1 9/3/2004 I. Discrete-Time Signals and Systems 1 MM2: Synthesis of IIR DT filters MM2: Synthesis of IIR DT filters Reading material: p.160-163, 439 – 458 and 824-829 1. Explanation of last exercise 2. Continuous time filters 3. Impulse-invariance method 4. Bilinear transformation method 9/3/2004 I. Discrete-Time Signals and Systems 2 Explanation of Exercise One Analog filter Discrete filter 9/3/2004 I. Discrete-Time Signals and Systems 3 Effect of Effect of Filter Filtering ing System frequency response: H(e jω ) = |H(e jω )| e H(ejω) Input and output relationship |Y(e jω )| = |H(e jω )| |X(e jω )| Y(e jω ) = H(e jω ) + X(e jω )
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9/3/2004 I. Discrete-Time Signals and Systems 1
MM2: Synthesis of IIR DT filtersMM2: Synthesis of IIR DT filters
Reading material: p.160-163, 439 – 458 and 824-829
1. Explanation of last exercise2. Continuous time filters3. Impulse-invariance method4. Bilinear transformation method
9/3/2004 I. Discrete-Time Signals and Systems 2
Explanation of Exercise One
Analog filter Discrete filter
9/3/2004 I. Discrete-Time Signals and Systems 3
Effect of Effect of FilterFiltering ing
System frequency response: H(ejωωωω) = |H(ejωωωω)| eH(ejωωωω)
Input and output relationship|Y(ejωωωω)| = |H(ejωωωω)| |X(ejωωωω)|
Y(ejωωωω) = H(ejωωωω) + X(ejωωωω)
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9/3/2004 I. Discrete-Time Signals and Systems 4
Requirements for Requirements for Filter DesignFilter Design
εεεε can be specified by the allowable passband ripple ΩΩΩΩc can be specified by the desired cutoff frequency N can be chosen that the stopband specification are met
Location of poles: on an ellipse Length of the minor axis – 2aΩΩΩΩc
Length of the major axis – 2bΩΩΩΩc
Type II chebyshev filters21
)(21
)(21
1
/1/1
/1/1
−++=
+=
−=
−
−
−
εεα
αα
αα
NN
NN
b
a
1222
)]/([11
|)(| −ΩΩ+=Ω
cNc V
jHε
9/3/2004 I. Discrete-Time Signals and Systems 15
Elliptic FiltersElliptic Filters Characteristics: equiripple in the passband and stopband Elliptic filters is the best that can be achieved for a given filter
order N, in the sense that for a given Ωp, δ1,δ2, the transition band (Ωs- Ωp) is as small as possible