2. Bohn 6-03 Rane Corporation Active Filters, EQs & Crossovers Dennis Bohn Rane Corporation

3. Bohn 6-03 Rane Corporation It’s All About the Mathematics Electronic filters are all about the mathematics. You cannot escape the math. We will study the math; … you will love the math.

4. Bohn 6-03 Rane Corporation Simplified Laplace Transforms Represents complex (frequency dependent) impedance, i.e., magnitude & phase Uses the Laplace Operator, s, where s = complex frequency variable = jω = j2πf –Resistor Impedance = R (freq. independent) –Capacitor Reactance = 1/sC –Inductor Reactance = sL Allows writing a circuit’s transfer function by summing circuit currents using Kirchoff’s Law

5. Bohn 6-03 Rane Corporation Transfer Functions (TF) Transfer functions mathematically describe the frequency domain behavior of filters. TF = ratio of Laplace Transforms of a circuit’s input and output voltages: T(s) = V out (s) / V in (s) Filter V in (s)V out (s)

6. Bohn 6-03 Rane Corporation Filter Transfer Functions General filter transfer function is the ratio of two polynomials:

7. Bohn 6-03 Rane Corporation TF Poles & Zeros “Zeros” = values that make numerator equal zero, i.e., the roots of the numerator. –Makes amplitude response rolloff 6 dB/oct. –Shifts phase +90°/zero (+45° @ fc) “Poles” = values that make denominator equal zero, i.e., the roots of the denominator. –Makes amplitude response rise 6 dB/oct. –Shifts phase –90°/zero (–45° @ fc)

8. Bohn 6-03 Rane Corporation Audio Filter Order The order or degree (equivalent terms) is the highest power of s in the transfer function. For analog circuits usually equals the number of capacitors (or inductors) in the circuit. 2nd-order most common. For common audio filters the order equals the rolloff rate divided by 6dB/oct, e.g. 24 dB/oct rolloff = 4th order (24  6 = 4)

9. Bohn 6-03 Rane Corporation Audio Filter Order (cont.) Rule: 6 dB/oct & 90° per order Examples: 1st-order = 6 dB/oct; θ = 90° (  45° @ fc) 2nd-order = 12 dB/oct; θ = 180° (  90° @ fc) 3rd-order = 18 dB/oct; θ = 270° (  135° @ fc) 4th-order = 24 dB/oct; θ = 360° (  180° @ fc) … etc.

10. Bohn 6-03 Rane Corporation Why 6 dB/octave Slope? The impedance of a capacitor is half with twice the frequency, i.e., X C = 1/sC = 1/2  fC The impedance of an inductor is twice when frequency doubles, i.e., X L = sL = 2  fL Twice or Half Impedance = 6 dB change Twice or Half Frequency = One Octave change

11. Bohn 6-03 Rane Corporation Why Phase Shift? Phase shift is the flip side of time It takes time to build up a charge on a capacitor -- that’s why you cannot change the voltage on a capacitor instantaneously. It takes time to build up a magnetic field (flux) in an inductor -- that’s why you cannot change the current through an inductor instantaneously. All this time = phase shift

12. Bohn 6-03 Rane Corporation Why 2nd-Order? Maximum phase shift is 180 degrees Guarantees circuit is unconditionally stable No oscillation problems under any conditions Get higher order circuits by cascading 2nd-order sections … or Design 4th-order section to mathematically emulate two cascaded 2nd-order (Rane’s L-R)

13. Bohn 6-03 Rane Corporation Filter Terminology Corner Frequency = –3 dB point = half power point Center Frequency (any 2nd-order BP) f C =  f H f L i.e., geometric mean, where f L & f H = –3 dB pts Q = Selectivity Factor = reciprocal of BW Q = f C / f H – f L = f C / BW Group Delay: rate of change of phase shift with respect to time, i.e., 1st derivative

14. Bohn 6-03 Rane Corporation Normalized Transfer Function Low-Pass (LP) = (2 poles) 2 poles = -12 dB/oct Frequency Amplitude

15. Bohn 6-03 Rane Corporation Normalized Transfer Function Bandpass (BP) = (1 zero, 2 poles) 1 pole = -6 dB/oct Frequency Amplitude 1 pole = -6 dB/oct 1 zero = +6 dB/oct

16. Bohn 6-03 Rane Corporation Normalized Transfer Function High-Pass (HP) = (2 zeros, 2 poles) 2 zeros = +12 dB/oct Frequency Amplitude 2 poles = -12 dB/oct

17. Bohn 6-03 Rane Corporation Normalized Transfer Function Notch = All-Pass = Poles & zeros cancel amplitude but add phase

18. Bohn 6-03 Rane Corporation Coefficients Determine Performance Butterworth: maximally flat passband s 2 + 1.414s + 1 Chebyshev: steeper rolloff w/magnitude ripples s 2 + 1.43s + 1.51 Bessel: best step response, but gentle rolloff s 2 + 3s + 3 LP ==

19. Bohn 6-03 Rane Corporation Response Comparison

20. Bohn 6-03 Rane Corporation Q Effects Butterworth Q = 0.707 Bessel Q = 0.5

21. Bohn 6-03 Rane Corporation Group Delay Comparison

22. Bohn 6-03 Rane Corporation Step Responses BesselButterworth

23. Bohn 6-03 Rane Corporation Active or Passive? There exists no sound quality attributable to active or passive circuits per se. TF determines the overshoot, ringing and phase shift regardless of implementation. A transfer function is a transfer function is a transfer function … no matter how it is implemented -- all produce the same fundamental results as long as the circuit stays linear: same magnitude response, same phase response, same time response; however there are secondary differences.

24. Bohn 6-03 Rane Corporation Active vs. Passive Passive Less noise No power supply More reliable Less EMI susceptible Better at RF frequency No oscillations No on/off transients No hard clipping Handles large V & I Active Gain & adjustable No loading effects Parameters adjustable Smaller Cs No inductors Smaller, lighter & cheaper No magnetic coupling High Q circuits easy

25. Bohn 6-03 Rane Corporation Audio Filter Applications The Heart of all Signal Processing Tools Loudspeaker Crossover Networks Analyzing Tools: SPL Meters, RTAs Equalizers, Tone Controls & Bandlimiting Dynamic Processors Feedback Suppressors Broadcast Pre-emphasis/De-emphasis Maximizing Recording Media Digital System Aliasing Control

26. Bohn 6-03 Rane Corporation Creating An Equalizer 1 BP Filter fcfc Input Signal BP Out In

27. Bohn 6-03 Rane Corporation Boost = Original + Bandpass 1 + BP 1 fcfc BP + Out In Boost (Lift)

28. Bohn 6-03 Rane Corporation Cut = Reciprocal 1 1+BP 1 fcfc BP + Out In Cut (Dip)

29. Bohn 6-03 Rane Corporation Why 1/3-Octave Centers? 1/3-Octave (2 1/3 oct = x1.26) approximately represents the smallest region humans reliably detect change. Relates to Critical Bands: a range of frequencies where interaction occurs; an auditory filter. About 1/3-octave wide above 500Hz (latest info says more like ~1/6-oct); 100 Hz below 500 Hz

30. Bohn 6-03 Rane Corporation Creating A Crossover: Use LP & HP To Split Signal HP 1 HP 2 High Out Mid Out Input LP 1 Low Out LP 2

31. Bohn 6-03 Rane Corporation 1st-Order & Butterworth Crossovers 1st-order plus 2nd through 4th- order Butterworth vector diagrams

32. Bohn 6-03 Rane Corporation Linkwitz-Riley Crossover Two Cascaded Butterworth Filters Outputs Down 6 dB at Crossover Frequency Both Outputs Always in Phase No Peaking or Lobing Error at Crossover Frequency

33. Bohn 6-03 Rane Corporation Creating A LR Crossover Cascaded Butterworth BW-HP BW-LP High Out Low Out Input BW-HP BW-LP

34. Bohn 6-03 Rane Corporation Linkwitz-Riley Crossovers LR-2 LR-4 LR-8

35. Bohn 6-03 Rane Corporation Ray Miller (Rane) Bessel Crossover

36. Bohn 6-03 Rane Corporation Successfully Crossing-Over Must know the exact amplitude and phase characteristics of the loudspeakers. Driver response strongly interacts with active crossover response. True response = loudspeaker + crossover DSP multiprocessors à la Drag Net allow custom tailoring the total response.

37. Bohn 6-03 Rane Corporation Accelerated-Slope Tone Controls

38. Bohn 6-03 Rane Corporation Stop Kidding Yourself (Rick Chinn Request) Why low-cut and high-cut filters are a must for sound system bandwidth control; or, Why cutting the end sliders on your EQ doesn’t do diddly-squat.

39. Bohn 6-03 Rane Corporation Analog vs. Digital Filters Digital Very complex filters Full adjustability Precision vs. cost Arbitrary magnitude Total linear phase EMI & magnetic noise immunity Stability (temp & time) Repeatability Analog Speed 10-100x faster Dynamic Range –Amplitude: 140 dB e.g., 12 V rms & 1  V noise –Frequency: 8 decades e.g., 0.01 Hz to 1 MHz Cheap, small, low power Precision limited by noise & component tolerances

40. Bohn 6-03 Rane Corporation Digital Filters and DSP Allow circuit designers to do new things. We can go back and solve old problems... like the truth-in-slider-position bugaboo of graphic equalizers: –Proportional-Q was good –Constant-Q was better –Perfect-Q is best

41. Bohn 6-03 Rane Corporation Truth in Slider Position Proportional-Q

42. Bohn 6-03 Rane Corporation Truth in Slider Position Constant-Q

43. Bohn 6-03 Rane Corporation Truth in Slider Position Perfect-Q

44. Bohn 6-03 Rane Corporation Truth in Slide Position Summary Perfect-Q Constant-Q Proportional-Q Any Questions?

45. Bohn 6-03 Rane Corporation PERFECT-Q ™ & DEQ 60 Rick Jeffs Sr. Design Engineer

46. Bohn 6-03 Rane Corporation DEQ 60 Graphic 1/3-Oct EQ

47. Bohn 6-03 Rane Corporation DEQ 60 Features

48. Bohn 6-03 Rane Corporation DEQ 60 Performance

49. Bohn 6-03 Rane Corporation Thanks! Any Questions?