Computer-Aided Design (CAD) of Recursive/Non-recursive Filters

. Engineering, computing curricula include courses on signals & systems as well as analysis/design of frequency selective circuits/systems called filters. The subject of filters (both analysis and design) are usually covered in several classes. Even at introductory level courses (e.g., circuits & systems, see, for example, [1]) whenever we are dealing with alternating current (AC) topics students are being introduced to simple forms of frequency selective structures. As example, in first order circuits (RC or RL) if we include AC excitations we are really dealing with filters: low-pass, high-pass, band-pass or band-stop (reject) devices. These simple circuits are examples of passive circuits & systems. With the introduction of operational amplifiers (OpAmps) integrated circuits, we start description of so-called active filters. Studying the input/output (I/O) models of these systems, students are gradually introduced to using computeraided design, CAD, or analysis tools that can be beneficial in their work. For example after “digitizing” the differential equation of the system, they can solve the I/O model (linear, constant-coefficient difference equation) of the system. The digital I/O models include both recursive and non-recursive systems. For the CAD design there is a need for utilizing window functions, one important class of windows require the use of high-level mathematical functions for which an algorithm is introduced so that students can come up with their final design and again use the software tool for the verification/validation step. Whenever possible students are given opportunity to make presentation in the class where more results can be discussed and a term paper/report submitted to receive credit toward their grades. Since the required subjects/topics are very complex by nature, these tools can be very beneficial for both learners and engineering educators. Finally there are cases that a regular design employing the so-called classical system approach may not be practical, however it will be possible to design the system using, for example, non-recursive, digital systems. The subject of digital signal processing (DSP) is covered in most computing and electrical engineering programs, as example see the textbook by Lathi & Green, [6]. We next show an important application of DSP in mobile computing, e.g., smartphones in simplified block diagram form, [2]: P ge 26385.2 As can be seen several of the system components are related to the subject of DSP. Design of FILTERS is a major topic in DSP including analog filter design (passive, active electronic circuits), classical filters, and digital filters. There are two main kinds of digital filters used in practice: filters with feedback (also known as IIR, ARMA filters) and filters without feedback (or FIR, MA filters.) IIR Filters. Analog filter design, a mature subject studied by several investigators and welldocumented (as example see [3]), is the starting point for design of IIR (infinite impulse response) digital filters. Basically we are dealing with K-th order differential equation input/output (I/O) model, with x(t) being the input (excitation) and y(t) the response or output: ∑ ak K k=0 dk dtk y(t) = ∑ bl L l=0 dl dtl x(t) Digital filter is then obtained by using several transformations as developed in the DSP literature: H(s) → H(z) (mapping from the s-plane to the z-plane), Laplace transforms, ztransforms Backward Difference Approximation, s → (1/T) [ 1 – z ] where 1/ T is the sampling frequency of DSP system Impulse Invariance, h(t) → h(nT) = h[n] → H(z) (also step invariance) Bilinear Transformation, s → (2/T) ( z – 1) / (z + 1) CAD tools, e.g., MATLAB, MATHCAD, WFilter Like in any engineering design problem there is a trade-off in the design process. The most optimum method of designing IIR filters is using the bilinear transformation, however this transformation creates a non-linear relationship between analog and digital frequencies. We need to use pre-warping techniques to avoid distortion in the final design. Impulse invariance method P ge 26385.3 also requires using a high sampling rate to avoid aliasing, this method is only used to design lowpass and ban-pass filters. From the stability point of view we need a transformation that results in a stable, causal system (filter.) While all methods mentioned above result in a stable final filter (the so-called forward difference transformation may result in unstable filter), the backward difference transformation is not widely used because pole locations do not occupy all the space within the unit circle (u.c.) in the z-plane but a sub-set inside u.c. To design digital filters with feedback we first introduce students to the subject of analog filter design, a difficult subject but most engineering curricula in affected disciplines do include courses (for example second electric circuit class) on this subject. Signals & systems courses also cover filters especially when studying system frequency response. In our school the recent book by Lathi & Green [6] is used in the digital signal processing senior, first-year graduate classes. Practical filter design discussion starts with the fact that ideal filters cannot be realized in practice. As example if we look at following ideal LPF, BPF, HPF and BSF frequency responses, we can show that their impulse response, h(t), will not represent a causal system. In case of LPF with cutoff frequency ωc and assuming zero phase response, it is easily seen that inverse Fourier transform gives: h(t) = (Aωc /π) Sa(ωct) ; -∞ < t < ∞ , where sampling function is defined as Sa(x) = sin(x) /x. On the other hand a simple RC LPF circuit shown below: