05.Filter design and approximation

5.2.1Low-Pass Filter

Low-pass filter networks are realized by using a cascade of series inductors and shunt capacitors. The number of these elements determines the steepness of the filter with the larger number resulting in a more complex and steep filter. The penalties are the complexity of the filter, the in-band loss, higher cost, and larger size. At low frequencies, series inductances produce low impedance, and shunt capacitors produce high impedance, thus allowing the signal to appear at the output of the filter. Above the cutoff frequency, the series inductors behave as large impedances and shunt capacitors as low impedances, thereby impeding the signal transfer to the load.

5.2.2High-Pass Filter

The high-pass filter shown in Fig. 5.3b allows signal frequencies higher than the cutoff frequency to pass through the filter to the load with a minimum loss and stops all frequencies below the cutoff frequency. This behavior is the reverse of the low-pass filter, and sometimes the high-pass filter is referred to as the complement of the low-pass filter. High-pass filter networks are realized by using a cascade of series capacitors and shunt inductors. Capacitors at high frequencies have low impedance, and inductors have high impedance. Thus the high-frequency signal passes through the filter to the output load with a minimum loss. Just the opposite happens at low frequencies, resulting in a high attenuation of the low frequencies.

5.2.3Band-Pass Filter

The band-pass filter shown in Fig. 5.3c shows the signal is transferred to the load in a band of frequencies between the lower cutoff frequency, ωc1, and the upper cutoff frequency, ωc2. Between the lower and upper cutoff frequency is the center frequency, ω, defined by the geometric mean of ωc1 and ωc2.

5.2.4Band-Stop Filter

The band-stop filter is a complement of the band-pass filter and is shown in Fig. 5.3d. The signal in a band-stop filter is transferred to the load in two frequency bands, one from a low frequency to a low cutoff frequency, ωc, and the other from the upper cutoff frequency, ωc2, to infinite frequency. The signal experiences high loss between ωc1 to ωc2, hence the name band stop or band reject.

5.2.5All-Pass Filter

The all-pass filter allows the signal amplitude for all frequencies to pass through the network without any significant loss (Fig. 5.3e). This network has no frequency selective pass band or stop band. The transmitted signal ideally experiences a linear phase shift or constant group delay with frequency.

88 FILTER DESIGN AND APPROXIMATION

Unfortunately, minimum phase networks do not have constant group delay: rather there are peaks near the corner frequency. All passive ladder networks, such as filters that have frequency selectivity, are minimum phase. In the design there is a trade-off between flat group delay and filter selectivity. However, a network that is nonminimum phase can be cascaded with a minimum phase network to achieve both flat group delay and selectivity. All pass networks with nonminimum phase are used as group delay compensation devices.

5.3TRANSFER FUNCTION AND BASIC FILTER CONCEPTS

Before proceeding with the design of filters, it is important to understand the transfer function in the complex frequency domain, s D C jω. The transfer function can easily be transformed from the time to the frequency domain when losses are small so that D 0 and s D jω. As described in Section 5.2, the filter transfer function is the ratio of the output signal voltage to the input signal voltage. One could also easily select the ratio of currents. The transfer function, in general, can be written as a ratio of two polynomials:

where polynomials P s and Q s in general are of order m and n. These polynomials are Hurwitz stable, which requires that the order of the numerator polynomial m be equal to or less than the denominator polynomial n, m < n. The order of polynomial Q s is the order of the filter as well. Polynomials P s and Q s can be factored and rewritten in the form

The values z1, z2, z3, . . . , zm, are called the zeros of the transfer function, or simply transmission zeros. The roots of Q s , p1, p2, p3, . . . , pn, are the poles of the transfer function. The poles and zeros can be real or complex, but complex poles and zeros must occur in conjugate pairs. That is, if 2 C j3 is a pole, then2 j3 must be a pole as well. The magnitude plot of voltage transfer function represents the loss or attenuation of the filter circuit, and in dB is given by

Poles and zeros of realizable passive networks must follow certain rules:

1.All poles of a transfer function occur in the left half s-plane. The left half s-plane includes the imaginary jω-axis.

2.Complex poles and zeros occur in complex conjugate pairs. However, on the imaginary axis, poles and zeros may exist singly.

90 FILTER DESIGN AND APPROXIMATION

ω D 0. The gain function for this type of filter is given by

where H0 1. The first 2n 1 derivatives of the denominator of this function are all zero at ω D 0, implying that it is maximally flat. The poles of this function all have a magnitude of 1 and are separated from one another on the unit circle by /n radians. Furthermore there are no poles on the jω-axis. At the edge of the pass band, the filter attenuates the power by 12 or 3 dB. A recursion formula for the filter elements that would produce this response can be found in a variety of references, one of which is [1]:

A set of filter values that is worth remembering is the three-pole Butterworth filter, where g1, g2, g3 D 1, 2, 1. The low-pass prototype starting with a shunt C has C1 D 1 F, L2 D 2 H, and C3 D 1 F.

Often minimum requirements are placed on the shape of the pass band. In this instance the minimum number of poles needed to produce a desired specifica-

tion is

log[ 10˛min/10 1 10˛max/10 1 ]

In this expression the maximum attenuation in the pass band 0 ω ωp is ˛max. The minimum attenuation in the stop band, ωs ω < 1, is ˛min.

5.4.2The Chebyshev Filter

The slope of the pass-band skirts for a given number of poles of a filter can be improved by allowing small ripples in the pass band. In antenna theory the DolphChebyshev amplitude weighting of the array elements provides the minimum beam width for a specified side lobe level. Similarly in filter design the Chebyshev function provides the maximum possible bandwidth for a given pass-band ripple or the minimum possible pass-band ripple for a given bandwidth. The Chebyshev (equal ripple) low-pass filter transducer gain function is

where ωc is the low-pass cutoff frequency. The value ε is a number <1 and is a measure of the pass-band ripple. The Chebyshev function, Tn x , oscillates

between C1 and 1 when its argument is less than 1. The poles of this transfer function lie on an ellipse with no jω axis poles.

For x > 1, Tn x rapidly becomes large. The Chebyshev function can be

Since Tn x < 1 in the pass band, the pass-band transfer function is

1 jH jω j2 1 1 C ε2

Outside the pass band, Tn x increases approximately exponentially.

The Chebyshev functions can be found in terms of a polynomial of its argument from a recursion formula:

The formula begins by setting T0 x D 1 and T1 x D x. Furthermore for n odd

If the maximum pass-band frequency is ωc and the minimum stop-band frequency beyond which the attenuation is always greater than ˛min is ωs, then the number of poles required in the function is n [1]:

92 FILTER DESIGN AND APPROXIMATION

Just as in the Butterworth approximation, there is a set of recursion formulas for the Chebyshev filter. Finding expressions for the g values for the filter requires first expanding the Chebyshev functions by its own set of recursion formulas. The low-pass prototype filter structure (for a given number n of reactive elements) is then equated to the nth-order filter function so that a correlation is made between the circuit and the function. Fortunately the hard work has been done in network synthesis books (e.g., [1,2]). The final recursion formulas are given below:

One important difference between the Butterworth and Chebyshev approximations is the value for gnC1. The unequal impedance levels for the even-order Chebyshev termination impedances is often avoided by simply restricting the choices of n for the Chebyshev function to odd values. The circuit element values for these two filter functions were found by using network synthesis techniques after determining the poles of the transfer function. Other filter functions are available and are briefly mentioned here.

5.4.3The Inverse Chebyshev Filter

In this filter, equal ripples are found in the stop band while the pass band is smooth. The inverse Chebyshev function is useful when it is necessary to control the attenuation over the entire stop band. The transfer function and the minimum number of poles needed to guarantee a maximum attenuation in the pass band