Husheng Li, UTK-EECS, Fall 2012 DISCRETE-TIME SIGNAL PROCESSING LECTURE 7 (FILER DESIGN) FILTER SPECIFICATIONS The specification of filter is usually given by the tolerance scheme. DETERMINING SPECIFICATIONS FOR A DISCRETE-TIME FILTER The above example uses a discrete-time filter to process a continuous-time signal after periodic sampling. In practice, many applications may not use this approach. FROM CONTINUOUS TIME IIR TO DISCRETE TIME IIR The art of continuous time IIR design is highly advanced. Many useful continuous-time IIR designs have relatively simple closed-form formulas. The standard approximation methods working well for continuous time IIR do not lead to simple closed-form design formulas when they are applied to discrete-time IIRs. DESIGN BY IMPULSE INVARIANCE Impulse invariance: a discrete-time system is defined by sampling the impulse response of a continuous-time system: ℎ = ℎ ( ) EXAMPLE BILINEAR TRANSFORMATION We use the following transformation from s-domain to zdomain: 2 1 − −1 = 1 + −1 MAPPING FREQUENCY WARPING The distortion in the frequency axis manifests itself as a warping of the phase response of the filter. DISCRETE TIME BUTTERWROTH, CHEBYSHEV AND ELLIPTIC FILTERS The most widely used classes of frequencyselective continuous-time filters are Butterworth, Chebyshev and elliptic filter designs. We expect the discrete Butterworth, Chebyshev and elliptic filters can retain the monotonicity and ripple characteristics of the corresponding continuous-time filters. RECAP: BUTTERWORTH Butterworth lowpass filters are defined by the property that the magnitude response is the maximally flat in the passband and that the magnitude response is monotonic in the passband and stopband. BUTTERWORTH The magnitude response of Butterworth is given by 1 2 |()| = 1 + (/ )2 EXAMPLE: BILINEAR TRANSFORMATION FOR BUTTERWORTH COMPARISON Butterworth Chebyshev I Chebyshev II FIR DESIGN BY WINDOWING Window method: We first obtain the ideal response ℎ , and then put a window on it: ℎ = ℎ () The simplest approach is truncation (rectangle window): ℎ , ℎ = 0, 0≤≤ ℎ EFFECT OF RECTANGLE WINDOW The rectangle window results in a smeared version of the ideal response. COMMONLY USED WINDOWS COMPARISON IN THE FREQUENCY DOMAIN Rectangle Hann Hamming Blackman Bartlett FUTURE COMPARISON GENERALIZED LINEAR PHASE Symmetry property: − , = 0, 0≤≤ ℎ The resulting frequency response will have a generalized linear phase: = −/ KAISER WINDOW FILTER DESIGN The Kaiser window can achieve near-optimality for the tradeoff between the main-lobe width and side-lobe area. EXAMPLE OF FIR DESIGN USING THE KAISER WINDOW METHOD OPTIMUM APPROXIMATIONS OF FIR It may not be good to simply minimize the error of approximation to an ideal filter. We consider a filter with ℎ = ℎ − , whose frequency response is given by = ℎ + 2ℎ cos() =1 PARKS-MCLELLAN ALGORITHM The frequency response can be rewritten as (cos ) = =1 The coefficients of the polynomial are optimized to minimize the error function: = [ − ()] in a minimax manner: ℎ [()] The Alternation Theorem in the theory of approxiamtion can be applied for the optimzation.