1. Normalized Lowpass Filters “All-pole” lowpass filters, such as Butterworth and Chebyshev filters, have transfer functions of the form: Where N is the filter order. Filter functions are tabulated in “normalized” form

2. Normalized Lowpass Filters Normalized form means the tabulated functions are for filter prototypes with: Ifwe can write so

3. Unity gain means that in the transfer function Normalized Lowpass Filters An N th order filter has N poles. If N is odd, one pole is purely real (It’s imaginary part is zero, so it lies on the real axis.

4. Normalized Lowpass Filters If N is even, no pole lies on the real axis. If a pole is not on the real axis (it’s imaginary part is not zero) then it’s complex conjugate is also a pole. If N is even, the transfer function may be factored into

5. Normalized Lowpass Filters If N is even, the transfer function may be factored into For Butterworth filters,  = 0

6. Normalized Lowpass Filters If N is odd, the transfer function may be factored into

7. Normalized Lowpass Filters x x jj  1 1 Second order

8. Normalized Lowpass Filters x x jj  1 1 Third order x

9. Normalized Lowpass Filters The pole locations are tabulated for Butterworth filters of other filter orders, and for Chebyshev filters of orders up to 8 and various ripple factors, in the textbook. Specialized filter references contain far more extensive tabulations for these and other filter types (Bessel, elliptic, etc.)

10. Lowpass to Lowpass Transformation Denormalizing the normalized filter We will denormalize a prototype lowpass filter (  c = 1) by scaling it so it’s cutoff frequency is  c. Take the normalized transfer function H(s), and replace s with

11. Lowpass to Lowpass Transformation Denormalizing the normalized filter For a second-order Butterworth, the normalized prototype is: If we’re designing a filter with

12. Lowpass to Lowpass Transformation Denormalizing the normalized filter

13. Lowpass to Lowpass Transformation Denormalizing the normalized filter This illustrates how we can denormalize a complex-conjugate pole pair, or second-order section.

14. Problems Volunteers????