Passive Filters Filter Transfer Function

Filtering of signals in telecommunications is necessary in order to select the desired signal from the range of signals transmitted and also to minimize the effects of noise and interference on the wanted signal. Electrical filters may be constructed using resistors and capacitors, resistors and inductors, or all three types of compo- nents, but it will be noticed that at least one reactive type of component must be present. The resonant circuits described in Sections 1.3 and 1.4 and also the tuned transformers described in Section 1.8 are all examples of filters. Many applications in telecommunications require filters with very sharply defined frequency character- istics, and the filter circuits are much more complex than simple tuned circuits. Most complex filters use all three types of components: inductors, capacitors, and resistors. The inductors tend to be large and costly, and these are now being replaced in many filter designs by electronic circuits that utilize operational amplifiers along with capacitors and resistors. Such filters are known as active filters. Active filters have many advantages over passive filters, the chief ones being that they are small in size, lightweight, and less expensive and offer more flexibility in filter design. The disadvantages are that they require external power supplies and are more sensitive to environmental changes, such as changes in temperature.

Filter design is a very extensive topic, embracing active filters, passive filters, and digital filters, and in this section only a brief introduction to passive filters will be given. In addition to passive filters designed using electrical components, various other types are available that utilize some form of electromechanical coupling.

These include piezoelectric filters and electromechanical filters.

A filter will alter both the amplitude and the phase of the sinusoidal signal passing through it. For audio applications, the effect on phase is seldom significant. Filters are classified by the general shape of the ampli- tude-frequency response into low-pass filters,high-pass filters,band-pass filters, and band-stop filters. These

dB 5 0 –5 –10

10¹ 10²

Frequency (logarithmic scale) 10³ 10⁴ 10⁵ Hz (a)

(b) r_p

C_p

L_p C_ps L_s

N_s C_s Z_L

N_p R_c

a = Np/Ns

L_c

r_s

Voltage gain (V2 ) dB V1

Figure 1.12.2 (a)Equivalent circuit for a low-frequency transformer. (b) Frequency response.

designations are also used for digital and video filtering, but the effect on phase is also very important in these applications. A further designation is the all-pass filter, which affects only the phase, and not the amplitude, of the signal.

The names given refer to the shape of the amplitude part of the filter transfer function. The filter transfer function is defined as the ratio of output voltage to input voltage (current, but not power, could be used instead of voltage) for a sinusoidal input. Thus, if the input to the filter is a sine wave having an amplitude of xand a phase angle of x, the output will also be a sine wave, but with a different amplitude and phase angle in general.

Let yrepresent the output amplitude and ythe output phase; then the filter transfer function is

(1.13.1) The modulus or amplitude-frequency part of the transfer function is H(f), and this is sketched in Fig. 1.13.1(a) for the various kinds of filters listed above. The low-pass filter (LPF) is seen to be character- ized by a passband of frequencies extending from zero up to some cut-off frequency f_c. Ideally, the response should drop to zero beyond the cut-off, but in practice there is a transition region leading to the edge of the stopband at f_s. The stopband is the region above f_swhere the transmission through the filter is ideally zero.

Again, in practice there will be a finite attenuation in the stopband and, also, ripple may be present in both the passband and stopband, as shown in Fig. 1.13.1(a).

The high-pass filter (HPF) characteristic is shown in Fig. 1.13.1(b). Here, the stopband is from zero up to some frequency f_s, the transition region from f_sup to the cut-off frequency f_c, and the passband from f_c onward. As with the LPF, ripple may appear in both the stopband and the passband.

H(f) H(f)

Figure 1.13.1 Amplitude response for basic filter designations: (a) low-pass, (b) high-pass, (c) band-pass, and (d) band-stop.

|H( f ) |

|H( f ) | |H( f ) |

|H( f ) |

0

(a) (b)

(c) (d)

fc

f_s1 f_c1 f_c2 f_s2 f_c1 f_s1 f_s2 f_c2

fs f 0 f_s f_c

f

f

f

The band-pass filter (BPF) characteristic is shown in Fig. 1.13.1(c). The passband is seen to be defined by two cut-off frequencies, a lower one atf_c1and an upper one atf_c2. There is a lower transition region lead- ing to a lower stopband frequency limitf_s1. The lower stopband is from zero up tof_s1. At the other end, the upper transition region leads from f_c2tof_s2, and then the upper stopband extends fromf_s2upward. The cou- pled tuned circuit response shown in Fig. 1.8.3 is an example of a band-pass response.

The band-stop filter (BSF), or band-reject filter, response is shown in Fig. 1.13.1(d). This has a lower passband extending from zero to f_c1, a lower transition region extending from f_c1to f_s1, a stopband extending from f_s1tof_s2, and then an upper transition region extending fromf_s2tof_c2and an upper passband extend- ing upward from f_c2.

A number of well-established filter designs are available, each design emphasizing some particular aspect of the response characteristic. Although these designs apply to all the categories mentioned previously, they will be illustrated here only with reference to the low-pass filter. In the following sections the response curves are normalized such that the maximum value is unity.

Butterworth response. The modulus of the Butterworth response is given by

(1.13.2) This gives what is termed a maximally flat response. The response is sketched in Fig. 1.13.2(a). The order of the filter is m, an integer, and the filter response approaches more closely to the ideal as mincreases.

H(f) 1

|H( f )|

|H( f )| |H( f )|

|H( f )|

l l

.707

0 f_c 0 f_c f

f f

(a)

(c) (d)

(b)

f

0 0

Figure 1.13.2 Sketches of the amplitude/frequency responses of several types of low-pass filters: (a) Butterworth;

(b) Chebyshev; (c) maximally flat time delay (MFTD); (d) Cauer, or elliptic.

Whatever the order of the filter, it will be seen from Eq. (1.13.2) that at the cut-off frequency f f_c the response is reduced by , or 3 dB. Thus, at the cut-off frequency the response is not abruptly “cut off.” The simple RClow-pass filter is an example of a first-order Butterworth filter.

Chebyshev (or Tchebycheff) response. The Chebyshev response is given by

(1.13.3) Here, C_m(f/f_c) is a function known as a Chebyshev polynomial, which for 1 f/f_c 1 is given by cos(mcos¹(f/f_c)). This is a rather formidable expression, but it can be seen that, for the range of f/f_cspec- ified, the Chebyshev polynomial oscillates between 1. This produces an equiripple response in the pass- band, and the coefficient can therefore be chosen to make the ripple as small as desired. The order of this filter response is also m, and this controls the sharpness of the transition region. The Chebyshev response is sketched in Fig. 1.13.2(b). It will be noticed that the cut-off frequency in this case defines the ripple passband. For f/f_c1, the Chebyshev polynomial is given by cosh(mcosh¹(f/f_c)).

Maximally flat time delay response. This type of filter is designed not for sharp cut-off, but to pro- vide a good approximation to a constant time delay or, equivalently, a linear phase-frequency response. In other words, the phase response, rather than the amplitude response, is of more importance. Such filters are required when handling video waveforms and pulses. The amplitude response is a monotonically decreasing function of frequency, meaning that it always decreases as frequency increases from zero, as sketched in Fig. 1.13.2(c).

The Cauer (or elliptic) filter. The filter response H(f) can be written generally as the ratio of two poly- nomials in frequency,N(f)/D(f). For the filters described so far, the numerator N(f) is made constant, and the filter response is shaped by the frequency dependence of the denominator D(f). In the Cauer filter, both the numerator and denominator are made to be frequency dependent, and although this leads to a more complicated filter design, the Cauer filter has the sharpest transition region from passband to stopband. Often, in telephony applications a sharp transition band is the most important requirement. The term elliptic filteris also widely used for this type of filter and comes about because the response can be expressed in terms of a mathematical function known as an elliptic function. The amplitude response of the Cauer filter has ripple in both the pass- band and the stopband, as sketched in Fig. 1.13.2(d).