Linear-phase Finite-length Impulse Response Research Articles

Unified Filter Order Estimate for Minimax-Designed Linear-Phase FIR Wideband and Lowpass Digital Differentiators

Digital differentiators enable the computation of the derivative of a continuous-time signal at discrete time instances, and they are used in many signal processing applications. This paper derives a unified filter order estimate for digital differentiators that are realized with linear-phase finite-length impulse response filters and designed in the minimax sense. The estimate is useful at the high-level system design when assessing the implementation complexity and it enables fewer designs when finding the minimal filter order required to satisfy a prescribed tolerable approximation error. The proposed unified estimate covers both wideband and lowpass differentiators of integer degrees up to ten. Furthermore, degree-individual filter order estimates are derived which improve and extend previous results. The performance of both the unified and degree-individual order estimates is evaluated through simulation examples and compared with previous estimates.

Two-Stage-Based Polyphase Structures for Arbitrary-Integer Sampling Rate Conversion

This brief derives efficient two-stage-based polyphase structures for arbitrary-integer sampling rate conversion. For even-integer conversions, the overall structures correspond to parallelized conventional two-stage structures, but the derivations in this brief offer further insights when comparing the two cases of odd- and even-integer conversions. For the class of linear-phase finite-length impulse response $M$ th-band filters, it is then demonstrated through design examples that conversions by odd factors are in fact more efficient than by even factors.

On Efficient Design of High-Order Filters With Applications to Filter Banks and Transmultiplexers With Large Number of Channels

This paper proposes a method for designing high-order linear-phase finite-length impulse response (FIR) filters which are required as, e.g., the prototype filters in filter banks (FBs) and transmultiplexers (TMUXs) with a large number of channels. The proposed method uses the Farrow structure to express the polyphase components of the desired filter. Thereby, the only unknown parameters, in the filter design, are the coefficients of the Farrow subfilters. The number of these unknown parameters is considerably smaller than that of the direct filter design methods. Besides these unknown parameters, the proposed method needs some predefined multipliers. Although the number of these multipliers is larger than the number of unknown parameters, they are known a priori. The proposed method is generally applicable to any linear-phase FIR filter irrespective of its order being high, low, even, or odd as well as the impulse response being symmetric or antisymmetric. However, it is more efficient for filters with high orders as the conventional design of such filters is more challenging. For example, to design a linear-phase FIR lowpass filter of order 131071 with a stopband attenuation of about 55 dB, which is used as the prototype filter of a cosine modulated filter bank (CMFB) with 8192 channels, our proposed method requires only 16 unknown parameters. The paper gives design examples for individual lowpass filters as well as the prototype filters for fixed and flexible modulated FBs.

Conditions for Lth-band filters of order 2N as cascades of identical linear-phase FIR spectral factors of order N

This paper presents formulas for the number of optimization parameters (degrees of freedom) when designing Type I linear-phase finite-length impulse response (FIR) Lth-band filters of order 2N as cascades of identical linear-phase FIR spectral factors of order N. We deal with two types of degrees of freedom referred to as (i) the total degrees of freedom DT, and (ii) the remaining degrees of freedom DR. Due to the symmetries or antisymmetries in the impulse responses of the spectral factors, DT roughly equals N/2. Some of these parameters are specifically needed to meet the Lth-band conditions because, in an Lth-band filter, every Lth coefficient is zero and the center tap equals 1/L. The remaining DR parameters can then be used to improve the stopband characteristics of the overall Lth-band filter. We derive general formulas for DR with given pairs of L and N. It is shown that for a fixed L, the choices of N, in a close neighborhood, may even decrease DR despite increasing the arithmetic complexity, order, and the delay.

A class of reconfigurable and low-complexity two-stage Nyquist filters

This paper introduces a class of reconfigurable two-stage Nyquist filters where the Farrow structure realizes the polyphase components of linear-phase finite-length impulse response (FIR) filters. By adjusting the variable predetermined multipliers of the Farrow structure, various linear-phase FIR Nyquist filters and integer interpolation/decimation structures are obtained, online. However, the filter design problem is solved only once, offline. Design examples, based on the reweighted ℓ1-norm minimization, illustrate the proposed method. Savings in the arithmetic complexity are obtained when compared to the reconfigurable single-stage structures.

On Two-Stage Nyquist Pulse Shaping Filters

This correspondence outlines a method for designing two-stage Nyquist filters. The Nyquist filter is split into two equal and linear-phase finite-length impulse response spectral factors. The per-time-unit multiplicative complexity, of the overall structure, is included as the objective function. Examples are then provided where Nyquist filters are designed so as to minimize the multiplicative complexity subject to the constraints on the overall Nyquist filter. In comparison to the single-stage case, the two-stage realization reduces the multiplicative complexity by an average of 48%. For two-stage sampling rate conversion (SRC), the correspondence shows that it is better to have a larger SRC ratio in the first stage.

A Class of Wide-Band Linear-Phase FIR Differentiators Using a Two-Rate Approach and the Frequency-Response Masking Technique

This paper introduces a class of wide-band linear-phase finite-length impulse response (FIR) differentiators. It is based on two-rate and frequency-response masking techniques. It is shown how to use these techniques to obtain all four types of linear-phase FIR differentiators. Design examples demonstrate that differentiators in this class can achieve substantial savings in arithmetic complexity in comparison with conventional direct-form linear-phase FIR differentiators. The savings achievable depend on the bandwidth and increase with increasing bandwidth beyond the break-even points which are in the neighborhood of 90% (80%) of the whole bandwidth for Type II and III (Type I and IV) differentiators. The price to pay for the savings is a moderate increase in the delay and number of delay elements. Further, in terms of structural arithmetic operations, the proposed filters are comparable to filters based on piecewise-polynomial impulse responses. The advantage of the proposed filters is that they can be implemented using non-recursive structures as opposed to the polynomial-based filters which are implemented with recursive structures.

A Complex Variable Fractional-Delay FIR Filter Structure

This brief introduces a structure for complex variable fractional delay (FD) finite-length impulse response (FIR) filters. The structure is derived from a real variable FD FIR filter and is constituted by a set of fixed real linear-phase FIR filters and two multiply-accumulate chains containing variable multipliers. In this way the implementation complexity and delay may be reduced in comparison with the cascade approach which hitherto has been used for the same purpose. A design example is included to demonstrate the benefits of the new structure.

Minimax design of adjustable-bandwidth linear-phase FIR filters

This paper considers the design of digital linear-phase finite-length impulse response (FIR) filters that have adjustable bandwidth(s) whereas the phase response is fixed. For this purpose, a structure is employed in which the overall transfer function is a weighted linear combination of fixed subfilters and where the weights are directly determined by the bandwidth(s). Minimax design techniques are introduced which generate globally optimal overall filters in the minimax (Chebyshev) sense over a whole set of filter specifications. The paper also introduces a new structure for bandstop and bandpass filters with individually adjustable upper and lower band edges, and with a substantially lower arithmetic complexity compared to structures that make use of two separate adjustable-bandwidth low-pass and high-pass filters in cascade or in parallel. Design examples are included in the paper.

Multirate IIR filter structures for arbitrary bandwidths

This paper introduces new multirate infinite-length impulse-response (IIR) filter structures for arbitrary bandwidths. The overall filters have the same input and output sample rates. Multirate techniques are only used internally in order to improve the efficiency. The overall filters make use of an IIR filter for the actual filtering, periodic allpass filters for constructing complementary filters, and linear-phase finite-length impulse-response (FIR) filters for the sampling rate alterations. There are the following two main reasons for introducing the new filters: 1) they can be used to obtain so called high-speed IIR filters (i.e., filters able to operate at a high sample rate) with lower complexity than that achievable with single-rate high-speed filters and 2) they can be used to obtain approximately linear-phase filters with substantially lower complexity than that achievable with single-rate filters. The overall filters are designed by separately optimizing a number of linear-phase (possibly half-band) FIR filters, and one IIR filter being realizable as a parallel connection of two allpass filters. Design examples are included illustrating the usefulness of the proposed filters.