This brief proposes a method for designing modulated filter banks (FBs) with a large number of channels. The impulse response of the long prototype filter is parameterized in terms of a few short impulse responses, thus significantly reducing the number of unknown parameters. The proposed method starts by first obtaining an FB with a few channels. The solution of this FB is then partly reused as an initial (very close to final) solution in the design of FBs with a large number of channels. The number of unknown parameters hence drastically reduces. For example, we can first design a cosine modulated FB (CMFB) with three channels whose prototype filter has a stopband attenuation of about 40 dB. We can then reuse the solution of this CMFB in the design of a CMFB with 16 384 channels whose prototype filter has a similar stopband attenuation. With our proposed method, we need to reoptimize only 14 parameters to design the CMFB with 16 384 channels.

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 l(1)-norm minimization, illustrate the proposed method. Savings in the arithmetic complexity are obtained when compared to the reconfigurable single-stage structures.

This paper introduces add-equalize structures for the implementation of linear-phase Nyquist (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 -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.

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 D-T, and (ii) the remaining degrees of freedom D-R. Due to the symmetries or antisymmetries in the impulse responses of the spectral factors, D-T 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 D-R parameters can then be used to improve the stopband characteristics of the overall Lth-band filter. We derive general formulas for D-R 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 D-R despite increasing the arithmetic complexity, order, and the delay.

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.

This paper proposes optimal finite-length impulse response (FIR) digital filters, in the least-squares (LS) sense, for compensation of chromatic dispersion (CD) in digital coherent optical receivers. The proposed filters are based on the convex minimization of the energy of the complex error between the frequency responses of the actual CD compensation filter and the ideal CD compensation filter. The paper utilizes the fact that pulse shaping filters limit the effective bandwidth of the signal. Then, the filter design for CD compensation needs to be performed over a smaller frequency range, as compared to the whole frequency band in the existing CD compensation methods. By means of design examples, we show that our proposed optimal LS FIR CD compensation filters outperform the existing filters in terms of performance, implementation complexity, and delay.

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(1)- 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.

This paper proposes a method to design variable fractional-delay (FD) filters using the Farrow structure. In the transfer function of the Farrow structure, different subfilters are weighted by different powers of the FD value. As both the FD value and its powers are smaller than 0.5, our proposed method uses them as diminishing weighting functions. The approximation error, for each subfilter, is then increased in proportion to the power of the FD value. This gives a new distribution for the orders of the Farrow subfilters which has not been utilized before. This paper also includes these diminishing weighting functions in the filter design so as to obtain their optimal values, iteratively. We consider subfilters of both even and odd orders. Examples illustrate our proposed method and comparisons, to various earlier designs, show a reduction of the arithmetic complexity.

This paper introduces a finite-length impulse response (FIR) digital filter having both a variable fractional delay (VFD) and a variable phase shift (VPS). The realization is reconfigurable online without redesign and without transients. It can be viewed as a generalization of the VFD Farrow structure that offers a VPS in addition to the regular VFD. The overall filter is composed of a number of fixed subfilters and a few variable multipliers whose values are determined by the desired FD and PS values. It is designed offline in an iterative manner, utilizing reweighted l(1)-norm minimization. This design procedure generates fixed subfilters with many zero-valued coefficients, typically located in the impulse response tails.

This paper introduces multimode transmultiplexers (TMUXs) in which the Farrow structure realizes the polyphase components of general lowpass interpolation/decimation filters. As various lowpass filters are obtained by one set of common Farrow subfilters, only one offline filter design enables us to cover different integer sampling rate conversion (SRC) ratios. A model of general rational SRC is also constructed where the same fixed subfilters perform rational SRC. These two SRC schemes are then used to construct multimode TMUXs. Efficient implementation structures are introduced and different filter design techniques such as minimax and least-squares (LS) are discussed. By means of simulation results, it is shown that the performance of the transmultiplexer (TMUX) depends on the ripples of the filters. With the error vector magnitude (EVM) as the performance metric, the LS method has a superiority over the minimax approach.