The Theory Of Fast Filter Banks And Its Applications In Multicarrier Communications

Many applications related to software defined radio (SDR) such as spectrum analysis, digital channel-ization and multicarrier are implemented with fast fourier transform (FFT) and inverse fast fourier transform (IFFT) based techniques, but the FFT/IFFT filter bank has poor frequency response due to the poor first side-lobe level with about-13 dB. It requires lots of resource to achieve the better performance. Base on the butterfly structure of FFT, professor Lim provided fast filter bank (FFB) technique to obtain the better performance between low complexity and good characteristics of frequency response. Furthermore, FFB has the property of the combinability of adjacent subchannels, which is not available in the polyphase filter bank. However, it does not favor of the development of FFB in theory and practice due to the absence of gener-al expression of transfer functions in FFB. Based on this, the research is pursued in the aspects including the general transfer function expressions in FFB, the design of the prototype filter, implementation with low complexity and the applications in some specific scenarios for theoretical innovation and application innova-tion.The main contents related to FFB in this paper are listed as follows:1) The transfer functions of the subfilters in a specific subchannel within analysis fast filter bank (AFFB) and the transfer function of the subchannel are expressed with a general form, which can be computed with the binary encoding of the subchannel and the transfer function of the prototype filter in each stage. The struc-ture of a synthesis fast (SFFB) filter bank is derived from IFFT and the relationship between the subfilter in the AFFB and the corresponding subfilter in the SFFB is presented. In addition, the relationship between the prototype filter in the AFFB and the corresponding prototype filter in the SFFB is also expressed. The transfer functions of the subfilters in a specific subchannel within SFFB and the transfer function of the subchannle are formulated with a general form. Especially, when the prototype filters are designed with the halfband method and the noncausality of the prototype filter are further assumed, the general expressions can be simplified, where if the corresponding transfer functions within AFFB and SFFB are same, it can be proved that AFFB has the same expressions of the corresponding transfer functions of the subfilters and the subchannels within SFFB.2) With the general expression of the transfer function of the subchannel within AFFB and SFFB, it is can be proved that the frequency response of the subchannels in AFFB and SFFB has the similar characteristics to that of the uniform filter bank, that is, the frequency response of a specific subchannel can be determined by shifting the frequency response of the prototype lowpass filter, in such a way that AFFB and SFFB can be decomposed with polyphase structure and the corresponding polyphase filter bank structure for AFFB and SFFB can be obtained.3) When overall specifications for FFB are given, the impulse response coefficients of the prototype filter for each stage can be obtained by using optimization technique defined by the optimization of the total multiplications required by FFB and the constraints of the allowable ripples in the passband and the stopband of the prototype lowpass filter, which has not been reported in the literature. On the basis of the design of the prototype filters within FFB, the reconfigurable variable bandwidth FIR is designed with the property of the combinability of adjacent subchannels. Furthermore, the variable bandwidth FIR, which provides to control the center frequency and the passband bandwidth continuously, is achieved by modifying the structure of the 0th prototype filter.4) FFB can provide a more accurate scheme in spectrum analysis compared to the FFT based solution because of high selectivity。Combining with digital phosphor (DPX) technology, FFB can be used to ana-lyze spectrum in real time. The spectrum analyzer is implemented on the platform constituted by modular instruments and the baseband unit in the analyzer based on FFB with field programmable gate array (FPGA) performs the functions of spectrum analysis and processing the DPX data. In the embedded controller, the spectrum of the signal is shown with colorful bitmaps to demonstrate the results of the DPX data from the baseband unit through high-speed data interface.5) Uniform and non-uniform channelizations can be achieved without hardware reimplementation by designing reasonable channel numbers and the prototype filters of FFB. Compared to the conventional meth-ods, low complexity is observed based on FFB scheme. Moreover, FFB base channelization can be used in the field of signal reconstruction, where a synthesis filter bank is not require resulting in low resource con-sumption.6) The possibility and feasibility of the usage of FFB in the filter bank based multicarrier system with offset quadrature amplitude modulation (OQAM) are investigated, which is called the FFBMC/OQAM sys-tem in this article. To meet the requirement of nearly-perfect reconstruction in the multicarrier system, the relationship between the two-path transfer functions of the prototype filter is expressed with a general form and the mathematical constraint equation is used to minimize the total distortion caused by interchannel in-terference (ICI) and intersymbol interference (ISI). By solving the constraint equation, the impulse response coefficients of the prototype filter for each stage can be computed by using optimization. It is can be proved that the frequency response of FFB has the similar characteristics to that of the uniform filter bank in the scenario of the multicarrier system in such manner that the corresponding structures based on polyphase filter bank for FFB are also obtained. Simulations are conducted to evaluate and compare the performance between FFBMC/OQAM and OFDM using protocol 802.11p for wireless access in the vehicular environment. The results show that the performance of FFBMC/OQAM is better than that of OFDM in the scenario of frequency shift, Doppler effect, multipath effect and white noise.