EL 6183 Week 6

EL 6183 Week 6 - EL 6183 EE 4163 EL Digital Signal...

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Unformatted text preview: EL 6183 EE 4163 EL Digital Signal Processing Lab Digital Monday 03/09/09 EL 6183 / EE 4163 DIGITAL SIGNAL PROCESSING LAB Filter Design Practice and LAB # 5 EL 6183 EE 4163 EL Digital Signal Processing Lab Digital Infinite Impulse Response Filters: Review y (n) = a x(n) + a x(n − 1) + ... + a x(n − N ) − b y (n − 1) − b y (n − 2) − ...b Y (n − M 0 1 N 1 2 M ∏ (z − z ) Y ( z ) a + a z − 1 + ... + a z − N H ( z) = = =K X ( z ) 1 + b z + ... + b z ∏ (z − z ) 0 1 N i −1 −M i =1 j =M j =1 1 M j i= N M poles and N zeros Stability: all poles are inside the unit circle: | p |= 1 The system oscillates if j | p |< 1 j The order M of the system is the order of the equation EL 6183 EE 4163 EL Digital Signal Processing Lab Digital Low Pass Digital IIR Filter Design Low We use a Low Pass Analog Filter H(s)=Ωp/(s+Ωp) And And 2*Fs*(z-1)/(z+1) 2*Fs*(z-1)/(z+1) The Bilinear Transformation EL 6183 EE 4163 EL Digital Signal Processing Lab Digital The Bilinear Transformation The In H(s), s is replaced by: In H(s) is z −1 s ⇔ 2.Fs z +1 to obtain H(z) H(z) Fs is the sampling rate EL 6183 EE 4163 EL Digital Signal Processing Lab Digital Problem: Bandwidth mismatch Problem: Cause: Warping effect of the transformation z −1 s ⇔ 2.Fs z +1 −1 ω Ω ⇔ 2 Fs tan( ) 2 Ω ω ⇔ 2 tan ( ) 2 Fs frequency scale [0 ∞], compresses to [0 π] to EL 6183 EE 4163 EL Digital Signal Processing Lab Digital Solution: Prewarp the analog Frequency Solution: before applying the transformation Replace all cut-off angular frequencies Ωc by Replace by their prewarp values their ω Ω ⇔ 2 Fs tan( ) 2 EL 6183 EE 4163 EL Digital Signal Processing Lab Digital Design Steps for Low Pass IIR Filter when H(s) is known. H(s) 1. 2. 2. 3. 3. 4. 5. 6. Convert the given (or identified) analog frequency values in Convert digital using the giving sampling rate. Compute the corresponding digital angular frequency value. Compute Prewarp the analog frequencies using corresponding to the Prewarp digital frequencies computed in Step 2. digital Replace the unwarped frequency values in H(s) by the warped Replace H(s) frequency values to get a new expression Hwp(s). Hwp(s) Apply the bilinear transformation on Hwp(s) Apply Hwp(s) Write the difference equation EL 6183 EE 4163 Digital Signal Processing Lab Digital Design Steps for High Pass, Band Pass, and Band Stop IIR filters IIR 1. 2. 3. Find a Low Pass analog filter with similar frequency Find response shape as the desired filter. response Prewarp all cut-off frequencies Convert the Low Pass analog filter into the desired Convert High Pass, Band Pass, or Band Stop filter, using the prewarp analog frequencies, as in the following table EL 6183 EE 4163 Digital Signal Processing Lab Digital EL 6183 EE 4163 Digital Signal Processing Lab Digital 1. Apply bilinear transformation to 1. Apply convert the analog filter to digital filter. convert 2. Write the corresponding difference Write equation equation EL 6183 EE 4163 EL Digital Signal Processing Lab Digital Infinite Impulse Response Filters Implementation: Direct Form I a0 + + Y ( n) Z-1 Z-1 x( n) a1 Z-1 + + -b1 Z-1 a2 . . Z-1 + . + . aN + + -b2 . + . + -bM . . Z-1 EL 6183 EE 4163 EL Digital Signal Processing Lab Digital Infinite Impulse Response Filters Implementation: Direct Form I A: a0 a1 … x(n) x(n-1) -b1 -b2 … … aN-1 x(n-(N-1)) aN x(n-N) -bM x(n-M) X: and B: Y: y(n-1) y(n-2) y1=A dot product X -bM-1 y(n-(M-1)) and y2=B dot product Y Finally y(n)=y1+y2 EL 6183 EE 4163 EL Digital Signal Processing Lab Digital Infinite Impulse Response Filters Implementation: Direct Form II w(n) = x(n) − b w(n − 1) − b w(n − 2) − ... − b w(n − M ) y ( n) = a w( n) + a w( n −1) +... + a w( n − N ) 1 2 M 0 1 N x( n ) + wn () a0 + Y( n ) + -b1 Z-1 a1 Z-1 + + -b2 a2 . + . + -bM Z-1 + . + . aN + EL 6183 EE 4163 EL Digital Signal Processing Lab Digital Infinite Impulse Response Filters Implementation: Direct Form II B: -b1 -b2 w(n-2) … -bM-1 w(n-(M-1)) -bM w(n-M) W1: w(n-1) and A: a1 a2 w(n)=x(n)+B dot product W1 … aN-1 aN and y(n)=a0w(n)+A dot product W1 EL 6183 EE 4163 EL Digital Signal Processing Lab Digital Lab # 6 Lab Activity 1 Activity 2 Test #3 on IIR Filter Design Monday 03/23/2009 ...
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This note was uploaded on 11/18/2009 for the course EL EL6183 taught by Professor Selesnick during the Fall '09 term at NYU Poly.

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