Design and Analysis of Variable Digital Filters using Fractional Fourier Transform

Abstract

The present thesis comprises research work carried out on the design and analysis of variable digital filters using Fractional Fourier Transform (FrFT). Variable Digital Filters (VDFs) are digital filters with controllable spectral characteristics. In many practical applications of digital signal processing, there is a need for real-time tuning of the frequency-domain characteristics of digital filters in order to meet the new desired specifications. Such tunable frequency selective digital filters are called VDFs. The frequency responses of VDFs can be varied instantaneously according to some tuning parameter. Different responses or delays can be immediately obtained by tuning the variable parameters without the need to design a new filter. Variable digital filters (also called tunable digital filters) have applications in different areas of signal processing and communications, e.g., digital tuners, audio tone control, software radio, digital hearing aids, sigma-delta modulators, multimedia signal enhancement and correction, in multi-standard wireless communication receiver systems, channel equalization, matched filtering and pulse shaping etc. VDFs are needed to perform a slightly modified task in accordance with the need that arises in real time processing.

Fractional Fourier transform is applied in the design of digital filters so that the frequency response characteristics of window-based fixed length FIR filter can be modified during operation simply by adjusting the fractional order parameter , while keeping the coefficients of the fixed length FIR filter unchanged. FrFT is having distinctive properties as one extra degree of freedom is possessed by the fractional order parameter (or rotation angle parameter , where ) of the transform. The adjustable parameter denoted by notation (or ) can be adjusted in different applications so that enhanced results are obtained in comparison to the other existing methods.

A design methodology based on FrFT is presented for designing window-based Low-Pass (LP) FIR filters with adjustable transition-width while keeping the length of the filter same. Sharp transition-width FIR filter is obtained by using fractional order of the transform as a tuning parameter. The transition width of the filter is controlled by FrFT order parameter with low computational cost.

A new closed-form expression of FrFT based frequency response function of a Rectangular window-based fixed length LP FIR filter is derived. The derived expression is analyzed for a range of fractional angle and on the basis of the analysis it is observed that transition width and pass-band width of FIR filter can be reduced, whereas, improved stop-band attenuation can be achieved by reducing the fractional angle from . Because , reducing the fractional angle from is analogous to reducing the fractional order parameter from . The performance comparison of the proposed FrFT based method over the ordinary method based on FFT (by varying the window length) in tuning FIR filters is also done.

Further, a new FrFT based design technique for obtaining the variable magnitude as well as the phase characteristics of fixed length window-based LP FIR filters is proposed. In the proposed design, the pass-band width and the phase response characteristics of a filter are modified simultaneously by using the fractional order parameter as a tuning parameter. The behaviour of desired impulse response of FIR filter is analyzed and it is observed that the pass-band width of a digital filter can be controlled using FrFT. An increase in pass-band width of FIR filter is experienced by reducing the FrFT order from to 0. Also, variations in phase response are observed for a range of fractional order of the transform.

Finally, a finite duration Chirp function is analyzed as windowing function by using fractional Fourier transform. A new mathematical model for obtaining the FrFT of Chirp function is developed and analyzed in order to add to the advantage of using FrFT in adjustable FIR filter design and in spectral analysis. It is put forward through the derived closed-form expression that FrFT of finite duration Chirp depends on the order of the transform and the Chirp parameter. The variations in the parameters such as Half Main-Lobe Width (HMLW), Maximum Side-Lobe Level (MSLL) & Side-Lobe Fall-Off Rate (SLFOR) of Chirp window are observed for a range of fractional orders. The performance of Chirp window under FrFT is compared with some of the existing windows and improvement in terms of main lobe and side ripples is observed for a range of fractional angle. It is found that the Chirp window is superior and provides better spectral parameters for some particular values of fractional angle . Also, the Chirp window-based LP FIR filter using FrFT is designed with modified transition-width characteristics.