Analog Filters and Applications

Chapter 1 INTRODUCTION TO DIGITAL SIGNAL PROCESSING 1.6 Analog Filters 1.7 Applications of Analog Filters

July 9, 2018

Frame # 1 Slide # 1 A. Antoniou Digital Filters – Secs. 1.6, 1.7 I It can be carried out by analog or digital means. I Analog ﬁlters have been in use since 1915 but with the emergence of digital technologies in the 1960s, they began to be replaced by digital ﬁlters in many applications.

I This presentation will provide a brief historical background on analog ﬁlters and their applications.

Introduction

I Filtering has found widespread applications in many areas such as communications systems, audio systems, speech synthesis, and many other areas.

Frame # 2 Slide # 2 A. Antoniou Digital Filters – Secs. 1.6, 1.7 I Analog ﬁlters have been in use since 1915 but with the emergence of digital technologies in the 1960s, they began to be replaced by digital ﬁlters in many applications.

I This presentation will provide a brief historical background on analog ﬁlters and their applications.

Introduction

I Filtering has found widespread applications in many areas such as communications systems, audio systems, speech synthesis, and many other areas.

I It can be carried out by analog or digital means.

Frame # 2 Slide # 3 A. Antoniou Digital Filters – Secs. 1.6, 1.7 I This presentation will provide a brief historical background on analog ﬁlters and their applications.

Introduction

I Filtering has found widespread applications in many areas such as communications systems, audio systems, speech synthesis, and many other areas.

I It can be carried out by analog or digital means. I Analog ﬁlters have been in use since 1915 but with the emergence of digital technologies in the 1960s, they began to be replaced by digital ﬁlters in many applications.

Frame # 2 Slide # 4 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Introduction

I Filtering has found widespread applications in many areas such as communications systems, audio systems, speech synthesis, and many other areas.

I This presentation will provide a brief historical background on analog ﬁlters and their applications.

Frame # 2 Slide # 5 A. Antoniou Digital Filters – Secs. 1.6, 1.7 I highpass ﬁltering can be used to pass a band of preferred high frequencies and reject a band of undesirable low frequencies

I bandpass ﬁltering can be used to pass a band of frequencies and reject certain low- and high-frequency bands, or

I bandstop ﬁltering can be used to reject a band of frequencies but pass certain low- and high-frequency bands.

Filtering

Filtering can be used to pass one or more desirable bands of frequencies and simultaneously reject one or more undesirable bands. For example,

I lowpass ﬁltering can be used to pass a band of preferred low frequencies and reject a band of undesirable high frequencies

Frame # 3 Slide # 6 A. Antoniou Digital Filters – Secs. 1.6, 1.7 I bandpass ﬁltering can be used to pass a band of frequencies and reject certain low- and high-frequency bands, or

I bandstop ﬁltering can be used to reject a band of frequencies but pass certain low- and high-frequency bands.

Filtering

Filtering can be used to pass one or more desirable bands of frequencies and simultaneously reject one or more undesirable bands. For example,

I lowpass ﬁltering can be used to pass a band of preferred low frequencies and reject a band of undesirable high frequencies

I highpass ﬁltering can be used to pass a band of preferred high frequencies and reject a band of undesirable low frequencies

Frame # 3 Slide # 7 A. Antoniou Digital Filters – Secs. 1.6, 1.7 I bandstop ﬁltering can be used to reject a band of frequencies but pass certain low- and high-frequency bands.

Filtering

Filtering can be used to pass one or more desirable bands of frequencies and simultaneously reject one or more undesirable bands. For example,

I lowpass ﬁltering can be used to pass a band of preferred low frequencies and reject a band of undesirable high frequencies

I highpass ﬁltering can be used to pass a band of preferred high frequencies and reject a band of undesirable low frequencies

I bandpass ﬁltering can be used to pass a band of frequencies and reject certain low- and high-frequency bands, or

Frame # 3 Slide # 8 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Filtering

Filtering can be used to pass one or more desirable bands of frequencies and simultaneously reject one or more undesirable bands. For example,

I lowpass ﬁltering can be used to pass a band of preferred low frequencies and reject a band of undesirable high frequencies

I highpass ﬁltering can be used to pass a band of preferred high frequencies and reject a band of undesirable low frequencies

I bandpass ﬁltering can be used to pass a band of frequencies and reject certain low- and high-frequency bands, or

I bandstop ﬁltering can be used to reject a band of frequencies but pass certain low- and high-frequency bands.

Frame # 3 Slide # 9 A. Antoniou Digital Filters – Secs. 1.6, 1.7 I Arbitrary amplitudes and phase angles can be assigned to the various sinusoidal components as shown in the next slide.

Filtering Cont’d

I To illustrate the ﬁltering process consider an arbitrary periodic signal which is made up of a sum of sinusoidal components such as 9 X x(t) = Ai sin(ωi t + θi ) i=1

where Ai is the amplitude and θi is the phase angle of the ith sinusoidal component.

Frame # 4 Slide # 10 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Filtering Cont’d

I To illustrate the ﬁltering process consider an arbitrary periodic signal which is made up of a sum of sinusoidal components such as 9 X x(t) = Ai sin(ωi t + θi ) i=1

where Ai is the amplitude and θi is the phase angle of the ith sinusoidal component.

I Arbitrary amplitudes and phase angles can be assigned to the various sinusoidal components as shown in the next slide.

Frame # 4 Slide # 11 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Filtering Cont’d

i ωi Ai θi 1 1 0.6154 0.0579 2 2 0.7919 0.3529 3 3 0.9218 −0.8132 4 4 0.7382 0.0099 5 5 0.1763 0.1389 6 6 0.4057 −0.2028 7 7 0.9355 0.1987 8 8 0.9169 −0.6038 9 9 0.4103 −0.2722

Frame # 5 Slide # 12 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Filtering Cont’d

Time-domain representation:

5 ) t

( 0 x

−5 −10 −5 0 5 10 15 Time, s

Frame # 6 Slide # 13 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Filtering Cont’d

Frequency-domain representation:

Amplitude spectrum Phase spectrum 1.0 0.4

0.2 0.8 0

0.6 −0.2

−0.4 Magnitude 0.4 Phase angle, rad −0.6 0.2 −0.8

0 −1.0 0 5 10 0 5 10 Frequency, rad/s Frequency, rad/s

Frame # 7 Slide # 14 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Filtering Cont’d

The filtering process can be represented by a block diagram as shown in the figure where x(t) is the input and y(t) is the output of the filtering process.

x(t) Filtering y(t)

x(t) y(t)

t t

Frame # 8 Slide # 15 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Lowpass Filtering

Lowpass ﬁltering will pass low frequencies and reject high frequencies as shown in the next two slides.

Frame # 9 Slide # 16 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Lowpass Filtering Cont’d

Amplitude spectrum Phase spectrum 1.0 0.4

0.2 0.8 0

0.6 − Input 0.2 −0.4 Magnitude 0.4 Phase angle, rad −0.6 0.2 −0.8

0 −1.0 0 5 10 0 5 10 Frequency, rad/s Frequency, rad/s

Amplitude spectrum Phase spectrum 1.0 0.4

0.2 0.8 0 rad

0.6 e, −0.2 angl Output nitude

se se −0.4

Mag 0.4 Pha −0.6 0.2 −0.8

0 −1.0 0 5 10 0 5 10 Frequency, rad/s Frequency, rad/s

Frame # 10 Slide # 17 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Lowpass Filtering Cont’d

5 ) t

( 0 Input x

−5 −10 −5 0 5 10 15 Time, s

5 ) t

( 0 Output y

−5 −10 −5 0 5 10 15 Time, s

Frame # 11 Slide # 18 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Highpass Filtering

Highpass ﬁltering will pass high frequencies and reject low frequencies as shown in the next two slides.

Frame # 12 Slide # 19 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Highpass Filtering Cont’d

Amplitude spectrum Phase spectrum 1.0 0.4

0.2 0.8 0

0.6 − Input 0.2 −0.4 Magnitude 0.4 Phase angle, rad −0.6 0.2 −0.8

0 −1.0 0 5 10 0 5 10 Frequency, rad/s Frequency, rad/s

Amplitude spectrum Phase spectrum 1.0 0.2

0.1

0.8 0

d −0.1 0.6 ra le, −0.2 nitude Output −0.3 se ang

Mag 0.4

Pha −0.4 − 0.2 0.5 −0.6 0 −0.7 0 5 10 0 5 10 Frequency, rad/s Frequency, rad/s

Frame # 13 Slide # 20 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Highpass Filtering Cont’d

5 ) t

( 0 Input x

−5 −10 −5 0 5 10 15 Time, s

5 ) t

( 0 Output y

−5 −10 −5 0 5 10 15 Time, s

Frame # 14 Slide # 21 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Bandpass Filtering

Amplitude spectrum Phase spectrum 1.0 0.4

0.2 0.8 0

0.6 − Input 0.2 −0.4 Magnitude 0.4 Phase angle, rad −0.6 0.2 −0.8

0 −1.0 0 5 10 0 5 10 Frequency, rad/s Frequency, rad/s

Amplitude spectrum Phase spectrum 0.8 0.15

0.7 0.10

0.6 0.05

0.5 0

0.4 ngle, rad −0.05

Output e a Magnitude 0.3 −0.10 Phas

0.2 −0.15

0.1 −0.20

0 −0.25 0 5 10 0 5 10 Frequency, rad/s Frequency, rad/s

Frame # 15 Slide # 22 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Bandstop Filtering

Amplitude spectrum Phase spectrum 1.0 0.4

0.2 0.8 0

0.6 − Input 0.2 −0.4 Magnitude 0.4 Phase angle, rad −0.6 0.2 −0.8

0 −1.0 0 5 10 0 5 10 Frequency, rad/s Frequency, rad/s Amplitude spectrum Phase spectrum 1.0 0.4

0.2 0.8 0 rad

0.6 e, −0.2 angl Output nitude − se 0.4

0.4Mag Pha −0.6 0.2 −0.8

0 −1.0 0 5 10 0 5 10 Frequency, rad/s Frequency, rad/s

Frame # 16 Slide # 23 A. Antoniou Digital Filters – Secs. 1.6, 1.7 I In this fairly broad definition, several other filtering processes can be identified such as – Differentiation – Integration

Filtering Cont’d

I In the ﬁrst presentation, the ﬁltering process was described as a process that will manipulate the spectrum of a signal in some way.

Frame # 17 Slide # 24 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Filtering Cont’d

I In the first presentation, the filtering process was described as a process that will manipulate the spectrum of a signal in some way. I In this fairly broad definition, several other filtering processes can be identified such as – Differentiation – Integration

Frame # 17 Slide # 25 A. Antoniou Digital Filters – Secs. 1.6, 1.7 We note that the amplitude and phase spectrums of the signal have become 1 {ωi Ai : i = 1, 2,..., 9} and {θi − 2 π : i = 1, 2,..., 9} respectively.

Diﬀerentiation

If we diﬀerentiate the signal 9 X x(t) = Ai sin(ωi t + θi ) i=1 with respect to t, we get 9 9 dx(t) X d X = [A sin(ω t + θ )] = ω A cos(ω t + θ ) dt dt i i i i i i i i=1 i=1 9 X 1 = ωi Ai sin(ωi t + θi − 2 π) i=1

Frame # 18 Slide # 26 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Diﬀerentiation

If we diﬀerentiate the signal 9 X x(t) = Ai sin(ωi t + θi ) i=1 with respect to t, we get 9 9 dx(t) X d X = [A sin(ω t + θ )] = ω A cos(ω t + θ ) dt dt i i i i i i i i=1 i=1 9 X 1 = ωi Ai sin(ωi t + θi − 2 π) i=1 We note that the amplitude and phase spectrums of the signal have become 1 {ωi Ai : i = 1, 2,..., 9} and {θi − 2 π : i = 1, 2,..., 9} respectively.

Frame # 18 Slide # 27 A. Antoniou Digital Filters – Secs. 1.6, 1.7 I an angle of π/2 has been subtracted from the phase spectrum.

Diﬀerentiation Cont’d ···

1 {ωi Ai : i = 1, 2,..., 9} and {θi − 2 π : i = 1, 2,..., 9} In eﬀect, I the amplitude spectrum has been multiplied by the frequency ωi , and

Frame # 19 Slide # 28 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Diﬀerentiation Cont’d ···

1 {ωi Ai : i = 1, 2,..., 9} and {θi − 2 π : i = 1, 2,..., 9} In eﬀect, I the amplitude spectrum has been multiplied by the frequency ωi , and I an angle of π/2 has been subtracted from the phase spectrum.

Frame # 19 Slide # 29 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Diﬀerentiation Cont’d

Amplitude spectrum Phase spectrum 1.0 0.4

0.2 0.8 0

0.6 − Input 0.2 −0.4 Magnitude 0.4 Phase angle, rad −0.6 0.2 −0.8

0 −1.0 0 5 10 0 5 10 Frequency, rad/s Frequency, rad/s Amplitude spectrum Phase spectrum 8 0

7 −0.5 6 rad 5 − e, 1.0 ude nit Output 4 angl se se

Mag − 3 1.5 Pha 2 −2.0 1

0 −2.5 0 5 10 0 5 10 Frequency, rad/s Frequency, rad/s

Frame # 20 Slide # 30 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Diﬀerentiation Cont’d

5 ) t

( 0 Input x

−5 −10 −5 0 5 10 15 Time, s

0 ) t ( Output y −10

−20

−30 −10 −5 0 5 10 15 Time, s

Frame # 21 Slide # 31 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Diﬀerentiation Cont’d

Evidently, diﬀerentiation tends to enhance high-frequency components and weaken low-frequency components somewhat like highpass ﬁltering.

Frame # 22 Slide # 32 A. Antoniou Digital Filters – Secs. 1.6, 1.7 The amplitude and phase spectrums of the signal have become

1 {Ai /ωi : i = 1, 2,..., 9} and {θi − 2 π : i = 1, 2,..., 9} respectively.

Integration

If we integrate the signal

9 X x(t) = Ai sin(ωi t + θi ) i=1 with respect to t, we get

Z 9 Z 9 X X Ai x(t) dt = Ai sin(ωi t + θi ) dt = − cos(ωi t + θi ) ωi i=1 i=1 9 X Ai 1 = sin(ωi t + θi − 2 π) ωi i=1

Frame # 23 Slide # 33 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Integration

If we integrate the signal

9 X x(t) = Ai sin(ωi t + θi ) i=1 with respect to t, we get

Z 9 Z 9 X X Ai x(t) dt = Ai sin(ωi t + θi ) dt = − cos(ωi t + θi ) ωi i=1 i=1 9 X Ai 1 = sin(ωi t + θi − 2 π) ωi i=1 The amplitude and phase spectrums of the signal have become

1 {Ai /ωi : i = 1, 2,..., 9} and {θi − 2 π : i = 1, 2,..., 9} respectively.

Frame # 23 Slide # 34 A. Antoniou Digital Filters – Secs. 1.6, 1.7 I an angle of π/2 has been subtracted from the phase spectrum.

Integration Cont’d ···

1 {Ai /ωi : i = 1, 2,..., 9} and {θi − 2 π : i = 1, 2,..., 9} Evidently, I the amplitude spectrum has been divided by the frequency ωi , and

Frame # 24 Slide # 35 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Integration Cont’d ···

1 {Ai /ωi : i = 1, 2,..., 9} and {θi − 2 π : i = 1, 2,..., 9} Evidently, I the amplitude spectrum has been divided by the frequency ωi , and I an angle of π/2 has been subtracted from the phase spectrum.

Frame # 24 Slide # 36 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Integration Cont’d

Amplitude spectrum Phase spectrum 1.0 0.4

0.2 0.8 0

0.6 − Input 0.2 −0.4 Magnitude 0.4 Phase angle, rad −0.6 0.2 −0.8

0 −1.0 0 5 10 0 5 10 Frequency, rad/s Frequency, rad/s

Amplitude spectrum Phase spectrum 0.7 0

0.6 −0.5 0.5

rad −1.0 0.4 le, ude

Output nit 0.3 e ang Mag −1.5 Phas 0.2 −2.0 0.1

0 −2.5 0 5 10 0 5 10 Frequency, rad/s Frequency, rad/s

Frame # 25 Slide # 37 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Integration Cont’d

5 ) t

( 0 Input x

−5 −10 −5 0 5 10 15 Time, s

5 ) t

( 0 Output y

−5 −10 −5 0 5 10 15 Time, s

Frame # 26 Slide # 38 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Integration Cont’d

Evidently, integration tends to enhance low-frequency components and weaken high-frequency components somewhat like lowpass ﬁltering.

Frame # 27 Slide # 39 A. Antoniou Digital Filters – Secs. 1.6, 1.7 N Electrical ﬁlters can be classiﬁed on the basis of their operating signals as analog or digital.

N In analog ﬁlters the input, output, and internal signals are in the form of continuous-time signals whereas in digital ﬁlters they are in the form of discrete-time signals.

Electrical Filters

N Electrical engineers have known about filtering processes for well over 80 years and through the years they invented a great variety of circuits and systems that can perform filtering, which are known collectively as filters.

Frame # 28 Slide # 40 A. Antoniou Digital Filters – Secs. 1.6, 1.7 N In analog ﬁlters the input, output, and internal signals are in the form of continuous-time signals whereas in digital ﬁlters they are in the form of discrete-time signals.

Electrical Filters

N Electrical ﬁlters can be classiﬁed on the basis of their operating signals as analog or digital.

Frame # 28 Slide # 41 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Electrical Filters

N Electrical ﬁlters can be classiﬁed on the basis of their operating signals as analog or digital.

N In analog ﬁlters the input, output, and internal signals are in the form of continuous-time signals whereas in digital ﬁlters they are in the form of discrete-time signals.

Frame # 28 Slide # 42 A. Antoniou Digital Filters – Secs. 1.6, 1.7 N Various families of analog filters have evolved over the years, which can be classified as follows on the basis of their constituent elements and the technology used: – Passive RLC filters – Discrete active RC filters – Integrated active RC filters – Switched-capacitor filters – Microwave filters

Analog Filters

N Analog ﬁlters were originally invented for use in radio receivers and long-distance telephone systems and continue to be critical components in all types of communication systems.

Frame # 29 Slide # 43 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Analog Filters

N Analog filters were originally invented for use in radio receivers and long-distance telephone systems and continue to be critical components in all types of communication systems. N Various families of analog filters have evolved over the years, which can be classified as follows on the basis of their constituent elements and the technology used: – Passive RLC filters – Discrete active RC filters – Integrated active RC filters – Switched-capacitor filters – Microwave filters

Frame # 29 Slide # 44 A. Antoniou Digital Filters – Secs. 1.6, 1.7 N They are made of interconnected resistors, inductors, and capacitors and are said to be passive in view of the fact that they do not require an energy source, like a power supply, to operate.

N Filtering action is achieved through the property of electrical resonance which occurs when an inductor and a capacitor are connected in series or in parallel.

N The importance of filtering in communications motivated engineers and mathematicians between the thirties and fifties to develop some very powerful and sophisticated methods for the design of passive RLC filters.

Passive RLC Filters

N Passive RLC ﬁlters began to be used extensively in the early twenties.

Frame # 30 Slide # 45 A. Antoniou Digital Filters – Secs. 1.6, 1.7 N Filtering action is achieved through the property of electrical resonance which occurs when an inductor and a capacitor are connected in series or in parallel.

Passive RLC Filters

N Passive RLC ﬁlters began to be used extensively in the early twenties. N They are made of interconnected resistors, inductors, and capacitors and are said to be passive in view of the fact that they do not require an energy source, like a power supply, to operate.

Frame # 30 Slide # 46 A. Antoniou Digital Filters – Secs. 1.6, 1.7 N The importance of filtering in communications motivated engineers and mathematicians between the thirties and fifties to develop some very powerful and sophisticated methods for the design of passive RLC filters.

Passive RLC Filters

N Filtering action is achieved through the property of electrical resonance which occurs when an inductor and a capacitor are connected in series or in parallel.

Frame # 30 Slide # 47 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Passive RLC Filters

N Filtering action is achieved through the property of electrical resonance which occurs when an inductor and a capacitor are connected in series or in parallel.

Frame # 30 Slide # 48 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Passive RLC Filters Cont’d

RS L1 L3 L5

L2 L4 V i Vo RL

C 2 C 4

Passive RLC lowpass ﬁlter

Frame # 31 Slide # 49 A. Antoniou Digital Filters – Secs. 1.6, 1.7 N They are made up of discrete resistors, capacitors, and amplifying electronic circuits.

N Inductors are absent and it is this feature that makes active RC ﬁlters attractive.

Discrete Active RC Filters

N Discrete active RC ﬁlters began to appear during the mid-ﬁfties and were a hot topic of research during the sixties.

Frame # 32 Slide # 50 A. Antoniou Digital Filters – Secs. 1.6, 1.7 N Inductors are absent and it is this feature that makes active RC ﬁlters attractive.

Discrete Active RC Filters

N Discrete active RC ﬁlters began to appear during the mid-ﬁfties and were a hot topic of research during the sixties.

N They are made up of discrete resistors, capacitors, and amplifying electronic circuits.

Frame # 32 Slide # 51 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Discrete Active RC Filters

N Discrete active RC ﬁlters began to appear during the mid-ﬁfties and were a hot topic of research during the sixties.

N They are made up of discrete resistors, capacitors, and amplifying electronic circuits.

N Inductors are absent and it is this feature that makes active RC ﬁlters attractive.

Frame # 32 Slide # 52 A. Antoniou Digital Filters – Secs. 1.6, 1.7 N Unfortunately, without inductors electrical resonance cannot be achieved and with just resistors and capacitors only crude types of ﬁlters can be designed.

N However, through the clever use of amplifying electronic circuits in RC circuits, it is possible to simulate resonance-like eﬀects that can be utilized to achieve ﬁltering of high quality.

N These ﬁlters are said to be active because the amplifying electronic circuits require an energy source in the form of a power supply.

Discrete Active RC Filters Cont’d

N Inductors have always been bulky, expensive, and generally less ideal than resistors and capacitors particularly for low-frequency applications.

Frame # 33 Slide # 53 A. Antoniou Digital Filters – Secs. 1.6, 1.7 N However, through the clever use of amplifying electronic circuits in RC circuits, it is possible to simulate resonance-like eﬀects that can be utilized to achieve ﬁltering of high quality.

N These ﬁlters are said to be active because the amplifying electronic circuits require an energy source in the form of a power supply.

Discrete Active RC Filters Cont’d

N Inductors have always been bulky, expensive, and generally less ideal than resistors and capacitors particularly for low-frequency applications.

N Unfortunately, without inductors electrical resonance cannot be achieved and with just resistors and capacitors only crude types of ﬁlters can be designed.

Frame # 33 Slide # 54 A. Antoniou Digital Filters – Secs. 1.6, 1.7 N These ﬁlters are said to be active because the amplifying electronic circuits require an energy source in the form of a power supply.

Discrete Active RC Filters Cont’d

N Inductors have always been bulky, expensive, and generally less ideal than resistors and capacitors particularly for low-frequency applications.

N Unfortunately, without inductors electrical resonance cannot be achieved and with just resistors and capacitors only crude types of ﬁlters can be designed.

N However, through the clever use of amplifying electronic circuits in RC circuits, it is possible to simulate resonance-like eﬀects that can be utilized to achieve ﬁltering of high quality.

Frame # 33 Slide # 55 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Discrete Active RC Filters Cont’d

N Inductors have always been bulky, expensive, and generally less ideal than resistors and capacitors particularly for low-frequency applications.

N Unfortunately, without inductors electrical resonance cannot be achieved and with just resistors and capacitors only crude types of ﬁlters can be designed.

N However, through the clever use of amplifying electronic circuits in RC circuits, it is possible to simulate resonance-like eﬀects that can be utilized to achieve ﬁltering of high quality.

N These ﬁlters are said to be active because the amplifying electronic circuits require an energy source in the form of a power supply.

Frame # 33 Slide # 56 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Discrete Active RC Filters Cont’d

R2 R1 - C 2 + Vi Vo

Discrete active bandpass ﬁlter

Frame # 34 Slide # 57 A. Antoniou Digital Filters – Secs. 1.6, 1.7 N Through the use of high-frequency amplifying circuits and suitable integrated-circuit elements, ﬁlters can be designed that can operate at frequencies as high as 15 GHz.

N Interest in these ﬁlters has been strong during the eighties and nineties and research is continuing.

Integrated-Circuit Active RC Filters

N Integrated-circuit active RC ﬁlters operate on the basis of the same principles as their discrete counterparts except that they are designed directly as complete integrated circuits.

Frame # 35 Slide # 58 A. Antoniou Digital Filters – Secs. 1.6, 1.7 N Interest in these ﬁlters has been strong during the eighties and nineties and research is continuing.

Integrated-Circuit Active RC Filters

N Integrated-circuit active RC ﬁlters operate on the basis of the same principles as their discrete counterparts except that they are designed directly as complete integrated circuits.

N Through the use of high-frequency amplifying circuits and suitable integrated-circuit elements, ﬁlters can be designed that can operate at frequencies as high as 15 GHz.

Frame # 35 Slide # 59 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Integrated-Circuit Active RC Filters

N Integrated-circuit active RC ﬁlters operate on the basis of the same principles as their discrete counterparts except that they are designed directly as complete integrated circuits.

N Through the use of high-frequency amplifying circuits and suitable integrated-circuit elements, ﬁlters can be designed that can operate at frequencies as high as 15 GHz.

N Interest in these ﬁlters has been strong during the eighties and nineties and research is continuing.

Frame # 35 Slide # 60 A. Antoniou Digital Filters – Secs. 1.6, 1.7 N These are essentially active RC ﬁlters except that switches are also utilized along with amplifying devices.

N In this family of ﬁlters, switches are used to simulate high resistance values which are diﬃcult to implement in integrated-circuit form.

N Like integrated active RC ﬁlters, switched-capacitor ﬁlters are compatible with integrated-circuit technology.

Switched-Capacitor Filters

N Switched-capacitor ﬁlters evolved during the seventies and eighties.

Frame # 36 Slide # 61 A. Antoniou Digital Filters – Secs. 1.6, 1.7 N In this family of ﬁlters, switches are used to simulate high resistance values which are diﬃcult to implement in integrated-circuit form.

N Like integrated active RC ﬁlters, switched-capacitor ﬁlters are compatible with integrated-circuit technology.

Switched-Capacitor Filters

N Switched-capacitor ﬁlters evolved during the seventies and eighties. N These are essentially active RC ﬁlters except that switches are also utilized along with amplifying devices.

Frame # 36 Slide # 62 A. Antoniou Digital Filters – Secs. 1.6, 1.7 N Like integrated active RC ﬁlters, switched-capacitor ﬁlters are compatible with integrated-circuit technology.

Switched-Capacitor Filters

N Switched-capacitor ﬁlters evolved during the seventies and eighties. N These are essentially active RC ﬁlters except that switches are also utilized along with amplifying devices.

N In this family of ﬁlters, switches are used to simulate high resistance values which are diﬃcult to implement in integrated-circuit form.

Frame # 36 Slide # 63 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Switched-Capacitor Filters

N Switched-capacitor ﬁlters evolved during the seventies and eighties. N These are essentially active RC ﬁlters except that switches are also utilized along with amplifying devices.

N In this family of ﬁlters, switches are used to simulate high resistance values which are diﬃcult to implement in integrated-circuit form.

N Like integrated active RC ﬁlters, switched-capacitor ﬁlters are compatible with integrated-circuit technology.

Frame # 36 Slide # 64 A. Antoniou Digital Filters – Secs. 1.6, 1.7 N Microwave ﬁlters are built from a variety of microwave components and devices such as waveguides, dielectric resonators, and surface acoustic devices.

Microwave Filters

N At microwave frequencies in the range 0.5 to 500 GHz, inductors and transistors do not work very well and, therefore, passive RLC or active RC ﬁlters have poor performance; hence, microwave ﬁlters are used.

Frame # 37 Slide # 65 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Microwave Filters

N Microwave ﬁlters are built from a variety of microwave components and devices such as waveguides, dielectric resonators, and surface acoustic devices.

Frame # 37 Slide # 66 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Microwave Filters Cont’d

Microwave bandpass ﬁlter

Frame # 38 Slide # 67 A. Antoniou Digital Filters – Secs. 1.6, 1.7 N Communication and radar systems N Telephone systems N Sampling systems N Audio equipment

Applications of Analog Filters

N Radios and TVs

Frame # 39 Slide # 68 A. Antoniou Digital Filters – Secs. 1.6, 1.7 N Telephone systems N Sampling systems N Audio equipment

Applications of Analog Filters

N Radios and TVs N Communication and radar systems

Frame # 39 Slide # 69 A. Antoniou Digital Filters – Secs. 1.6, 1.7 N Sampling systems N Audio equipment

Applications of Analog Filters

N Radios and TVs N Communication and radar systems N Telephone systems

Frame # 39 Slide # 70 A. Antoniou Digital Filters – Secs. 1.6, 1.7 N Audio equipment

Applications of Analog Filters

N Radios and TVs N Communication and radar systems N Telephone systems N Sampling systems

Frame # 39 Slide # 71 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Applications of Analog Filters

N Radios and TVs N Communication and radar systems N Telephone systems N Sampling systems N Audio equipment

Frame # 39 Slide # 72 A. Antoniou Digital Filters – Secs. 1.6, 1.7 N The same principle can be used to prevent radar signals from interfering with communications channels and vice-versa at an airport.

Applications of Analog Filters Cont’d

N When we select our favorite radio station or TV channel, we are actually tuning a bandpass ﬁlter inside the radio or TV to the frequencies of the radio or TV station. The signal from our favorite radio station is the desirable signal and the signals from all the other stations are undesirable.

Frame # 40 Slide # 73 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Applications of Analog Filters Cont’d

N The same principle can be used to prevent radar signals from interfering with communications channels and vice-versa at an airport.

Frame # 40 Slide # 74 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Applications of Analog Filters Cont’d

N Signals are often corrupted by spurious signals known collectively as noise. Such signals may originate from a large number of sources, e.g., lightnings, electrical motors, transformers, and power lines. Noise signals are characterized by frequency spectrums that stretch over a wide range of frequencies. They can be eliminated through the use of bandpass ﬁlters that would pass the desired signal but reject everything else, namely, the noise content.

Frame # 41 Slide # 75 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Applications of Analog Filters Cont’d

N We all talk to our friends and relatives, who may live in another city or another country, almost daily through the telephone system. The telephone signals are transmitted through expensive communications channels. If these channels were to carry just a single voice, as in the days of Alexander Graham Bell, no one would ever be able to aﬀord a telephone call to anyone, even the very rich.

Frame # 42 Slide # 76 A. Antoniou Digital Filters – Secs. 1.6, 1.7 N This is achieved through the use of a so-called frequency-division multiplex (FDM) communications system.

N An FDM communication system requires a multitude of ﬁlters to operate properly.

Frequency-Division Multiplex System

N What makes long-distance calls aﬀordable is our ability to transmit thousands upon thousands of conversations through one and the same communications channel.

Frame # 43 Slide # 77 A. Antoniou Digital Filters – Secs. 1.6, 1.7 N An FDM communication system requires a multitude of ﬁlters to operate properly.

Frequency-Division Multiplex System

N What makes long-distance calls aﬀordable is our ability to transmit thousands upon thousands of conversations through one and the same communications channel.

N This is achieved through the use of a so-called frequency-division multiplex (FDM) communications system.

Frame # 43 Slide # 78 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Frequency-Division Multiplex System

N What makes long-distance calls aﬀordable is our ability to transmit thousands upon thousands of conversations through one and the same communications channel.

N This is achieved through the use of a so-called frequency-division multiplex (FDM) communications system.

N An FDM communication system requires a multitude of ﬁlters to operate properly.

Frame # 43 Slide # 79 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Frequency-Division Multiplex System Cont’d

The operation of a typical FDM communications system is as follows: 1. At the transmit end, the different voice signals are superimposed on different carrier frequencies using a process known as modulation. 2. The different carrier frequencies are combined by using an adder circuit. 3. At the receive end, carrier frequencies are separated using bandpass filters. 4. The voice signals are then extracted from the carrier frequencies through demodulation. 5. The voice signals are distributed to the appropriate persons through the local telephone lines.

Frame # 44 Slide # 80 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Frequency-Division Multiplex System Cont’d

Transmitter

Modulator 1

ω1

Modulator g(t) 2 Bandpass ω2 filters Demodulator ω Modulator 1 1 m

ωm Demodulator ω 2 2 g(t)

Demodulator ωm m

Receiver (a) Basic FDM system

G(ω)

ω ω1 ω2 ωm (b) Voice signals arranged into a group

Frame # 45 Slide # 81 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Frequency-Division Multiplex System Cont’d

N The transmit section adds the frequency of a unique carrier to the frequencies of each voice signal, thereby, shifting its frequency spectrum by the frequency of the carrier. In this way, the frequency spectrums of the diﬀerent voice signals are arranged one after the other to form the composite signal g(t) shown in ﬁgure (a) of the previous slide, which is referred to as a group by telephone engineers. The amplitude spectrum of g(t), designated as G(ω), is illustrated in Fig. (b).

Frame # 46 Slide # 82 A. Antoniou Digital Filters – Secs. 1.6, 1.7 N The FDM system requires as many bandpass ﬁlters as there are voice signals, and this is why thousands upon thousands of bandpass ﬁlters are required.

N The FDM system also uses a large number of modulators and demodulators and these devices, as it turns out, also need ﬁlters to operate properly.

N In short, communications systems are simply not feasible without ﬁlters.

Frequency-Division Multiplex System Cont’d

N The receive section separates the translated voice signals and restores their original spectrums.

Frame # 47 Slide # 83 A. Antoniou Digital Filters – Secs. 1.6, 1.7 N The FDM system also uses a large number of modulators and demodulators and these devices, as it turns out, also need ﬁlters to operate properly.

N In short, communications systems are simply not feasible without ﬁlters.

Frequency-Division Multiplex System Cont’d

N The receive section separates the translated voice signals and restores their original spectrums.

N The FDM system requires as many bandpass ﬁlters as there are voice signals, and this is why thousands upon thousands of bandpass ﬁlters are required.

Frame # 47 Slide # 84 A. Antoniou Digital Filters – Secs. 1.6, 1.7 N In short, communications systems are simply not feasible without ﬁlters.

Frequency-Division Multiplex System Cont’d

N The receive section separates the translated voice signals and restores their original spectrums.

N The FDM system requires as many bandpass ﬁlters as there are voice signals, and this is why thousands upon thousands of bandpass ﬁlters are required.

N The FDM system also uses a large number of modulators and demodulators and these devices, as it turns out, also need ﬁlters to operate properly.

Frame # 47 Slide # 85 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Frequency-Division Multiplex System Cont’d

N The receive section separates the translated voice signals and restores their original spectrums.

N The FDM system requires as many bandpass ﬁlters as there are voice signals, and this is why thousands upon thousands of bandpass ﬁlters are required.

N The FDM system also uses a large number of modulators and demodulators and these devices, as it turns out, also need ﬁlters to operate properly.

N In short, communications systems are simply not feasible without ﬁlters.

Frame # 47 Slide # 86 A. Antoniou Digital Filters – Secs. 1.6, 1.7 N At the receiving end, a supergroup is subdivided into the individual groups by a bank of bandpass ﬁlters. The groups are, in turn, subdivided into the individual voice signals by appropriate banks of bandpass ﬁlters.

N Similarly, several supergroups can be combined into a mastergroup, and so on, until the bandwidth capacity of the cable or microwave link is completely ﬁlled.

Frequency-Division Multiplex System Cont’d

N A more complex FDM system can be constructed by modulating several groups individually as if they were voice signals and then adding them up to form a supergroup to increase the number of voice signals transmitted over an intercity cable or microwave link.

Frame # 48 Slide # 87 A. Antoniou Digital Filters – Secs. 1.6, 1.7 N Similarly, several supergroups can be combined into a mastergroup, and so on, until the bandwidth capacity of the cable or microwave link is completely ﬁlled.

Frequency-Division Multiplex System Cont’d

N At the receiving end, a supergroup is subdivided into the individual groups by a bank of bandpass ﬁlters. The groups are, in turn, subdivided into the individual voice signals by appropriate banks of bandpass ﬁlters.

Frame # 48 Slide # 88 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Frequency-Division Multiplex System Cont’d

N Similarly, several supergroups can be combined into a mastergroup, and so on, until the bandwidth capacity of the cable or microwave link is completely ﬁlled.

Frame # 48 Slide # 89 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Frequency-Division Multiplex System Cont’d

Transmitter 11 12 Modulator 1 1m

ω1′

21 22 Modulator 2 m 2 Bandpass ω ′2 filters 11 k Demodulator 1 ω 12 k2 Modulator 1′ 1 k 1m km

ωk′ 21 Demodulator 22 ω 2′ 2 2m

k1 Demodulator k2 ωk ′ k km Receiver FDM system with two levels of modulation.

Frame # 49 Slide # 90 A. Antoniou Digital Filters – Secs. 1.6, 1.7 N In situations where the sampling frequency is ﬁxed and the highest frequency present in the signal can exceed half the sampling frequency, it is crucial to bandlimit the signal to be sampled to prevent a certain type of signal distortion known as aliasing.

N This bandlimiting process, must be carried out through the use of a lowpass analog ﬁlter.

N A sampling system also requires an analog lowpass ﬁlter at the output of the D/A converter to serve as smoothing device.

Use of Analog Filters in Sampling Systems

N In a sampling system, the sampling frequency must be at least twice the highest frequency present in the spectrum of the signal. This is known as the sampling theorem.

Frame # 50 Slide # 91 A. Antoniou Digital Filters – Secs. 1.6, 1.7 N This bandlimiting process, must be carried out through the use of a lowpass analog ﬁlter.

N A sampling system also requires an analog lowpass ﬁlter at the output of the D/A converter to serve as smoothing device.

Use of Analog Filters in Sampling Systems

N In a sampling system, the sampling frequency must be at least twice the highest frequency present in the spectrum of the signal. This is known as the sampling theorem.

N In situations where the sampling frequency is ﬁxed and the highest frequency present in the signal can exceed half the sampling frequency, it is crucial to bandlimit the signal to be sampled to prevent a certain type of signal distortion known as aliasing.

Frame # 50 Slide # 92 A. Antoniou Digital Filters – Secs. 1.6, 1.7 N A sampling system also requires an analog lowpass ﬁlter at the output of the D/A converter to serve as smoothing device.

Use of Analog Filters in Sampling Systems

N In a sampling system, the sampling frequency must be at least twice the highest frequency present in the spectrum of the signal. This is known as the sampling theorem.

N This bandlimiting process, must be carried out through the use of a lowpass analog ﬁlter.

Frame # 50 Slide # 93 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Use of Analog Filters in Sampling Systems

N In a sampling system, the sampling frequency must be at least twice the highest frequency present in the spectrum of the signal. This is known as the sampling theorem.

N This bandlimiting process, must be carried out through the use of a lowpass analog ﬁlter.

N A sampling system also requires an analog lowpass ﬁlter at the output of the D/A converter to serve as smoothing device.

Frame # 50 Slide # 94 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Use of Analog Filters in Sampling Systems Cont’d

x(t) xˆ(t) x(nT) y(nT) ˆy(t) y(t)

FLP A/D DF D/A FLP 1 2 3 4 5 6 7

c(t)

y(nT) y'(t)

t nT (a) (b)

Frame # 51 Slide # 95 A. Antoniou Digital Filters – Secs. 1.6, 1.7 N This is due to the fact that the enclosure or cabinet used can often exhibit mechanical resonances that are superimposed on the audio signal.

N This is one of the reasons why diﬀerent makes of loudspeaker systems often produce their own characteristic sound.

Use of Analog Filters as Equalizers

N Loudspeaker systems behave very much like ﬁlters and, consequently, they tend to change the spectrum of an audio signal.

Frame # 52 Slide # 96 A. Antoniou Digital Filters – Secs. 1.6, 1.7 N This is one of the reasons why diﬀerent makes of loudspeaker systems often produce their own characteristic sound.

Use of Analog Filters as Equalizers

N Loudspeaker systems behave very much like ﬁlters and, consequently, they tend to change the spectrum of an audio signal.

N This is due to the fact that the enclosure or cabinet used can often exhibit mechanical resonances that are superimposed on the audio signal.

Frame # 52 Slide # 97 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Use of Analog Filters as Equalizers

N Loudspeaker systems behave very much like ﬁlters and, consequently, they tend to change the spectrum of an audio signal.

N This is due to the fact that the enclosure or cabinet used can often exhibit mechanical resonances that are superimposed on the audio signal.

N This is one of the reasons why diﬀerent makes of loudspeaker systems often produce their own characteristic sound.

Frame # 52 Slide # 98 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Use of Analog Filters as Equalizers Cont’d

N To correct for mechanical resonances and other imperfections, sound reproduction equipment, such as stereos, is often equipped with equalizers that can be used to reshape the spectrum of the audio signal. These subsystems typically incorporate a number of sliders that can be adjusted to modify the quality of the sound reproduced. One can, for example, strengthen or weaken the low-frequency (bass) or high-frequency (treble) content of the audio signal.

Frame # 53 Slide # 99 A. Antoniou Digital Filters – Secs. 1.6, 1.7 N Through the use of an equalizer, one could adjust the spectrum of the audio signal to one’s preference.

N A thick carpet can actually absorb a lot of the high-frequency content of the audio signal, i.e., the room would behave very much like a lowpass ﬁlter. In such a situation, one might need to boost the treble a bit to restore some of the lost high-frequency content of the music.

Use of Analog Filters as Equalizers Cont’d

N Since an equalizer is a device that can modify the spectrum of a signal, equalizers are filters in terms of the broader definition adopted in the textbook. What the sliders do is to alter the parameters of the filter that performs the equalization.

Frame # 54 Slide # 100 A. Antoniou Digital Filters – Secs. 1.6, 1.7 N A thick carpet can actually absorb a lot of the high-frequency content of the audio signal, i.e., the room would behave very much like a lowpass ﬁlter. In such a situation, one might need to boost the treble a bit to restore some of the lost high-frequency content of the music.

Use of Analog Filters as Equalizers Cont’d

N Through the use of an equalizer, one could adjust the spectrum of the audio signal to one’s preference.

Frame # 54 Slide # 101 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Use of Analog Filters as Equalizers Cont’d

N Through the use of an equalizer, one could adjust the spectrum of the audio signal to one’s preference.

Frame # 54 Slide # 102 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Use of Analog Filters as Equalizers Cont’d

N Transmission cables, telephone lines, and communication channels often behave very much like ﬁlters and, as a result, they tend to reshape the spectrums of the signals transmitted through them. The local telephone lines are particularly notorious in this respect – we often do not even recognize the voice of the person at the other end only because the spectrum of the signal has been changed by the telephone line.

Frame # 55 Slide # 103 A. Antoniou Digital Filters – Secs. 1.6, 1.7 N These equalizers are incorporated in the modems at either end of a telephone line. So-called ADSL Internet service available though telephone companies is achieved by means of adaptive ﬁlters.

Use of Analog Filters as Equalizers Cont’d

N Like the performance of loudspeaker systems, that of telephone lines and communication channels can be improved by using suitable equalizers. In fact, it is through the use of sophisticated equalizers, in the form of adaptive ﬁlters, that it is possible to achieve high data transmission rates through local telephone lines.

Frame # 56 Slide # 104 A. Antoniou Digital Filters – Secs. 1.6, 1.7 Use of Analog Filters as Equalizers Cont’d

N These equalizers are incorporated in the modems at either end of a telephone line. So-called ADSL Internet service available though telephone companies is achieved by means of adaptive ﬁlters.

Frame # 56 Slide # 105 A. Antoniou Digital Filters – Secs. 1.6, 1.7 This slide concludes the presentation. Thank you for your attention.

Frame # 57 Slide # 106 A. Antoniou Digital Filters – Secs. 1.6, 1.7