### lecture 7

```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.

```