N-1-n] (even symmetry) or antisymmetric: h[n] = -h[N-1-n]. For an N-tap symmetric FIR filter, the group delay is exactly (N-1)/2 samples. This is a constant, independent of frequency, which is the definition of linear phase.

Four Types of Linear Phase FIR Filters

Linear phase FIR filters are classified into four types based on the length N and the symmetry condition. Type I has odd N and even symmetry. It has no restrictions on frequency response and can realize all four classical filter types. Type II has even N and even symmetry, but H(pi) = 0 always, so it cannot be used for HPF or BSF. Type III has odd N and odd (antisymmetric) symmetry, with H(0) = H(pi) = 0, suitable only for BPF-like designs. Type IV has even N and antisymmetric symmetry with H(0) = 0.

Stability

FIR filters are unconditionally stable. Stability requires that all poles of H(z) lie strictly inside the unit circle. Since the FIR transfer function has no denominator polynomial (or equivalently, all poles are at the origin z=0, which is inside the unit circle), the stability condition is always satisfied regardless of coefficient values. This is a major advantage over IIR filters, which can become unstable if coefficients are not chosen carefully.

Mathematical Expression

The output of an N-tap FIR filter is y[n] = sum over k from 0 to N-1 of h[k]*x[n-k]. The frequency response is H(e^jw) = sum over k of h[k]*e^(-jwk). For a symmetric filter with h[n] = h[N-1-n], the frequency response can be expressed as H(e^jw) = e^(-jw*(N-1)/2) * A(w), where A(w) is a real-valued function. This confirms the linear phase: the e^(-jw*(N-1)/2) term represents the constant group delay of (N-1)/2.

Numerical Example

Given:
FIR filter with h[0]=1, h[1]=2, h[2]=3, h[3]=2, h[4]=1 (N=5, Type I)

Why this formula applies:
Symmetric h[n] guarantees linear phase with group delay = (N-1)/2

Formula:
Group delay = (N-1)/2 samples
Check symmetry: h[0]=h[4]=1, h[1]=h[3]=2, h[2]=3 (center)

Substitution:
Group delay = (5-1)/2 = 4/2 = 2 samples
Verify: h[n] = h[4-n] → 1=1, 2=2, 3=3 (symmetric confirmed)

Calculation:
Frequency response at w=0:
H(e^j0) = 1+2+3+2+1 = 9
Group delay = 2 samples at all frequencies (constant = linear phase)

Final Answer:
Group delay = 2 samples (constant at all frequencies).
The filter is Type I (N=5 odd, symmetric). H(0)=9. Linear phase is confirmed.

Exam Tip: For GATE, always check symmetry to confirm linear phase. Group delay = (N-1)/2. Type II cannot design HPF because H(pi)=0 always. Type III cannot design LPF or HPF because both H(0) and H(pi) are zero.

Implementation Advantages

• No feedback path means no recursive computation and no risk of overflow accumulation over time.
• Symmetric coefficients allow efficient implementation using a folded structure that halves the number of multiplications: (N+1)/2 multiplications for odd N.
• FIR filters can be implemented using fast convolution via FFT for very long filters, reducing computation from O(N) per sample to O(log N) per sample asymptotically.
• Fixed-point quantization of FIR coefficients does not cause instability, unlike IIR filters where coefficient quantization can move poles outside the unit circle.

Quick Revision

• FIR: finite impulse response, no feedback, all poles at z=0, always stable.
• Linear phase requires symmetric (or antisymmetric) impulse response h[n].
• Group delay = (N-1)/2 samples for symmetric FIR. Constant at all frequencies.
• Type I (N odd, symmetric): no restrictions. Type II (N even, symmetric): H(pi)=0.
• Type III (N odd, antisymmetric): H(0)=H(pi)=0. Type IV (N even, antisymmetric): H(0)=0.
• Folded structure reduces multiplications to ceil(N/2) for symmetric FIR.
• FIR filters are preferred when linear phase and guaranteed stability are required.