Active Filters, EQs

1
Active Filters, EQs Crossovers

• Dennis Bohn
• Rane Corporation

2
Its 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.

3
Simplified Laplace Transforms

• Represents complex (frequency dependent)
  impedance, i.e., magnitude phase
• Uses the Laplace Operator, s, where
• s complex frequency variable j? j2pf
• Resistor Impedance R (freq. independent)
• Capacitor Reactance 1/sC
• Inductor Reactance sL
• Allows writing a circuits transfer function by
  summing circuit currents using Kirchoffs Law

4
Transfer Functions (TF)

• Transfer functions mathematically describe the
  frequency domain behavior of filters.
• TF ratio of Laplace Transforms of a circuits
  input and output voltages
• T(s) Vout(s) / Vin(s)

Filter
Vin(s)
Vout(s)
5
Filter Transfer Functions

• General filter transfer function is the ratio of
  two polynomials

6
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 _at_ 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 _at_ fc)

7
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)

8
Audio Filter Order (cont.)

• Rule 6 dB/oct 90 per order
• Examples1st-order 6 dB/oct ? 90 (?
  45 _at_ fc)
• 2nd-order 12 dB/oct ? 180 (? 90 _at_ fc)
• 3rd-order 18 dB/oct ? 270 (?135 _at_ fc)
  4th-order 24 dB/oct ? 360 (?180 _at_ fc)
  etc.

9
Why 6 dB/octave Slope?

• The impedance of a capacitor is half with twice
  the frequency, i.e., XC 1/sC 1/2?fC
• The impedance of an inductor is twice when
  frequency doubles, i.e., XL sL 2?fL
• Twice or Half Impedance 6 dB change
• Twice or Half Frequency One Octave change

10
Why Phase Shift?

• Phase shift is the flip side of time
• It takes time to build up a charge on a capacitor
  -- thats why you cannot change the voltage on a
  capacitor instantaneously.
• It takes time to build up a magnetic field (flux)
  in an inductor -- thats why you cannot change
  the current through an inductor instantaneously.
• All this time phase shift

11
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 (Ranes L-R)

12
Filter Terminology

• Corner Frequency 3 dB point half power point
• Center Frequency (any 2nd-order BP)
• fC ?fHfL
  i.e., geometric mean, where fL fH
  3 dB pts
• Q Selectivity Factor reciprocal of BW Q
  fC / fH fL fC / BW
• Group Delay rate of change of phase shift with
  respect to time, i.e., 1st derivative

13
Normalized Transfer Function

• Low-Pass (LP) (2 poles)

2 poles -12 dB/oct
Amplitude
Frequency
14
Normalized Transfer Function

• Bandpass (BP) (1 zero, 2 poles)

1 pole -6 dB/oct
1 pole -6 dB/oct
Amplitude
1 zero 6 dB/oct
Frequency
15
Normalized Transfer Function

• High-Pass (HP) (2 zeros, 2 poles)

2 poles -12 dB/oct
Amplitude
2 zeros 12 dB/oct
Frequency
16
Normalized Transfer Function

• Notch
• All-Pass

Poles zeros cancel amplitude but add phase
17
Coefficients Determine Performance
LP

• Butterworth maximally flat passband s2
  1.414s 1
• Chebyshev steeper rolloff w/magnitude ripples
  s2 1.43s 1.51
• Bessel best step response, but gentle rolloff
  s2 3s 3

18
Response Comparison
19
Q Effects

Butterworth Q 0.707 Bessel Q 0.5
20
Group Delay Comparison

21
Step Responses

Bessel
Butterworth
22
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.

23
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

24
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

25
Creating An Equalizer
Input Signal
Out
In
1
BP
BP Filter
fc
26
Boost Original Bandpass
Boost (Lift)
1 BP
Out

In
BP
1
fc
27
Cut Reciprocal
Out

In
Cut (Dip)
BP
1

1 1BP
fc
28
Why 1/3-Octave Centers?

• 1/3-Octave (21/3 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

29
Creating A CrossoverUse LP HP To Split Signal
HP1
High Out
Input
HP2
LP2
Mid Out
LP1
Low Out
30
1st-Order Butterworth Crossovers

1st-order plus 2nd through 4th-order Butterworth
vector diagrams
31
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

32

Creating A LR CrossoverCascaded Butterworth
BW-HP
BW-HP
High Out
Input
BW-LP
BW-LP
Low Out
33
Linkwitz-Riley Crossovers

LR-4
LR-2
LR-8
34
Ray Miller (Rane)Bessel Crossover

35
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.

36
Accelerated-Slope Tone Controls
37
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 doesnt do
  diddly-squat.

38
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 Vrms 1 ?V noise
• Frequency 8 decades
• e.g., 0.01 Hz to 1 MHz
• Cheap, small, low power
• Precision limited by noise component tolerances

39
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

40
Truth in Slider PositionProportional-Q
41
Truth in Slider PositionConstant-Q
42
Truth in Slider PositionPerfect-Q
43
Truth in Slide PositionSummary

• Perfect-Q
• Constant-Q
• Proportional-Q
• Any Questions?

44
PERFECT-Q DEQ 60

• Rick Jeffs
• Sr. Design Engineer

45
DEQ 60 Graphic 1/3-Oct EQ
46
DEQ 60 Features
47
DEQ 60 Performance
48
Thanks! Any Questions?