1. EE421, Fall 1998 Michigan Technological University Timothy J. Schulz 10-Sept., 1998EE421, Lecture 21 Lecture 2: Continuation of Sampling Butterworth Filters The frequency response magnitude for an N th order Butterworth filter is of the form: –The cutoff frequency is f 0. –The roll-off rate is 20N dB/decade.

2. EE421, Fall 1998 Michigan Technological University Timothy J. Schulz 10-Sept., 1998EE421, Lecture 22 Butterworth Filters l 2nd order Sallen-Key low-pass filter (can be cascaded to create 4th, 6th, 8th, … order filters) C C RR (1-K)R f RfRf + - + + - - inputoutput

3. EE421, Fall 1998 Michigan Technological University Timothy J. Schulz 10-Sept., 1998EE421, Lecture 23 Butterworth Filters Example 2.1: Suppose H 0 = 1 and you want to select f 0 for 1 dB (or less) attenuation over the band from 0 to 4 kHz. at f = 4 kHz. So, or in kHz

4. EE421, Fall 1998 Michigan Technological University Timothy J. Schulz 10-Sept., 1998EE421, Lecture 24 Ideal Reconstruction signal spectrum sampled-signal spectrum ideal reconstruction filter Reconstructed-signal spectrum f f f

5. EE421, Fall 1998 Michigan Technological University Timothy J. Schulz 10-Sept., 1998EE421, Lecture 25 Ideal Reconstruction l The impulse response of an ideal reconstructor is a sinc function, therefore an ideal reconstructor uses sinc interpolation. sampled signalInterpolation signals Interpolated signal

6. EE421, Fall 1998 Michigan Technological University Timothy J. Schulz 10-Sept., 1998EE421, Lecture 26 Practical Reconstruction l The staircase reconstructor is the simplest and most widely used reconstructor in practice. Basically, it is a sample and hold. sampled signalInterpolation signals Interpolated signal The impulse response for a staircase reconstructor is a rectangle. Therefore, its frequency response is a sinc!

7. EE421, Fall 1998 Michigan Technological University Timothy J. Schulz 10-Sept., 1998EE421, Lecture 27 Staircase Reconstruction signal spectrum staircase reconstruction filter Reconstructed-signal spectrum f f f sampled-signal spectrum surviving replicas (images)

8. EE421, Fall 1998 Michigan Technological University Timothy J. Schulz 10-Sept., 1998EE421, Lecture 28 Staircase Reconstruction l The staircase reconstructor should be followed by a low-pass filter to remove the residual images. l Also, a staircase reconstructor suffers from nonuniform pass-band attenuation. This can be corrected with a digital equalization filter with a response equal to an inverse sinc function over the pass-band.