Chapter 5 THE APPLICATION OF THE Z TRANSFORM 5.5.4 Aliasing Copyright c 2005 Andreas Antoniou Victoria, BC, Canada Email: [email protected] July 14, 2018 Frame # 1 Slide # 1 A. Antoniou Digital Signal Processing { Sec. 5.5.4 Aliasing is objectionable in practice and it must be prevented from occurring. This presentation explores the nature of aliasing in DSP. Aliasing can occur in other types of systems where sampled signals are involved, for example, in videos and movies, as will be demonstrated. Introduction When a continuous-time signal that contains frequencies outside the baseband −!s =2 < ! < !s =2 is sampled, a phenomenon known as aliasing will arise whereby frequencies outside the baseband impersonate frequencies within the baseband. Frame # 2 Slide # 2 A. Antoniou Digital Signal Processing { Sec. 5.5.4 This presentation explores the nature of aliasing in DSP. Aliasing can occur in other types of systems where sampled signals are involved, for example, in videos and movies, as will be demonstrated. Introduction When a continuous-time signal that contains frequencies outside the baseband −!s =2 < ! < !s =2 is sampled, a phenomenon known as aliasing will arise whereby frequencies outside the baseband impersonate frequencies within the baseband. Aliasing is objectionable in practice and it must be prevented from occurring. Frame # 2 Slide # 3 A. Antoniou Digital Signal Processing { Sec. 5.5.4 Aliasing can occur in other types of systems where sampled signals are involved, for example, in videos and movies, as will be demonstrated. Introduction When a continuous-time signal that contains frequencies outside the baseband −!s =2 < ! < !s =2 is sampled, a phenomenon known as aliasing will arise whereby frequencies outside the baseband impersonate frequencies within the baseband. Aliasing is objectionable in practice and it must be prevented from occurring. This presentation explores the nature of aliasing in DSP. Frame # 2 Slide # 4 A. Antoniou Digital Signal Processing { Sec. 5.5.4 Introduction When a continuous-time signal that contains frequencies outside the baseband −!s =2 < ! < !s =2 is sampled, a phenomenon known as aliasing will arise whereby frequencies outside the baseband impersonate frequencies within the baseband. Aliasing is objectionable in practice and it must be prevented from occurring. This presentation explores the nature of aliasing in DSP. Aliasing can occur in other types of systems where sampled signals are involved, for example, in videos and movies, as will be demonstrated. Frame # 2 Slide # 5 A. Antoniou Digital Signal Processing { Sec. 5.5.4 By sampling the signal using a sampling frequency of !s rad/s, a discrete-time sinusoidal signal with frequency ! + !s rad/s, i.e., x(nT ) = sin[(! + !s )nT ] will be generated. Aliasing in DSP Consider a continuous-time sinusoidal signal with frequency ! + !s rad/s, i.e., x(t) = sin[(! + !s )t] where 0 < ! < !s =2. Frame # 3 Slide # 6 A. Antoniou Digital Signal Processing { Sec. 5.5.4 Aliasing in DSP Consider a continuous-time sinusoidal signal with frequency ! + !s rad/s, i.e., x(t) = sin[(! + !s )t] where 0 < ! < !s =2. By sampling the signal using a sampling frequency of !s rad/s, a discrete-time sinusoidal signal with frequency ! + !s rad/s, i.e., x(nT ) = sin[(! + !s )nT ] will be generated. Frame # 3 Slide # 7 A. Antoniou Digital Signal Processing { Sec. 5.5.4 Evidently, sampling a sinusoidal signal of frequency ! + !s will produce a discrete-time signal which is numerically identical to the discrete-time signal obtained by sampling a sinusoidal signal of frequency !. In effect, the sampled version of signal sin[(! + !s )t] will be aliased to the sampled version of sin(!)t or frequency ! + !s will be aliased to !. Aliasing in DSP Cont'd ··· x(nT ) = sin[(! + !s )nT ] Signal x(nT ) can be expressed as x(nT ) = sin !nT cos !s nT + cos !nT sin !s nT 2π 2π = sin !nT cos · nT + cos !nT sin · nT T T = sin !nT cos 2nπ + cos !nT sin 2nπ = sin !nT Frame # 4 Slide # 8 A. Antoniou Digital Signal Processing { Sec. 5.5.4 In effect, the sampled version of signal sin[(! + !s )t] will be aliased to the sampled version of sin(!)t or frequency ! + !s will be aliased to !. Aliasing in DSP Cont'd ··· x(nT ) = sin[(! + !s )nT ] Signal x(nT ) can be expressed as x(nT ) = sin !nT cos !s nT + cos !nT sin !s nT 2π 2π = sin !nT cos · nT + cos !nT sin · nT T T = sin !nT cos 2nπ + cos !nT sin 2nπ = sin !nT Evidently, sampling a sinusoidal signal of frequency ! + !s will produce a discrete-time signal which is numerically identical to the discrete-time signal obtained by sampling a sinusoidal signal of frequency !. Frame # 4 Slide # 9 A. Antoniou Digital Signal Processing { Sec. 5.5.4 Aliasing in DSP Cont'd ··· x(nT ) = sin[(! + !s )nT ] Signal x(nT ) can be expressed as x(nT ) = sin !nT cos !s nT + cos !nT sin !s nT 2π 2π = sin !nT cos · nT + cos !nT sin · nT T T = sin !nT cos 2nπ + cos !nT sin 2nπ = sin !nT Evidently, sampling a sinusoidal signal of frequency ! + !s will produce a discrete-time signal which is numerically identical to the discrete-time signal obtained by sampling a sinusoidal signal of frequency !. In effect, the sampled version of signal sin[(! + !s )t] will be aliased to the sampled version of sin(!)t or frequency ! + !s will be aliased to !. Frame # 4 Slide # 10 A. Antoniou Digital Signal Processing { Sec. 5.5.4 Aliasing in DSP Cont'd sin ωnT 3T 4T 5T T 2T nT sin[(ω+ωs)nT] Frame # 5 Slide # 11 A. Antoniou Digital Signal Processing { Sec. 5.5.4 As before, we can write x(nT ) = sin(!s − !)nT = sin !s nT cos !nT − cos !s nT sin !nT 2π 2π = sin · nT cos !nT − cos · nT sin !nT T T = sin 2nπ cos !nT − cos 2nπ sin !nT = − sin !nT = sin(−!nT ) In this case, frequency !s − ! will be aliased to frequency −!. Aliasing in DSP Cont'd Now consider a sinusoidal signal x(nT ) = sin[(!s − !)nT ] where 0 < ! < !s =2. Frame # 6 Slide # 12 A. Antoniou Digital Signal Processing { Sec. 5.5.4 In this case, frequency !s − ! will be aliased to frequency −!. Aliasing in DSP Cont'd Now consider a sinusoidal signal x(nT ) = sin[(!s − !)nT ] where 0 < ! < !s =2. As before, we can write x(nT ) = sin(!s − !)nT = sin !s nT cos !nT − cos !s nT sin !nT 2π 2π = sin · nT cos !nT − cos · nT sin !nT T T = sin 2nπ cos !nT − cos 2nπ sin !nT = − sin !nT = sin(−!nT ) Frame # 6 Slide # 13 A. Antoniou Digital Signal Processing { Sec. 5.5.4 Aliasing in DSP Cont'd Now consider a sinusoidal signal x(nT ) = sin[(!s − !)nT ] where 0 < ! < !s =2. As before, we can write x(nT ) = sin(!s − !)nT = sin !s nT cos !nT − cos !s nT sin !nT 2π 2π = sin · nT cos !nT − cos · nT sin !nT T T = sin 2nπ cos !nT − cos 2nπ sin !nT = − sin !nT = sin(−!nT ) In this case, frequency !s − ! will be aliased to frequency −!. Frame # 6 Slide # 14 A. Antoniou Digital Signal Processing { Sec. 5.5.4 If a signal has several frequency components outside the baseband, then all the frequencies beyond the Nyquist frequency will be aliased and a rather serious type of signal distortion will occur, which is known as aliasing distortion. Aliasing can be prevented by filtering out all frequency components outside the baseband using a lowpass filter. Filtering out the high-frequency content of the signal with a lowpass filter will, of course, distort the signal but the distortion introduced is less serious than aliasing distortion (see Chap. 6). Aliasing in DSP Cont'd Aliasing has occurred in the sinusoidal signals considered because the frequencies of the signals were outside the baseband −!s =2 < ! < !s =2. Frame # 7 Slide # 15 A. Antoniou Digital Signal Processing { Sec. 5.5.4 Aliasing can be prevented by filtering out all frequency components outside the baseband using a lowpass filter. Filtering out the high-frequency content of the signal with a lowpass filter will, of course, distort the signal but the distortion introduced is less serious than aliasing distortion (see Chap. 6). Aliasing in DSP Cont'd Aliasing has occurred in the sinusoidal signals considered because the frequencies of the signals were outside the baseband −!s =2 < ! < !s =2. If a signal has several frequency components outside the baseband, then all the frequencies beyond the Nyquist frequency will be aliased and a rather serious type of signal distortion will occur, which is known as aliasing distortion. Frame # 7 Slide # 16 A. Antoniou Digital Signal Processing { Sec. 5.5.4 Filtering out the high-frequency content of the signal with a lowpass filter will, of course, distort the signal but the distortion introduced is less serious than aliasing distortion (see Chap. 6). Aliasing in DSP Cont'd Aliasing has occurred in the sinusoidal signals considered because the frequencies of the signals were outside the baseband −!s =2 < ! < !s =2. If a signal has several frequency components outside the baseband, then all the frequencies beyond the Nyquist frequency will be aliased and a rather serious type of signal distortion will occur, which is known as aliasing distortion. Aliasing can be prevented by filtering out all frequency components outside the baseband using a lowpass filter. Frame # 7 Slide # 17 A. Antoniou Digital Signal Processing { Sec. 5.5.4 Aliasing in DSP Cont'd Aliasing has occurred in the sinusoidal signals considered because the frequencies of the signals were outside the baseband −!s =2 < ! < !s =2.