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.