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DISCRETE-TIME SIGNAL PROCESSINGLECTURE 7 (FILER DESIGN)
Husheng Li, UTK-EECS, Fall 2012
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:
EXAMPLE
BILINEAR TRANSFORMATION
We use the following transformation from s-domain to z-domain:
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 frequency-selective 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
EXAMPLE: BILINEAR TRANSFORMATION FOR BUTTERWORTH
COMPARISON
Butterworth Chebyshev I Chebyshev II
FIR DESIGN BY WINDOWING
Window method: We first obtain the ideal response , and then put a window on it:
The simplest approach is truncation (rectangle window):
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
Bartlett
Hann
Hamming
Blackman
FUTURE COMPARISON
GENERALIZED LINEAR PHASE
Symmetry property:
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 whose frequency response is given by
PARKS-MCLELLAN ALGORITHM
The frequency response can be rewritten as
The coefficients of the polynomial are optimized to minimize the error function:
in a minimax manner:
The Alternation Theorem in the theory of approxiamtion can be applied for the optimzation.