Chapter 1 INTRODUCTION TO DIGITAL SIGNAL PROCESSING 1.6 Analog Filters 1.7 Applications of Analog Filters Copyright c 2018 Andreas Antoniou Victoria, BC, Canada Email: [email protected] 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 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. 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(!i 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(!i 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 !i 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 −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.