Chapter 5 the APPLICATION of the Z TRANSFORM 5.5.4 Aliasing

Chapter 5 THE APPLICATION OF THE Z TRANSFORM 5.5.4 Aliasing

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 −ωs /2 < ω < ωs /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

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

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

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 ωs rad/s, a discrete-time sinusoidal signal with frequency ω + ωs rad/s, i.e., x(nT ) = sin[(ω + ωs )nT ] will be generated.

Aliasing in DSP

Consider a continuous-time sinusoidal signal with frequency ω + ωs rad/s, i.e.,

x(t) = sin[(ω + ωs )t]

where 0 < ω < ωs /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 ω + ωs rad/s, i.e.,

x(t) = sin[(ω + ωs )t]

where 0 < ω < ωs /2.

By sampling the signal using a sampling frequency of ωs rad/s, a discrete-time sinusoidal signal with frequency ω + ωs rad/s, i.e., x(nT ) = sin[(ω + ωs )nT ] will be generated.

Frame # 3 Slide # 7 A. Antoniou Digital Signal Processing – Sec. 5.5.4 Evidently, sampling a sinusoidal signal of frequency ω + ωs will produce a discrete-time signal which is numerically identical to the discrete-time signal obtained by sampling a sinusoidal signal of frequency ω.

In eﬀect, the sampled version of signal sin[(ω + ωs )t] will be aliased to the sampled version of sin(ω)t or frequency ω + ωs will be aliased to ω.

Aliasing in DSP Cont’d

··· x(nT ) = sin[(ω + ωs )nT ]

Signal x(nT ) can be expressed as

x(nT ) = sin ωnT cos ωs nT + cos ωnT sin ωs nT 2π 2π = sin ωnT cos · nT + cos ωnT sin · nT T T = sin ωnT cos 2nπ + cos ωnT sin 2nπ = sin ωnT

Frame # 4 Slide # 8 A. Antoniou Digital Signal Processing – Sec. 5.5.4 In eﬀect, the sampled version of signal sin[(ω + ωs )t] will be aliased to the sampled version of sin(ω)t or frequency ω + ωs will be aliased to ω.

Aliasing in DSP Cont’d

··· x(nT ) = sin[(ω + ωs )nT ]

Signal x(nT ) can be expressed as

x(nT ) = sin ωnT cos ωs nT + cos ωnT sin ωs nT 2π 2π = sin ωnT cos · nT + cos ωnT sin · nT T T = sin ωnT cos 2nπ + cos ωnT sin 2nπ = sin ωnT

Evidently, sampling a sinusoidal signal of frequency ω + ωs will produce a discrete-time signal which is numerically identical to the discrete-time signal obtained by sampling a sinusoidal signal of frequency ω.

Frame # 4 Slide # 9 A. Antoniou Digital Signal Processing – Sec. 5.5.4 Aliasing in DSP Cont’d

··· x(nT ) = sin[(ω + ωs )nT ]

Signal x(nT ) can be expressed as

x(nT ) = sin ωnT cos ωs nT + cos ωnT sin ωs nT 2π 2π = sin ωnT cos · nT + cos ωnT sin · nT T T = sin ωnT cos 2nπ + cos ωnT sin 2nπ = sin ωnT

In eﬀect, the sampled version of signal sin[(ω + ωs )t] will be aliased to the sampled version of sin(ω)t or frequency ω + ωs will be aliased to ω.

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(ωs − ω)nT

= sin ωs nT cos ωnT − cos ωs nT sin ωnT 2π 2π = sin · nT cos ωnT − cos · nT sin ωnT T T = sin 2nπ cos ωnT − cos 2nπ sin ωnT = − sin ωnT = sin(−ωnT )

In this case, frequency ωs − ω will be aliased to frequency −ω.

Aliasing in DSP Cont’d

Now consider a sinusoidal signal

x(nT ) = sin[(ωs − ω)nT ]

where 0 < ω < ωs /2.

Frame # 6 Slide # 12 A. Antoniou Digital Signal Processing – Sec. 5.5.4 In this case, frequency ωs − ω will be aliased to frequency −ω.

Aliasing in DSP Cont’d

Now consider a sinusoidal signal

x(nT ) = sin[(ωs − ω)nT ]

where 0 < ω < ωs /2.

As before, we can write

x(nT ) = sin(ωs − ω)nT

= sin ωs nT cos ωnT − cos ωs nT sin ωnT 2π 2π = sin · nT cos ωnT − cos · nT sin ωnT T T = sin 2nπ cos ωnT − cos 2nπ sin ωnT = − sin ωnT = sin(−ωnT )

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[(ωs − ω)nT ]

where 0 < ω < ωs /2.

As before, we can write

x(nT ) = sin(ωs − ω)nT

= sin ωs nT cos ωnT − cos ωs nT sin ωnT 2π 2π = sin · nT cos ωnT − cos · nT sin ωnT T T = sin 2nπ cos ωnT − cos 2nπ sin ωnT = − sin ωnT = sin(−ωnT )

In this case, frequency ωs − ω will be aliased to frequency −ω.

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 −ωs /2 < ω < ωs /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 −ωs /2 < ω < ωs /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 −ωs /2 < ω < ωs /2.

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 −ωs /2 < ω < ωs /2.

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).

Frame # 7 Slide # 18 A. Antoniou Digital Signal Processing – Sec. 5.5.4 A movie is a sequence of still photographs which constitutes a sampled signal. If the sampling rate (frames/second) is not fast enough to deal with the images projected, funny things can happen.

For example, as the cowboy wagon accelerates into the sunset, the wheels of the wagon appear to accelerate in the forward direction, then reverse, slow down, stop momentarily, and after that they accelerate again in the forward direction. This is what should be seen, it’s not an illusion!

Aliasing at the Movies

Aliasing can occur in other types of systems that rely on sampling.

Frame # 8 Slide # 19 A. Antoniou Digital Signal Processing – Sec. 5.5.4 For example, as the cowboy wagon accelerates into the sunset, the wheels of the wagon appear to accelerate in the forward direction, then reverse, slow down, stop momentarily, and after that they accelerate again in the forward direction. This is what should be seen, it’s not an illusion!

Aliasing at the Movies

Aliasing can occur in other types of systems that rely on sampling.

A movie is a sequence of still photographs which constitutes a sampled signal. If the sampling rate (frames/second) is not fast enough to deal with the images projected, funny things can happen.

Frame # 8 Slide # 20 A. Antoniou Digital Signal Processing – Sec. 5.5.4 Aliasing at the Movies

Aliasing can occur in other types of systems that rely on sampling.

A movie is a sequence of still photographs which constitutes a sampled signal. If the sampling rate (frames/second) is not fast enough to deal with the images projected, funny things can happen.

Frame # 8 Slide # 21 A. Antoniou Digital Signal Processing – Sec. 5.5.4 The number of ﬁlm frames per second, N, times the number of wheel revolutions per second is analogous to the sampling frequency ωs , i.e.,

ωs = Nω or ω = ωs /N

Rotation in the clockwise or counterclockwise direction is analogous to positive or negative frequency.

Aliasing at the Movies Cont’d

The slides that follow show a wagon wheel with a black dot.

The number of wheel revolutions per second is analogous to frequency, say, ω.

Frame # 9 Slide # 22 A. Antoniou Digital Signal Processing – Sec. 5.5.4 Rotation in the clockwise or counterclockwise direction is analogous to positive or negative frequency.

Aliasing at the Movies Cont’d

The slides that follow show a wagon wheel with a black dot.

The number of wheel revolutions per second is analogous to frequency, say, ω.

The number of ﬁlm frames per second, N, times the number of wheel revolutions per second is analogous to the sampling frequency ωs , i.e.,

ωs = Nω or ω = ωs /N

Frame # 9 Slide # 23 A. Antoniou Digital Signal Processing – Sec. 5.5.4 Aliasing at the Movies Cont’d

The slides that follow show a wagon wheel with a black dot.

The number of wheel revolutions per second is analogous to frequency, say, ω.

The number of ﬁlm frames per second, N, times the number of wheel revolutions per second is analogous to the sampling frequency ωs , i.e.,

ωs = Nω or ω = ωs /N

Rotation in the clockwise or counterclockwise direction is analogous to positive or negative frequency.

Frame # 9 Slide # 24 A. Antoniou Digital Signal Processing – Sec. 5.5.4 ω = ωs/4, Frame #1

Frame # 10 Slide # 25 A. Antoniou Digital Signal Processing – Sec. 5.5.4 ω = ωs/4, Frame #2

Frame # 11 Slide # 26 A. Antoniou Digital Signal Processing – Sec. 5.5.4 ω = ωs/4, Frame #3

Frame # 12 Slide # 27 A. Antoniou Digital Signal Processing – Sec. 5.5.4 ω = ωs/4, Frame #4

Frame # 13 Slide # 28 A. Antoniou Digital Signal Processing – Sec. 5.5.4 ω = ωs/4, Frame #5

Frame # 14 Slide # 29 A. Antoniou Digital Signal Processing – Sec. 5.5.4 ω = ωs/2, Frame #1

Frame # 15 Slide # 30 A. Antoniou Digital Signal Processing – Sec. 5.5.4 ω = ωs/2, Frame #2

Frame # 16 Slide # 31 A. Antoniou Digital Signal Processing – Sec. 5.5.4 ω = ωs/2, Frame #3

Frame # 17 Slide # 32 A. Antoniou Digital Signal Processing – Sec. 5.5.4 ω = 3ωs/4, Frame #1

Frame # 18 Slide # 33 A. Antoniou Digital Signal Processing – Sec. 5.5.4 ω = 3ωs/4, Frame #2

Frame # 19 Slide # 34 A. Antoniou Digital Signal Processing – Sec. 5.5.4 ω = 3ωs/4, Frame #3

Frame # 20 Slide # 35 A. Antoniou Digital Signal Processing – Sec. 5.5.4 ω = 3ωs/4, Frame #4

Frame # 21 Slide # 36 A. Antoniou Digital Signal Processing – Sec. 5.5.4 ω = 3ωs/4, Frame #5

Frame # 22 Slide # 37 A. Antoniou Digital Signal Processing – Sec. 5.5.4 ω = 7ωs/8, Frame #1

Frame # 23 Slide # 38 A. Antoniou Digital Signal Processing – Sec. 5.5.4 ω = 7ωs/8, Frame #2

Frame # 24 Slide # 39 A. Antoniou Digital Signal Processing – Sec. 5.5.4 ω = 7ωs/8, Frame #3

Frame # 25 Slide # 40 A. Antoniou Digital Signal Processing – Sec. 5.5.4 ω = 7ωs/8, Frame #4

Frame # 26 Slide # 41 A. Antoniou Digital Signal Processing – Sec. 5.5.4 ω = 7ωs/8, Frame #5

Frame # 27 Slide # 42 A. Antoniou Digital Signal Processing – Sec. 5.5.4 ω = 7ωs/8, Frame #6

Frame # 28 Slide # 43 A. Antoniou Digital Signal Processing – Sec. 5.5.4 ω = 7ωs/8, Frame #7

Frame # 29 Slide # 44 A. Antoniou Digital Signal Processing – Sec. 5.5.4 ω = 7ωs/8, Frame #8

Frame # 30 Slide # 45 A. Antoniou Digital Signal Processing – Sec. 5.5.4 ω = 7ωs/8, Frame #9

Frame # 31 Slide # 46 A. Antoniou Digital Signal Processing – Sec. 5.5.4 ω = ωs, Frame #1

Frame # 32 Slide # 47 A. Antoniou Digital Signal Processing – Sec. 5.5.4 ω = ωs, Frame #2

Frame # 33 Slide # 48 A. Antoniou Digital Signal Processing – Sec. 5.5.4 ω = ωs, Frame #3

Frame # 34 Slide # 49 A. Antoniou Digital Signal Processing – Sec. 5.5.4 ω = 9ωs/8, Frame #1

Frame # 35 Slide # 50 A. Antoniou Digital Signal Processing – Sec. 5.5.4 ω = 9ωs/8, Frame #2

Frame # 36 Slide # 51 A. Antoniou Digital Signal Processing – Sec. 5.5.4 ω = 9ωs/8, Frame #3

Frame # 37 Slide # 52 A. Antoniou Digital Signal Processing – Sec. 5.5.4 ω = 9ωs/8, Frame #4

Frame # 38 Slide # 53 A. Antoniou Digital Signal Processing – Sec. 5.5.4 ω = 9ωs/8, Frame #5

Frame # 39 Slide # 54 A. Antoniou Digital Signal Processing – Sec. 5.5.4 ω = 9ωs/8, Frame #6

Frame # 40 Slide # 55 A. Antoniou Digital Signal Processing – Sec. 5.5.4 ω = 9ωs/8, Frame #7

Frame # 41 Slide # 56 A. Antoniou Digital Signal Processing – Sec. 5.5.4 ω = 9ωs/8, Frame #8

Frame # 42 Slide # 57 A. Antoniou Digital Signal Processing – Sec. 5.5.4 ω = 9ωs/8, Frame #9

Frame # 43 Slide # 58 A. Antoniou Digital Signal Processing – Sec. 5.5.4 This slide concludes the presentation. Thank you for your attention.

Frame # 44 Slide # 59 A. Antoniou Digital Signal Processing – Sec. 5.5.4