EE2S11 Signals and Systems, part 2. Ch.7.3 Analog filter design. Note differences in notation. We often write. transition band passband.

Transcription

1 Note differences in notation. We often write EE2S11 Signals and Systems, part 2 Ch.7.3 How can I design an analog filter that meets certain specifications? passband ripple transition band passband stopband Stopband ripple instead of. 1

2 Continuous-time filter functions General form: Stability and causality: poles of in left half plane zeros of in left half plane Frequency spectrum: Damping (loss): usually specified in db: ; 2 db

3 Filter specifications Example: specification of a low-pass filter passband ripple transition band passband stopband Stopband ripple In the same manner: high-pass, band-pass, band-stop An ideal filter does not have ripples and no transition band. But, a causal filter has a finite number of zeros and cannot be an ideal filter (Paley-Wiener). cannot be constant over an interval. Usually, only the amplitude spectrum is specified, because the phase spectrum is (almost) completely determined by this (cf. the Hilbert transform or the causality requirement) 3

4 : minimal squared-amplitude in the pass-band, or Practical design We limit ourselves to design techniques based on amplitude specifications. We start with low-pass filters (the other types are derived from these). Specifications: like before, also written as : pass-band frequency maximal damping : stop-band frequency : maximal squared-amplitude ( minimal damping) in the stop-band 4

5 derivatives equal to zero in derivatives equal to zero in is rational, order Butterworth filter We start from the following characteristics so that is an even function The Butterworth filter is obtained if we require and : to be maximally flat for or 5

6 Example design Butterworth filter Filter parameters are and ; but usually is specified Parameter is the cutoff frequency: db We can design such that this specification is met: Example (or normalized Butterworth filter): We often use one of these forms for the design: 6

7 What is From for the Butterworth filter? it follows The poles of follow as These are located on a circle with radius s-plane Stable: poles of are the left-half plane values in the 7

8 Example 1: Design Butterworth filter Determine the minimal order of the Butterworth filter with pass-band frequency db, stop-band frequency khz, maximal damping in the pass-band khz, and minimal damping in the stop-band db Solution: We start from From we derive : with From, and we derive the minimal : 8

9 In the pass-band we must have:. Elsewhere: Chebyshev filter The Butterworth filter has maximal error in the pass-band at, elsewhere the error is smaller. Perhaps the filter order can be made smaller (or the response sharper for the same filter order) by distributing the error more uniformly over the pass-band? We keep the maximal flatness in : derivatives zero in In the case of Chebyshev this is written more specifically as is an even or odd polynomial of order (because has to be even) 9

10 Chebyshev filter From now on, normalize the pass-band to. with 10

11 is an even or odd polynomial in of order Chebyshev polynomials Idea: How do we design with such that is an even or odd polynomial of order From the property and Repeat this to obtain This gives. if Also valid: same recursion! 11 ( we obtain the recursion has to oscillate between -1 and 1 in the pass-band, hence set? ).

12 Chebyshev polynomials The recursion becomes: : 2 T T T T Ω 12

13 Resulting filters: 13

14 What is for the Chebyshev filter? Like with the Butterworth filter we look for for which Poles of These turn out to lie on an ellipse. Poles of are the poles in the left-half plane satisfy s-plane Calculation of the cut-off frequency (3 db level) : 14

15 Example 2: Design Chebyshev filter Determine the minimal order of a Chebyshev filter with pass-band frequency khz, maximal damping in the pass-band db, stop-band frequency khz, and minimal damping in the stop-band db Solution: We start from From we derive : with From, and we derive the minimal order : 15

16 Elliptic filter Generalization of the Chebyshev filter: with an arbitrary rational function in We will not discuss this any further. 16

17 to Frequency transformations To transform a prototype filter into a desired filter we use transformations of the frequency axis: low-pass to low-pass : shift a frequency from : substitute: More generally: shift a frequency of to : substitute: 17

18 Example: low-pass to low-pass Suppose we have a template filter with cut-off frequency : Mapping to a filter with cut-off frequency : transform Mapping to a filter with cut-off frequency : transform 18

19 Frequency transforms (2) low-pass to high-pass: mapping, and More generally: mapping with reversal of the frequency axis 19

20 Example: low-pass to high-pass Suppose the template low-pass filter has cut-off frequency : Transform gives a high-pass filter with cut-off frequency : 20

21 Example 3: use of the low-to-high transform We require an analog high-pass filter design with the following specifications: Pass-band: starting at Hz; ripple in the pass-band: 1 db Stop-band: until Hz; stop-band damping: 30 db. We start with a Butterworth low-pass filter structure of the form which we design such that, and equal to -1 db. 21

22 Next, we apply to a low-to-high transform: gives This is a high-pass filter with pass-band. From the transformation, it follows that the stop-band frequency in the design of should be. Instead of first designing, we can also directly determine and for 22

23 by evaluation at Determine by evaluation at : Determine : We take filter order. 23

24 (Advanced material) Frequency transformations (3) low-pass to band-pass: Suppose the template filter has cut-off frequency ( ) ( ) This transformation should map,, and. Band center and scale factor are computed based on the desired cutoff frequency and : ( ( ) ) 24

25 The pass-band is geometrically symmetric around : Derivation After transformation, we must have: as this gives. We find ( ) 25

26 Frequency transformations (4) low-pass to band-pass (general) Suppose the template filter has cut-off frequency, the new filter has cut-off frequencies and : Verification: Evaluate for : this gives Evaluate for : this gives Evaluate for : this gives. Note that this transformation doubles the filter order! 26

27 low-pass to band-stop (template cut-off frequency 1) ( ) ( ) and are calculated based on and : ( ) ( ) Band-stop characteristic is geometrically symmetric around : 27

28 Example 4: Frequency transformations Design a band-pass Chebyshev filter with pass-band khz until khz, maximal damping db in de pass-band, and minimal damping db for khz and khz. Solution: We don t have a transformation with 4 frequencies as parameters. We select a transformation based on the pass-band frequencies and then check the stop-band. We will use the following transformation: ( ) ( ) with and derived from the pass-band frequencies. 28

29 Example 4 (cont d) Determine the geometric center of the pass-band and the scale factor: Use the property to determine which side gives the strongest damping requirements in the stop-band: db, khz db, khz db, khz db, khz Therefore we calculate our low-pass characteristic based on khz: if we meet 40 db damping here, then we will also have this at khz and certainly at khz. 29

30 Thus, we have to design a template low-pass Chebyshev filter with at Example 4 (cont d) The transform gives ( ) ( a damping of db, and at a damping of Like before, we find and Next, determine 30 and insert the transformation to obtain the desired filter. ) db. khz