Frequency Response Masking: FIR Filter Design

• FRM is a digital filter design methodology that creates sharp linear-phase FIR filters by interpolating a low-order prototype and applying low-order masking filters.
• The technique significantly reduces hardware complexity by offering up to 76–89% multipliers savings compared to direct FIR designs, making it ideal for hearing aids and SDR.
• FRM-based filter banks support both uniform and non-uniform spectral decompositions through reconfigurable architectures, enabling precise subband extraction with minimal cost.

Frequency Response Masking (FRM) is a digital filter design methodology that enables the realization of sharp, linear-phase finite impulse response (FIR) filters with extremely narrow transition bands using substantially lower computational complexity compared to direct FIR implementations. The technique exploits the spectral periodicity introduced by interpolating a prototype FIR filter and isolates the desired frequency band(s) using low-order masking filters. FRM and its variants are foundational in applications requiring highly selective subband decomposition with stringent resource constraints, such as digital hearing aids and software-defined radio (SDR) channelizers (Sebastian et al., 2020, K. et al., 2020).

1. Fundamental Principles of Frequency Response Masking

The central concept of FRM is to synthesize narrow transition-band filters from wideband, low-order prototype FIR filters through interpolation (delay expansion) and subsequent masking. The process can be summarized as follows:

• Prototype Expansion: A prototype FIR filter $H_p(z)$ of low order and broad transition width is stretched via interpolation by replacing each $z^{-1}$ delay with $z^{-M}$, yielding $H_p(z^M)$. This generates a frequency response consisting of multiple, widely spaced images of the prototype's spectrum across $[0, 2\pi)$.
• Image Selection by Masking: Sparse masking filters $H_{m1}(z), H_{m2}(z),\ldots$ are designed to carve out one or more desired spectral replicas, while attenuating the remaining images. When perfect reconstruction is required, the complementary prototype $H_p^c(z) = z^{-(N_p-1)} - H_p(z)$ is processed in parallel, with its own masking chain.
• Block Diagram Structure:
  • Path A: $$X(z) \rightarrow H_p(z^M) \rightarrow H_{m1}(z) \rightarrow + \rightarrow Y(z)$$
  • Path B: $$X(z) \rightarrow H_p^c(z^M) \rightarrow H_{m2}(z) \rightarrow + \rightarrow Y(z)$$
  • $M$ (interpolation factor) determines the number of images and thus the granularity of achievable subband edges (Sebastian et al., 2020).

This approach enables the creation of extremely sharp transition bands with significantly fewer filter taps by confining most selectivity to the masking filters, whose orders remain low due to the wide separation of images.

2. Mathematical Formulation and Filter Optimization

Prototype Filter Design: The prototype FIR filter $H_p(z)$ of length $N_p$ is optimized to meet passband and stopband specifications:

• Passband: $\Omega \in [0, \Omega_p]$, $|H_p(e^{j\Omega}) - 1| \leq \delta_p$
• Stopband: $\Omega \in [\Omega_s, \pi]$, $|H_p(e^{j\Omega})| \leq \delta_s$
• Design criteria typically follow min-max (Chebyshev, equiripple) or least-squares optimization.

Spectral Imaging and Masking:

• Interpolation by $M$ generates $M$ periodic images: transitions shift to $\omega = M \cdot \Omega_p \pmod{2\pi}$
• The desired target filter is formed as:

$$H_{des}(e^{j\omega}) = H_p(e^{jM\omega}) H_{m1}(e^{j\omega}) + H_p^c(e^{jM\omega}) H_{m2}(e^{j\omega})$$

• For configurations with $K$ masking stages:

$$H_{des}(z) = \sum_{k=0}^{K} H_p^{(k)}(z^M) H_{m_k}(z)$$

where $H_p^{(0)} = H_p$, $H_p^{(1)} = H_p^c$, and each $H_{m_k}(z)$ isolates a specific band (Sebastian et al., 2020).

3. FRM-Based Filter Bank Synthesis for Non-Uniform and Uniform Decomposition

Non-Uniform Filter Banks (NUFBs):

• Subband edges $\{\omega_1, \omega_2, ..., \omega_L\}$ are chosen for applications such as audiogram-based hearing loss compensation.
• $M$ is selected so that the images of $H_p(z^M)$ align with these subband boundaries. Passband edges for the prototype are fixed to $\Omega_p = \min_i (\omega_i)/M$.
• Low-order masking filters selectively extract the desired subbands, each targeting a corresponding image.

Uniform Filter Banks (UFBs) and ModFRM Architecture:

• The ModFRM technique replaces classical model filters with power-complementary, DFT-modulated filter bank (FB) channels derived from a prototype $h_{a,0}(n)$.
• Alternate masking of even and odd channel groups using two base masking filters ($H_{Ma}(z), H_{Mc}(z)$) yields all $M$ subbands:

$$H_k(z) = \begin{cases} H_{even}(z) H_{Ma}(z W_M^k) + H_{odd}(z) H_{Mc}(z W_M^k), & k\ \text{even} \ H_{even}(z) H_{Mc}(z W_M^k) + H_{odd}(z) H_{Ma}(z W_M^k), & k\ \text{odd} \end{cases}$$

where $W_M = e^{-j2\pi/M}$ and $H_{even}$, $H_{odd}$ are sums over interpolated channels (K. et al., 2020).

• Non-uniform FBs are constructed by merging adjacent subbands post-analysis, preserving linear phase through sign adjustments.

4. Computational Complexity and Hardware Implications

A major advantage of FRM is substantial multiplier savings:

Configuration	Prototype Order(s)	Mask Filters	Total Multipliers	Attenuation (dB)	Reference
8-band NUFB (Lian & Wei)	2 × 16	10	15	80	(Sebastian et al., 2020)
16-band NUFB (Wei & Lian)	34	--	34	60	(Sebastian et al., 2020)
32-channel ModFRM+IFIR	Prototype + IFIR-masked	--	137	--	(K. et al., 2020)

• In direct $N$-tap FIR realization with $L$ subbands, computation requires $L\cdot N$ multipliers per input sample.
• In FRM, a half-band prototype uses $N_p/2$ multiplies, masks typically $6$–$12$ multiplies each, yielding nearly an order-of-magnitude overall reduction (Sebastian et al., 2020).
• ModFRM with IFIR masking achieves even lower counts; for a 32-channel filter bank with transition width $0.0025\pi$, the total is reduced to $137$ multipliers—76–89% savings relative to prior reconfigurable FRM or NMDFB hybrids (K. et al., 2020).
• The masking and prototype structures exploit coefficient symmetry, further minimizing hardware.
• Increased group delay, generally on the order of $(N_p \cdot M)/2$, is the trade-off; this is acceptable in hearing aids and SDR provided total latency is within a few milliseconds (Sebastian et al., 2020).

5. Design Steps and Reconfigurability

The procedure for FRM-based non-uniform FIR filter bank construction comprises:

a. Subband Edge Specification: Definition based on application-specific frequency grid (e.g., audiogram points for hearing loss). b. Interpolation Factor Selection: $M$ is chosen so prototype images correspond to desired subbands. c. Prototype Filter Design: Low-order, often half-band for linear phase and zero odd-index coefficients. d. Masking Filter Design: Each subband uses a dedicated low-order mask. e. Parallel Implementation: All masking-filtered paths summed to produce the composite output. f. Subband Gain Control: Per-subband gains applied for spectral shaping (e.g., audiogram matching) (Sebastian et al., 2020).

Reconfigurability: In ModFRM, the interpolation factor $L$ governs subband widths and center frequencies. Changing $L$ adapts the bank without redesigning $h_{a,0}(n)$ or masking filters. Uniform-to-non-uniform adaptation proceeds by grouping adjacent channels, preserving linear phase with appropriate sign corrections (K. et al., 2020). This flexibility is essential for SDR and multistandard extraction where frequency plans are dynamic.

6. Practical Applications and Measured Outcomes

• Hearing Aids: FRM-based non-uniform filter banks achieve high stopband attenuation (60–80 dB) and audiogram-matching error as low as $0.4$–$4$ dB with a multiplier count one to two orders of magnitude below conventional direct FIR banks. Specific results:
  • 8-band: $<4$ dB matching error (reduced to $1.25$ dB via least-squares gain optimization) (Sebastian et al., 2020).
  • 16-band: $0.4$ dB matching error at 60 dB attenuation.
• Software Defined Radio (SDR) Channelizers: ModFRM allows extraction of multiple communication standards with a single prototype and two masks. For 9 simultaneous standards, only $137$ multipliers are needed, demonstrating scalable hardware efficiency (K. et al., 2020).
• Scalability: By adjusting $M$ or $L$, FRM can target both narrow and wide bands, variable subband widths, and arbitrary spectral partitions at low cost.

7. Advantages, Limitations, and Context

Advantages:

• Sharp, linear-phase selectivity with filter orders $10$–$20\times$ lower than direct design.
• Flexibility in realizing both uniform and non-uniform filter banks.
• Modular, reconfigurable architecture; especially in ModFRM, changing the interpolation factor enables rapid frequency plan adaptation.
• Half-band prototypes exploit multiplier-less arithmetic (odd-order coefficients zero-valued).

Limitations:

• Group delay is increased by the stretch factor, which may constrain use in ultralow-latency systems.
• Design of masking filters becomes increasingly complex as the number of required subbands and spectral irregularities rises.
• The number of masking filters or channel-merging steps scales with subband count for non-uniform decompositions.

In summary, FRM remains a critical technique for low-complexity, sharp linear-phase FIR filter and filter bank design in constrained hardware environments. ModFRM and IFIR-based optimizations further advance this paradigm, supporting high channel counts, reconfiguration, and fine-grained spectral control in hearing aid and SDR applications (Sebastian et al., 2020, K. et al., 2020).