3 Analog filters. 3.1 Analog filter characteristics

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1 Chapter 3, page 1 of 11 3 Analog filters This chapter deals with analog filters and the filter approximations of an ideal filter. The filter approximations that are considered are the classical analog filter types Butterworth, Chebyshev, Causer, and Bessel. 3.1 Analog filter characteristics A filter is a system with a frequency dependent response. Signals within one or several frequency bands are passed through almost unaffected, while signals of other frequencies are dampened. The frequency characteristics of a filter are best given by the frequency response or the transfer function and the corresponding s- plane. The most common filter types are the low pass (LP), high pass (HP), band pass (BP), and band stop (BS) filters. Allpass (AP) filters are also an important group of filters. Figure 3.1 shows frequency characteristics of a general low pass filter with the magnitude in decibel H db versus angular frequency ω. It is divided in three bands, the pass band, the transition band, and the stop band. In the pass band, all frequencies are passed without any attenuation. However, this is not the case for a real filter and therefore a small attenuation is accepted. The maximum attenuation that is allowed in the passband is called A max. The variations in attenuation in the passband is called pass band ripple. Often a choice is made A max 3 db and the pass band reaches until this criterion is no longer fulfilled. This happens at the cutoff frequency ω c. In the stop band all frequencies are ideally completely attenuated. However, for a real filter this is not possible and therefore a certain amount of ripple is accepted in the stop band (stop band ripple). The minimum attenuation that is at least

2 Chapter 3, page of 11 required in the stop band is called A min. Typical choice is A min 0 db. The stop band starts at the frequency when the above criterion is met. This frequency is called ω s and thus for ω > ω s the attenuation is greater than A min. The transition band is the region between ω c and ω s. It is characterized by the inclination or the steepness, i.e. how much the magnitude decreases from ω c to ω s. This is called roll off and it is usually expressed in db/decade or db/octave. An octave represents a doubling in frequency and decade means and tenfold increase in frequency. For example, a roll off in H db of -6dB/octave means that the magnitude has decreased by a factor two for an increase in frequency by a factor two. This can also be expressed as -0 db/decade which means that the magnitude has decreased by a factor ten for a tenfold increase in frequency. Figure 3.1. Frequency charactersitics of a general low pass filter First order transfer function A first order transfer function is of the following general form H(s) = b 1s + b 0 s + a 0. For low pass frequency characteristics, the transfer function has the form H LP (s) = b 0 s + a 0

3 Chapter 3, page 3 of 11 and for high pass characteristics H HP (s) = b 1s s + a 0 The pole of the first order transfer function is on the real axis (σ-axis) in the s- plane. The pole is in the negative half plane if a 0 > 0 and in the positive half plane if a 0 < Second order transfer function A second order transfer function is of the following general form H(s) = b s + b 1 s + b 0 s + a 1 s + a 0 For low pass frequency characteristics, the transfer function has the form b 0 H LP (s) = s + a 1 s + a 0 and for high pass characteristics b s H HP (s) = s + a 1 s + a 0 Band pass characteristics can be described by a second order transfer function b 1 s H BP (s) = s + a 1 s + a 0 The second order transfer function has a pair of complex conjugated poles. The denominator polynomial can therefore be expressed in its poles D(s) = s + a 1 s + a 0 D(s) = (s p 1 )(s p )

4 Chapter 3, page 4 of 11 and if we assume that the complex conjugate poles are p 1 = σ p + jω p and p = σ p jω p and perform the multiplication, then D(s) becomes D(s) = s + σ p s + σ p + ω p where the last two terms constitute = σ p + ω p which is the distance between the poles and the origo (figure 3.) in the s-plane and it is called the corner frequency or cutoff frequency (also center frequency for a bandpass filter; also eigenfrequency). It can also be considered the magnitude of the poles. The dampening factor is defined as the ratio d = σ p and the quality factor (or Q-value) is defined as (figure 3.) Q = σ p. The angle between the σ-axis and the pole is α and thus cos(α) = σ p = d = 1 Q. Another parameter for the pair of poles is the bandwidth B defined as B = σ p and thus we have B = Q.

5 Chapter 3, page 5 of 11 If a pair of poles is close to the jω-axis then the Q-value becomes large due to small σ p (figure 3.). The Q-value is also a measure of selectivity or the steepness of the transition band of the frequency response. Figure 3.. Definition of and Q in the s-plane for the pair of poles σ p ± jω p. The three common forms of the denominator polynomial are D(s) = s + σ p s +, D(s) = s + d s +, D(s) = s + Q s +. The effect of the dampening factor on system behavior can be analyzed by analyzing the location of the poles for d > 1, d = 1, 0 < d < 1 and d = 0 (and d < 0). Similar analysis can be done for the Q-value for Q > 1/, Q = 1/, and Q < 1/. The general expression for a :nd order filter is thus H(s) = b s + b 1 s + b 0 s + s Q + It has two quadratic polynomials, therefore these filters are called biquadratic or biquad filters. The denominator polynomial determine the location of the poles

6 Chapter 3, page 6 of 11 ( resonance peaks), but it is the numerator polynomial that determine if it is a LP, HP, or BP filter. The general expression for the :nd order LP filter is b = b 1 = 0 b 0 = H 0 H LP (s) = H 0 s + s Q + where H 0 is the pass band gain (in this case the DC-gain). The general expression for the :nd order HP filter is b 1 = b 0 = 0 b = H 0 where H 0 is the pass band gain. H HP (s) = H 0 s The a :nd order BP filter the expression is H BP (s) = s + s Q + b = b 0 = 0 b 1 = H 0 B b 1 s s + s Q Higher order Higher order filters and thus transfer functions are obtained by multiplying first and/or second order transfer functions. For instance a fourth order transfer function consists of two second order transfer functions multiplied together.

7 Chapter 3, page 7 of All-pole filter An all-pole filter lacks finite zeros, i.e. all of its zeros are in the infinity. This means that the denominator polynomial is independent of s, i.e. a constant. In general the roll off of an n:th order all-pole filter is -0n db/decade or -6n db/octave. 3. Butterworth The Butterworth filter is designed to give maximum flat magnitude of the frequency response in the pass band while the dampening in the stop band is as large as possible. For the normalized Butterworth filter the poles are located along a circle in the left half of the s-plane. It has no zeros. Figure 3.3 shows the location of the poles and the magnitude of the frequency response for a normalized 5:th order low pass Butterworth filter. Figure 3.3. A 5:th order normalized low pass Butterworth filter. (left) poles, (right) magnitude of the frequency response. 3.3 Chebyshev There are two types of Chebyshev filters, type 1 and type. Chebyshev filters are used when a higher dampening in the stop band is required. The drawback is that ripple is introduced in the pass band (type 1) or in the stop band (type ). The type

8 Chapter 3, page 8 of 11 1 filter has the poles along an ellipse in left half of the s-plane. The type filter also has zeros. See figure 3.4 for a normalized 5:th order lowpass Chebyshev type 1 filter with 3 db ripple. See figure 3.5 for a normalized 5:th order lowpass Chebyshev type filter with 40 db ripple. Figure 3.4. A 5:th order normalized low pass Chebyshev type 1 filter with 3 db ripple. (left) poles, (right) magnitude of the frequency response. Figure 3.5. A 5:th order normalized low pass Chebyshev type filter with 40 db ripple. (left) poles and zeros, (right) magnitude of the frequency response. 3.4 Cauer The causer filter is also called an elliptic filter and it has ripple in both the pass band and the stop band. The dampening in the transition band is very high, i.e. the magnitude curve of the frequency response is very steep in the transition band. See figure 3.6 for a normalized 5:th order low pass Cauer filter with 5 db ripple in the pass band and 0 db ripple in the stop band.

9 Chapter 3, page 9 of 11 Figure 3.6. A 5:th order normalized low pass Cauer filter. (left) poles and zeros, (right) magnitude of the frequency response. 3.5 Bessel Bessel filters are a type of filter with a very linear phase response in the pass band. See figure 3.7 for a normalized 5:th order low pass Bessel filter. Figure 3.7. A 5:th order normalized low pass Bessel filter. (top left) poles, (top right) magnitude of the frequency response, and (bottom left) group delay.

10 Chapter 3, page 10 of Filter transformations When designing and constructing a filter, the first step is usually to construct a normalized LP filter, a so called prototype filter. Thereafter transformations are used to obtain a LP, HP, BP, or BS filter with desired frequency characteristics Low Pass-to-Low Pass (LPLP) Suppose that the poles (and zeros) or the transfer function are known for a LP filter with cutoff frequency, then a transformation can be done to obtain a LP filter with the same frequency characteristics, but with a different cutoff frequency found at ω 0. This transformation is done by the following substitution in the transfer function s s For example, the general :nd order LP-filter has a transfer function H(s) = s + s Q + If the above substitution is made, then the transfer function becomes H (s ω s) = 0 ( ω 0 s) + ( s) ω = 0 Q + 1 = ( 1 s) + ( 1 s) 1 = Q ( 1 s) + ( 1 s) 1 Q + 1 = = s + s Q + Thus, the substitution has changed the cutoff frequency from to. This means that each pole will be moved from the origo to a new location

11 Chapter 3, page 11 of 11 p x p ω x Low Pass-to-High Pass (LPHP) Suppose that the transfer function for a LP filter is known and you wish to construct a HP filter with similar frequency characteristics, but with a cutoff frequency. Then the transformation can be done by the following substitution s ω 0 s This means that for each pole, a zero will be introduced in the origo. Each pole will be moved to a new location p x p x Low Pass-to-Band Pass (LPBP) A band pass filter has two cutoff frequencies, a lower (ω 1 ) and an upper (ω ) cutoff frequency. The transformation from a LP filter with cutoff frequency to a BP filter with ω 1 and ω is done by the substitution s s + ω ω 1 s (ω ω 1 ) Low Pass-to-Band Stop (LPBS) A band stop filter has also two cutoff frequencies, a lower (ω 1 ) and an upper (ω ) cutoff frequency. The transformation from a LP filter with cutoff frequency to a BS filter with ω 1 and ω is done by the substitution s s (ω ω 1 ) s + ω ω 1

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