Timo I. Laakso

Splitting the unit delay [FIR/all pass filters design]

A fractional delay filter is a device for bandlimited interpolation between samples. It finds applications in numerous fields of signal processing, including communications, array processing, speech processing, and music technology. We present a comprehensive review of FIR and allpass filter design techniques for bandlimited approximation of a fractional digital delay. Emphasis is on simple and efficient methods that are well suited for fast coefficient update or continuous control of the delay value. Various new approaches are proposed and several examples are provided to illustrate the performance of the methods. We also discuss the implementation complexity of the algorithms. We focus on four applications where fractional delay filters are needed: synchronization of digital modems, incommensurate sampling rate conversion, high-resolution pitch prediction, and sound synthesis of musical instruments.

Principles of fractional delay filters

In numerous applications, such as communications, audio and music technology, speech coding and synthesis, antenna and transducer arrays, and time delay estimation, not only the sampling frequency but the actual sampling instants are of crucial importance. Digital fractional delay (FD) filters provide a useful building block that can be used for fine-tuning the sampling instants, i.e., implement the required bandlimited interpolation. In this paper an overview of design techniques and applications is given.

Eigenfilter approach for the design of allpass filters approximating a given phase response

We apply the eigenfilter method to design an allpass filter that approximates a given phase response in the least-squares (LS) sense. As it is not possible to express the exact LS phase error as a quadratic form suitable for eigenfilter formulation, alternative error measures that approximate the ideal LS error are proposed. For each of these new formulations, the allpass coefficients are obtained as the elements of the eigenvector corresponding to the minimum eigenvalue of a real, symmetric, and positive definite matrix. We propose a fast-converging iterative technique to approximate the ideal LS phase error solution. By employing an iterative weighting technique, the phase error can he made approximately equiripple. The design methods are illustrated with various practical examples and the results are compared to allpass filters designs reported in the literature. >

Simple and robust method for the design of allpass filters using least-squares phase error criterion

We consider a simple scheme for the design of allpass filters for approximation (or equalization) of a given phase function using a least-squares error criterion. Assuming that the desired phase response is prescribed at a discrete set of frequency points, we formulate a general least-squares equation-error solution with a possible weight function. Based on the general formulation and detailed analysis of the introduced error, we construct a new algorithm for phase approximation. In addition to iterative weighting of the equation error, the nominal value of the desired group delay is also adjusted iteratively to minimize the total phase error measure in equalizer applications. This new feature essentially eliminates the difficult choice of the nominal group delay which is known to have a profound effect on the stability of the designed allpass filter. The proposed method can be used for highpass and bandpass equalization as well, where the total phase error can be further reduced by introducing an adjustable-phase offset in the optimization. The performance of the algorithm is analyzed in detail with examples. First we examine the approximation of a given phase function. Then we study the equalization of the nonlinear phase of various lowpass filters. Also, a bandpass example is included. Finally we demonstrate the use of the algorithm for the design of approximately linear-phase recursive filters as a parallel connection of a delay line and an allpass filter. >

Noise reduction in recursive digital filters using high-order error feedback

The problem of solving the optimal (minimum-noise) error feedback coefficients for recursive digital filters is addressed in the general high-order case. It is shown that when minimum noise variance at the filter output is required, the optimization problem leads to set of familiar Wiener-Hopf or Yule-Walker equations, demonstrating that the optimal error feedback can be interpreted as a special case of Wiener filtering. As an alternative to the optimal solution, the formulas for suboptimal error feedback with symmetric or antisymmetric coefficients are derived. In addition, the design of error feedback using power-of-two coefficients is discussed. The efficiency of high order error feedback is examined by test implementations of the set of standard filters. It is concluded that error feedback is a very powerful and versatile method for cutting down the quantization noise in any classical infinite impulse response (IIR) filter implemented as a cascade of second-order direct form sections. The high-order schemes are attractive for use with high-order direct form sections. >

Delta operator realizations of direct-form IIR filters

The use of the delta operator in the realizations of digital filters has recently gained interest due to its good finite-word-length performance under fast sampling. We studied efficient direct form structures, and show that only some of them can be used in delta configurations, while others are evidently unstable. In this paper, we focus on the roundoff noise analysis. Of all the direct-form structures, the direct form II transposed (DFIIt) delta structure has the lowest quantization noise level at its output. This structure outperforms both the conventional direct-form (delay) structures, as well as the state-space structures for narrow-band low-pass filters with respect to output roundoff noise. Excellent roundoff noise performance is achieved at the cost of only a minor additional implementation complexity when compared with the corresponding delay realization. Complexity of a signal processor implementation of the DFIIt delta structure, which was found to be the most suitable delta structure for signal processors, is compared with those of the direct form and state-space delay structures. In addition, some hardware implementation aspects are discussed, including the minimization of the internal word length.

Receiver Cancellation Technique for Nonlinear Power Amplifier Distortion in SDMA–OFDM Systems

Space-division multiple access (SDMA) and orthogonal frequency-division multiplexing (OFDM) can be combined to design a robust communications system with increased spectral efficiency and system capacity. This combination is one of the most promising candidates for future wireless local area network implementations. However, one drawback of OFDM systems is the high peak-to-average power ratio, which imposes strong requirements on the linearity of power amplifiers (PAs). Such linearity requirements translate into high back-off that results in low power efficiency. In order to improve power efficiency, a PA nonlinearity cancellation (PANC) technique is introduced in this paper. This technique reduces the nonlinear distortion effects on the received signal. The performance of the new technique is evaluated with simulations, which show significant power efficiency improvements. To obtain meaningful results for comparison purposes, we derive a theoretical upper bound on the bit error rate performance of an SDMA-OFDM system subject to PA nonlinearities. In addition, a novel channel estimation technique that combines frequency- and time-domain channel estimation with PANC is also presented. Simulation results show the robustness of the cancellation method also when channel estimation is included.

Suppression of transients in variable recursive digital filters with a novel and efficient cancellation method

A new method for suppressing transients in recursive infinite impulse response (IIR) digital filters is proposed. The technique is based on modifying the state (delay) variables of the filter when coefficients are changed so that the filter enters a new state smoothly without transient attacks, as originally proposed by Zetterberg and Zhang (1988). In this correspondence, we modify the Zetterberg-Zhang algorithm to render it feasible for efficient implementation. We define a mean square error (MSE) measure for transients and determine the optimal transient suppressor to cancel the transients down to a desired level at the minimum complexity of implementation. The application of the method to all-pole and direct-form II (DF II) IIR filter sections is studied in detail. Time-varying recursive filtering with transient elimination is illustrated for tunable fractional delay filters and variable-bandwidth lowpass filters.

Design and implementation of efficient IIR notch filters with quantization error feedback

Straightforward methods for the design of digital notch filters are presented. The design method is based on setting a zero of the filter at a notch frequency and placing a pole in its neighborhood such that the notch width is narrow enough while keeping the group delay of the filter sufficiently flat. A technique for efficient and well-behaved implementation with fixed-point signal processors is advanced, based on the use of quantization error feedback for roundoff noise reduction. The design approach is illustrated with numerical examples, and an assembly-language program for the family of TMS320 signal processors is provided. >

Closed-form design of tunable fractional-delay allpass filter structures

Controllable fractional delay (FD) elements are needed in numerous applications, e.g., in timing adjustment of digital receivers, where the received signal is usually not sampled in synchronism with the incoming data stream-the optimal sampling is obtained by interpolation. Tunable FD structures have been known only for FIR filters, most notably the Farrow structure. In this paper we introduce a computationally efficient, tunable FD allpass filter structure. The polynomial coefficients are obtained in closed form using the Thiran allpass filter design method with modifications which completely eliminate the division operations.