Finite-length Impulse Response Filters Research Articles

Analysis, Design, and Order Estimation of Least-Squares FIR Equalizers for Bandwidth Extension of ADCs

In modern mixed-signal systems, it is important to build the conversion components with a flat frequency response over their full Nyquist frequency band. However, with increasing circuit speed, it is becoming more difficult to achieve this, due to limitations of the analog front-end circuits. This paper considers finite-length impulse-response (FIR) filters, designed in the least-squares sense, for the bandwidth extension of analog-to-digital converters, which is one of the most important applications in frequency response equalization. The main contributions of this paper are as follows: Firstly, based on extensive simulations, filter order-estimation expressions of the least-squares designed equalizers are derived. It appears to be the first time that order-estimation expressions are presented for any least-squares designed FIR filter. These expressions accurately estimate the order required for given specifications on the targeted extended bandwidth systems. Secondly, based on the derived order-estimation expressions, systematic design procedures are presented, which contribute to reducing the design time. Finally, a relation between the dynamic-range degradation and the system parameters is also derived and verified in the paper.

An Improved Digital Predistortion in Wideband Wireless Transmitters Using an Under-Sampled Feedback Loop

This letter presents an improved digital predistortion (DPD) method to optimize the linearization of power amplifiers (PAs) when an under-sampled feedback loop is used in the transmitter. The proposed approach is based on an adaptive piecewise Lagrange (APL) basis digital predistorter and a variable fractional-delay (VFD) finite-length impulse response (FIR) filter. A normalized least mean square (NLMS) algorithm with a direct learning structure is used to update the DPD parameters, and the VFD FIR filter is used to periodically vary the fractional-delay of the transmitted signal. This changes the initial phase of the under-sampled feedback signal to preserve the peak power information from the full sampling rate signal. Experiment results show that there is 10 dB improvement in the normalized mean square error (NMSE) for a 70 MHz Long Term Evolution-advanced signal, even when the sampling rate of feedback is reduced from 491.52 Msps to 122.88 Msps.

Add-Equalize Structures for Linear-Phase Nyquist FIR Filter Interpolators and Decimators

This paper introduces add-equalize structures for the implementation of linear-phase Nyquist ( $M$ th-band) finite-length impulse response (FIR) filter interpolators and decimators. The paper also introduces a systematic design technique for these structures based on iteratively reweighted $\ell _{1}$ -norm minimization. In the proposed structures, the polyphase components share common parts which leads to a considerably lower implementation complexity as compared to conventional single-stage converter structures. The complexity is comparable to that of multi-stage Nyquist structures. A main advantage of the proposed structures is that they work equally well for all integer conversion factors, thus including prime numbers which cannot be handled by the regular multi-stage Nyquist converters. Moreover, the paper shows how to utilize the frequency-response masking approach to further reduce the complexity for sharp-transition specifications. It also shows how the proposed structures can be used to reduce the complexity for reconfigurable sampling rate converters. Several design examples are included to demonstrate the effectiveness of the proposed structures.

Two Polynomial FIR Filter Structures With Variable Fractional Delay and Phase Shift

This paper introduces two polynomial finite-length impulse response (FIR) digital filter structures with simultaneously variable fractional delay (VFD) and phase shift (VPS). The structures are reconfigurable (adaptable) online without redesign and do not exhibit transients when the VFD and VPS parameters are altered. The structures can be viewed as generalizations of VFD structures in the sense that they offer a VPS in addition to the regular VFD. The overall filters are composed of a number of fixed subfilters and a few variable multipliers whose values are determined by the desired FD and PS values. A systematic design algorithm, based on iteratively reweighted ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm minimization, is proposed. It generates fixed subfilters with many zero-valued coefficients, typically located in the impulse response tails. The paper considers two different structures, referred to as the basic structure and common-subfilters structure, and compares these proposals as well as the existing cascaded VFD and VPS structures, in terms of arithmetic complexity, delay, memory cost, and transients. In general, the common-subfilters structure is superior when all of these aspects are taken into account. Further, the paper shows and exemplifies that the VFDPS filters under consideration can be used for simultaneous resampling and frequency shift of signals.

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.

On FIR Filter Approximation of Fractional-Order Differentiators and Integrators

This paper considers finite-length impulse response (FIR) filter approximation of differentiators and integrators, collectively called differintegrators. The paper introduces and compares three different FIR filter structures for this purpose, all of which are optimized in the minimax sense using iterative reweighted l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm minimization. One of the structures is the direct-form structure, but featuring equal-valued taps and zero-valued taps, the latter corresponding to sparse filters. The other two structures comprise two subfilters in parallel and cascade, respectively. In their basic forms, nothing is gained by realizing the filters in parallel or in cascade, instead of directly. However, as the paper will show, these forms enable substantial further complexity reductions, because they comprise symmetric and antisymmetric subfilters of different orders, and also features additional equal-valued and zero-valued taps. The cascade structure employs a structurally sparse filter. The additional sparsity, as well as tap equalities, are for all three structures found automatically in the design via the l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm minimization. Design examples included reveal feasible multiplication complexity savings of more than 50% in comparison with regular (unconstrained) direct-form structures. In addition, an example shows that the proposed designs can even have lower complexity than existing infinite-length impulse response filter designs.

Integer Linear Programming-Based Bit-Level Optimization for High-Speed FIR Decimation Filter Architectures

Analog-to-digital converters based on sigma-delta modulation have shown promising performance, with steadily increasing bandwidth. However, associated with the increasing bandwidth is an increasing modulator sampling rate, which becomes costly to decimate in the digital domain. Several architectures exist for the digital decimation filter, and among the more common and efficient are polyphase decomposed finite-length impulse response (FIR) filter structures. In this paper, we consider such filters implemented with partial product generation for the multiplications, and carry-save adders to merge the partial products. The focus is on the efficient pipelined reduction of the partial products, which is done using a bit-level optimization algorithm for the tree design. However, the method is not limited only to filter design, but may also be used in other applications where high-speed reduction of partial products is required. The presentation of the reduction method is carried out through a comparison between the main architectural choices for FIR filters: the direct-form and transposed direct-form structures. For the direct-form structure, usage of symmetry adders for linear-phase filters is investigated, and a new scheme utilizing partial symmetry adders is introduced. The optimization results are complemented with energy dissipation and cell area estimations for a 90 nm CMOS process.

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.

Reconstruction of Nonuniformly Sampled Bandlimited Signals by Means of Time-Varying Discrete-Time FIR Filters

This paper deals with reconstruction of nonuniformly sampled bandlimited continuous-time signals using time-varying discrete-time finite-length impulse response (FIR) filters. The main theme of the paper is to show how a slight oversampling should be utilized for designing the reconstruction filters in a proper manner. Based on a time-frequency function, it is shown that the reconstruction problem can be posed as one that resembles an ordinary filter design problem, both for deterministic signals and random processes. From this facts, an analytic least-square design technique is then derived. Furthermore, for an important special case, corresponding to periodic nonuniform sampling, it is shown that the reconstruction problem alternatively can be posed as a filter bank design problem, thus with requirements on a distortion transfer function and a number of aliasing transfer functions. This eases the design and offers alternative practical design methods as discussed in the paper. Several design examples are included that illustrate the benefits of the proposed design techniques over previously existing techniques.

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.

Adaptive FIR filter structure based on the generalized subband decomposition of FIR filters

An adaptive structure based on a generalized structural subband decomposition of FIR (finite-impulse-response) filters is presented. The proposed structure implements an adaptive FIR filter of length N as a parallel connection of L branches, with each branch composed of a cascade of a fixed interpolator and a sparse adaptive subfilter containing at least L nonzero coefficients. There is no sampling rate alteration in this structure, and therefore no problems with aliasing occur. The interpolators are implemented by a computationally efficient transform (e.g., DCT, DFT). The proposed structure presents superior convergence performance for colored input signals when compared to the conventional direct-form LMS (least-mean-square) structure, with a very small increase in the number of operations. The advantages of using the subband structure in the adaptive line enhancer, acoustic echo canceller, and channel equalizer applications are shown through computer simulations. >

Complex correction of data acquisition channels using FIR equalizer filters

The common conviction about FIR (finite impulse response) digital filters is that the number of necessary taps, to reach the same performance as provided by IIR (infinite impulse response) digital filters, is usually too large. Moreover, the standard FIR filter design algorithm (Remez exchange) allows the design of linear-phase filters only. Therefore, IIR filters are often preferred over FIR ones, without any further investigations. This paper presents a case study of complex (amplitude and phase) equalization of the passband of a commercial anti-aliasing filter. The novelty is the usage of an FIR filter for this purpose (or an FIR one, combined with a low-order IIR filter), and a thorough discussion of the special design aspects. It turns out that for the given anti-aliasing filter (a Cauer filter of order 11) an FIR filter of length 60...100 can perform as well as a 26/26 (numerator order/denominator order) IIR one. The properties are even better if a low-order IIR filter is used in combination with an FIR one (orders, e.g., 1/1+40/0). Because of the absence of stability problems and the ease of implementation, the use of FIR filters is suggested. >