The electrical engineering handbook CH010

The electrical engineering handbook

Dorf, R.C., Wan, Z “Transfer Functions of Filters”

The Electrical Engineering Handbook

Ed Richard C Dorf

Boca Raton: CRC Press LLC, 2000

10 Transfer Functions

of Filters

10.1 Introduction 10.2 Ideal Filters 10.3 The Ideal Linear-Phase Low-Pass Filter 10.4 Ideal Linear-Phase Bandpass Filters 10.5 Causal Filters

10.6 Butterworth Filters 10.7 Chebyshev Filters

10.1 Introduction

Filters are widely used to pass signals at selected frequencies and reject signals at other frequencies An electrical filter is a circuit that is designed to introduce gain or loss over a prescribed range of frequencies In this section,

we will describe ideal filters and then a selected set of practical filters

10.2 Ideal Filters

An ideal filter is a system that completely rejects sinusoidal inputs of the form x(t) = A cos wt, –¥ < t < ¥, for w in certain frequency ranges and does not attenuate sinusoidal inputs whose frequencies are outside these ranges There are four basic types of ideal filters: low-pass, high-pass, bandpass, and bandstop The magnitude functions of these four types of filters are displayed in Fig 10.1 Mathematical expressions for these magnitude functions are as follows:

(10.1)

(10.2)

(10.3)

(10.4)

Ideal low-pass: * *

* *

B

( ) ,

,

w

>

ì í ï î

1 0

Ideal high-pass: * *

* *

B

( ) ,

,

w

= - < <

³

ì í ï î

0 1

Ideal bandpass:

all other

* H ( ) * , B * * B

,

w

= ì í ï £ £ î

1 0

Ideal bandstop:

all other

* H ( ) * , B * * B

,

w

= ì í ï £ £ î

0 1

Richard C Dorf

University of California, Davis

Zhen Wan

University of California, Davis

The stopband of an ideal filter is defined to be

the set of all frequencies w for which the filter

completely stops the sinusoidal input x(t) = A cos

wt, –¥ < t < ¥ The passband of the filter is the

set of all frequencies w for which the input x(t) is

passed without attenuation

More complicated examples of ideal filters can

be constructed by cascading ideal low-pass,

high-pass, bandhigh-pass, and bandstop filters For instance,

by cascading bandstop filters with different values

of B1 and B2, we can construct an ideal comb filter,

whose magnitude function is illustrated in Fig 10.2

10.3 The Ideal Linear-Phase Low-Pass Filter

Consider the ideal low-pass filter with the frequency function

(10.5)

where t d is a positive real number Equation (10.5) is the polar-form representation of H(w) From Eq (10.5)

we have

and

FIGURE 10.1 Magnitude functions of ideal filters:(a) low-pass; (b) high-pass; (c) bandpass; (d) bandstop.

|H |

B 1

0 –B

(a)

B1 1

0 – B1

(c)

B2 – B2

1

0 (d)

1

0 (b)

w

|H |

|H |

|H |

B –B

w w

w

B1

– B2

FIGURE 10.2 Magnitude function of an ideal comb filter.

|H |

1

0

j t d

, ,

w

< - >

ì í ï î

-0

( ) ,

, ,

< - >

ì í ï î

1 0

, ,

d

< - >

ì í ï î ï0

The phase function /H(w) of the filter is plotted in Fig 10.3 Note that over the frequency range 0 to B, the phase function of the system is linear with slope equal to –t d

The impulse response of the low-pass filter defined by Eq (10.5) can be computed by taking the inverse Fourier transform of the frequency function H(w) The impulse response of the ideal lowpass filter is

(10.6)

where Sa(x) = (sin x)/x The impulse response h(t) of the ideal low-pass filter is not zero for t < 0 Thus, the filter has a response before the impulse at t = 0 and is said to be noncausal As a result, it is not possible to build an ideal low-pass filter

10.4 Ideal Linear-Phase Bandpass Filters

One can extend the analysis to ideal phase bandpass filters The frequency function of an ideal linear-phase bandpass filter is given by

where t d, B1, and B2 are positive real numbers The magnitude function is plotted in Fig 10.1(c) and the phase function is plotted in Fig 10.4 The passband of the filter is from B1 to B2 The filter will pass the signal within the band with no distortion, although there will be a time delay of t d seconds

FIGURE 10.3 Phase function of ideal low-pass filter defined by Eq (10.5).

FIGURE 10.4 Phase function of ideal linear-phase bandpass filter.

H ( w )

Btd

–B

–Btd Slope = – td

w

H ( w ) B2td

0

w

Slope = – td

B1td

h t ( ) [ ( )], = B Sa B t t t - d - ¥ < < ¥

p

j t d

,

w

w

= ì í ï £ £ î

0

* * all other

10.5 Causal Filters

As observed in the preceding section, ideal filters cannot be utilized in real-time filtering applications, since

they are noncausal In such applications, one must use causal filters, which are necessarily nonideal; that is,

the transition from the passband to the stopband (and vice versa) is gradual In particular, the magnitude

functions of causal versions of low-pass, high-pass, bandpass, and bandstop filters have gradual transitions

from the passband to the stopband Examples of magnitude functions for the basic filter types are shown in

Fig 10.5

For a causal filter with frequency function H(w), the passband is defined as the set of all frequencies w for

which

(10.7)

where wp is the value of w for which *H(w)* is maximum Note that Eq (10.7) is equivalent to the condition

that *H(w)*dB is less than 3 dB down from the peak value *H(wp)*dB For low-pass or bandpass filters, the width

of the passband is called the 3-dB bandwidth

A stopband in a causal filter is a set of frequencies w for which *H(w)*dB is down some desired amount (e.g., 40

or 50 dB) from the peak value *H(wp)*dB The range of frequencies between a passband and a stopband is called a

transition region In causal filter design, a key objective is to have the transition regions be suitably small in extent

10.6 Butterworth Filters

The transfer function of the two-pole Butterworth filter is

Factoring the denominator of H(s), we see that the poles are located at

FIGURE 10.5 Causal filter magnitude functions: (a) low-pass; (b) high-pass; (c) bandpass; (d) bandstop.

wp -wp

1 0.707

(a)

1

(b)

1

(c)

1

(d)

* H ( ) ( w * ³ 1 * H wp) ( * * H wp) *

H s

n

( ) =

w

2

2

s = - ± wn j wn

Note that the magnitude of each of the poles is equal to wn.

Setting s = jw in H(s), we have that the magnitude function of the two-pole Butterworth filter is

(10.8)

From Eq (10.8) we see that the 3-dB bandwidth of the Butterworth filter is equal to wn For the case wn = 2 rad/s, the frequency response curves of the Butterworth filter are plotted in Fig 10.6 Also displayed are the

frequency response curves for the one-pole low-pass filter with transfer function H(s) = 2/(s + 2), and the

two-pole low-pass filter with z = 1 and with 3-dB bandwidth equal to 2 rad/s Note that the Butterworth filter has the sharpest cutoff of all three filters

10.7 Chebyshev Filters

The magnitude function of the n-pole Butterworth filter has a monotone characteristic in both the passband and stopband of the filter Here monotone means that the magnitude curve is gradually decreasing over the

passband and stopband In contrast to the Butterworth filter, the magnitude function of a type 1 Chebyshev filter has ripple in the passband and is monotone decreasing in the stopband (a type 2 Chebyshev filter has the opposite characteristic) By allowing ripple in the passband or stopband, we are able to achieve a sharper transition between the passband and stopband in comparison with the Butterworth filter

The n-pole type 1 Chebyshev filter is given by the frequency function

(10.9)

where T n(w/w1) is the nth-order Chebyshev polynomial Note that e is a numerical parameter related to the

level of ripple in the passband The Chebyshev polynomials can be generated from the recursion

Tn(x) = 2xTn – 1(x) – Tn – 2(x)

where T0(x) = 1 and T1(x) = x The polynomials for n = 2, 3, 4, 5 are

T2(x) = 2x(x) – 1 = 2x2 – 1

T3(x) = 2x(2x2 – 1) – x = 4x3 – 3x

T4(x) = 2x(4x3 – 3x) – (2x2 – 1) = 8x4 – 8x2 + 1

T5(x) = 2x(8x4 – 8x2 + 1) – (4x3 – 3x) = 16x5 – 20x3 + 5x (10.10)

FIGURE 10.6 Magnitude curves of one- and two-pole low-pass filters.

2

s + 2

w Two-pole Butterworth filter Two-pole filter with z = 1 One-pole filter H(s) =

1 2 3 4 5 6 7 8 9

0.707

1 0.8 0.6 0.4 0.2 0

Passband

|H ( w )|

* H *

n

( )

( / )

w

w w

= +

1

* H *

Tn

( )

w

w w

= + 1

1 e /2 2 1

Using Eq (10.10), the two-pole type 1 Chebyshev filter has the following frequency function

For the case of a 3-dB ripple (e = 1), the transfer functions of the two-pole and three-pole type 1 Chebyshev filters are

where wc = 3-dB bandwidth The frequency curves for these two filters are plotted in Fig 10.7 for the case wc

= 2.5 rad

The magnitude response functions of the three-pole Butterworth filter and the three-pole type 1 Chebyshev filter are compared in Fig 10.8 with the 3-dB bandwidth of both filters equal to 2 rad Note that the transition from passband to stopband is sharper in the Chebyshev filter; however, the Chebyshev filter does have the

3-dB ripple over the passband

FIGURE 10.7 Frequency curves of two- and three-pole Chebyshev filters with wc = 2.5 rad/s: (a) magnitude curves; (b) phase curves.

w Three-pole filter

Two-pole filter

1 2 3 4 5 6 7 8 9

0.707

1 0.8 0.6 0.4 0.2 0

Passband

|H( w )|

(a)

w

Three-pole filter Two-pole filter

1 2 3 4 5 6 7 8 9

0

|H( w )|

(b)

* H ( ) *

[ ( / ) ]

w

w w

=

-1

1 e2 2 1 2 12

H s

H s

c

c

=

=

0 50

0 645 0 708

0 251

0 597 0 928 0 251

2

3

w

w

Defining Terms

filter is realizable

which *H(w)*dB is less than 3 dB down from the peak value *H(wp)*dB

–¥ < t < ¥, for w within a certain frequency range, and does not attenuate sinusoidal inputs whose frequencies are outside this range There are four basic types of ideal filters:low-pass, high-pass, bandpass, and bandstop

Related Topics

4.2 Low-Pass Filter Functions • 4.3 Low Pass Filters • 11.1 Introduction • 29.1 Synthesis of Low-Pass Forms

References

R.C Dorf, Introduction to Electrical Circuits, 3rd ed., New York: Wiley, 1996.

E.W Kamen, Introduction to Signals and Systems, 2nd ed., New York: Macmillan, 1990.

G.R Cooper and C.D McGillem, Modern Communications and Spread Spectrum, New York: McGraw-Hill, 1986.

Further Information

IEEE Transactions on Circuits and Systems, Part I: Fundamental Theory and Applications.

IEEE Transactions on Circuits and Systems, Part II: Analog and Digital Signal Processing.

Available from IEEE

FIGURE 10.8 Magnitude curves of three-pole Butterworth and three-pole Chebyshev filters with 3-dB bandwidth equal

to 2.5 rad/s.

w Three-pole Chebyshev

1 2 3 4 5 6 7 8 9

0.707

1 0.8 0.6 0.4 0.2 0

Passband