3  Filter Design

4 Introduction to Filters

  • Filtering = Frequency-selective signal processing
    • It‘s the most common type of signal processing
    • Examples
      • Extract desired signal from many (radio)
      • Seperate signal from noise
      • Amplifier bandwidth limitation
  • Where to start
    • Perfectionist: Ideal (low-pass) filter
    • Engineer: Continuous-time, first-order lowpass filter

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5 Let‘s do something useful …

  • Ideal LPF (recall)
  • RC LPF (recall)
  • Specification
    • Quality factor
    • Frequency characteristics
    • Poles and zeros
    • Group delay, impulse response
  • Approximation problem
    • Butterworth
    • Chebychev
    • Elliptic
    • Bessel

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6 First-order RC Filter

Steady-state response

7 s-Plane / Pole-Zero Map

8 Frequency Response

Question: Can we really get more than 100 dB damping at GHz?

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9 Parasitics

  • Analog passive components aren‘t ideal.
    • Extra real pole/zeros result from parasitics which limit the performance of analog components.
    • Parasitic effects begin to appear „50 dB beyond“ desired component characteristics.
    • Common sense helps anticipate them.
  • Possible parasitics
    • Shunt capacitance
    • Feed-through capacitance
    • Finite inductor, capacitor Q
  • Digital filters do not suffer from these effects.
    • Which effects do they suffer from?

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10 First-order RC Filter incl. Parasitics

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11 Frequency Response incl. Parasitics

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12 Nomenclature Filter Types

Source: H. Khorramabadi, Univ. Berkeley

13 The Filter Approximation Problem

  • Practical filter
    • Ripple in either or both the passband and stopband
    • Limited attenuation in the stopband
  • Ideal Filter
    • Brick-wall characteristic
    • Flat magnitude response in the passband
    • Infinite attenuation in the stopband

Source: EE315a, B. Murmann, Stanford Univ.

14 Filter Design

  • Ideal filters are non-causal or otherwise impractical
  • No global optimization techniques known
  • In practice, chose from several known solutions
    • Butterworth, Elliptic, Bessel, …
  • The overall goal of filter design is to approximate the ideal response by one that implements a reasonable compromise between filter complexity (number of poles and zeros) and approximation error
  • Filter design, in general, requires a compromise between magnitude response, phase response, step response, complexity, etc.

Source: EE315a, B. Murmann, Stanford Univ.

15 Filter Specifications

  • Frequency characteristics (low-pass filter)
    • Passband ripple
    • Cutoff frequency or -3dB frequency
    • Stopband rejection
    • Passband gain
  • Phase characteristics
    • Group delay
  • SNR, dynamic range, SNDR, signal-to-noise + distortion ratio
  • Linearity measures
    • IM3 (intermodulation distortion)
    • HD3 (harmonic distortion)
    • IIP3 or OIP3 (input-referred/output-referred third order intercept point)

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16 Design Parameter Q and w0

  • The term quality factor has different definitions in different contexts
    • Component quality factor (inductor & capacitor)
    • Pole quality factor
    • Bandpass filter quality factor

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17 Inductor / Capacitor Q

Inductor Q

Capacitor Q

Bandpass Q

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18 Pole Q

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19 Lowpass Filter Template

Max./min. passband gain

Magnitude response is fully specified by Apmin, Apmax, As, ωp, ωs

Source: EE315a, B. Murmann, Stanford Univ.

20 Second Order Lowpass Filter

Magnitude response is “maximally flat” (no peaking) for

Source: EE315a, B. Murmann, Stanford Univ.

21 Pole Positions

The poles are the roots of the denominator polynomial

Location in the s-plane for QP > 0.5

Complex conjugate poles

Source: Carusone, p. 164

Source: EE315a, B. Murmann, Stanford Univ.

Source: EE315a, B. Murmann, Stanford Univ.

Source: EE315a, B. Murmann, Stanford Univ.

22 Improvements

  • A maximally flat response is great, but how can we make the roll-off steeper?
  • Let’s look at
    • Imaginary zeros
    • Increasing the filter order
    • High-Q poles
    • High-Q poles and imaginary zeros
  • Different filter approximation
    • Butterworth
    • Chebychev (type I and II)
    • Cauer (Elliptic)

Source: EE315a, B. Murmann, Stanford Univ.

23 Bode Plot

Steeper roll-off at the expense of reduced stopband rejection

Source: EE315a, B. Murmann, Stanford Univ.

24 Adding Another Pole

As expected, steeper roll-off, but transition is not all that sharp

Can fix this issue by increasing QP of H1(s)!

Source: EE315a, B. Murmann, Stanford Univ.

25 Utilizing Peaking in H1(s)

  • Win-win improvement
    • Passband flat, roll-off steeper

Source: EE315a, B. Murmann, Stanford Univ.

26 nth Order Generalization

Stephen Butterworth showed in 1930 that the magnitude response of an nth order maximally flat lowpass filter is given by

This magnitude response is monotonically decreasing and satisfies

The corresponding pole locations can be determined using

Source: EE315a, B. Murmann, Stanford Univ.

27 Pole Locations

The poles lie equally spaced (in angle) on a circle in the s-plane centered at the origin with radius ωp

The LHP roots are taken to be the poles of H(s), while those in the RHP are regarded as the poles of H(–s)

Source: EE315a, B. Murmann, Stanford Univ.

28 A Closer Look at n=4

Source: EE315a, B. Murmann, Stanford Univ.

29 Increasing QP2

Helps make the roll-off steeper, but introduces peaking

We can try to alleviate this problem this by reducing ωP1

Source: EE315a, B. Murmann, Stanford Univ.

30 Increased QP2, Reduced ωP1

This may not a bad choice of we can tolerate some peaking or ripple

Source: EE315a, B. Murmann, Stanford Univ.

31 Comparison with Original Butterworth

How can we optimize this situation, i.e. minimize the transition band for a given tolerable peaking (or “ripple”) in the passband?

Source: EE315a, B. Murmann, Stanford Univ.

32 Matlab Code

Source: EE315a, B. Murmann, Stanford Univ.

33 Matlab Code

Source: EE315a, B. Murmann, Stanford Univ.

34 Design Example

Source: EE315a, B. Murmann, Stanford Univ.

35 Step Response of Design Example

  • Overshoot and other forms of pulse deformation can be problematic
    • Consider e.g. oscilloscopes, pulse-based data links, etc.
  • The pulse deformation is mostly due to the fact that different frequency components pass the filter with different time delays
    • This is called phase distortion
  • Let’s first have a look at the phase response of our filter

Source: EE315a, B. Murmann, Stanford Univ.

36 Phase Distortion (1)

Consider a filter with transfer function

Apply two sine waves at different frequencies (two-tone test)

Source: EE247, B. Boser, Univ. Berkeley

Assuming that the difference between |H(jω1)| and |H(jω2)| is small, the “shape” of the time-domain output signal will be preserved as long as

This condition is satisfied for

A filter with this characteristic is called “linear phase”.

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37 Delay with Linear Phase

Source: U. Moon, Stanford Univ.

38 Delay with Nonlinear Phase

Phase distortion occurs whenever the phase is nonlinear, i.e. the derivative of the phase is not constant. The (negative) derivative of the phase is also called “group delay” or τg Note that for a linear phase filter, we have τg = τd = const.

Source: U. Moon, Stanford Univ.

39 Comparison: Bessel vs. Chebyshev I

Lowpass filters with 100 kHz passband

Both filters are 4th order with the same -3 dB frequency

Passband ripple of 1dB for Chebyshev I

Source: H. Khorramabadi, Univ. Berkeley

40 Phase and Group Delay

Source: H. Khorramabadi, Univ. Berkeley

41 Step Response

42 Pulse Response

43 What about other filters than LP?

  • Frequency Transformation
    • Lowpass-to-lowpass transformation
    • Lowpass-to-highpass transformation
    • Lowpass-to-bandpass transformation
    • Lowpass-to-bandstop transformation
  • Cmp.

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44 Element Transformation (1)

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