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Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.341: Discrete-Time Signal Processing OpenCourseWare 2006 Lecture Minimum-Phase and All-Pass Systems Reading: Sections 5.5 and 5.6 in Oppenheim, Schafer Buck (OSB). All-Pass Systems Deﬁnition of an all-pass system AP is as follows: AP j The gain of an all-pass system is real constant (A doesn’t necessarily need to be 1). In order to satisfy the above deﬁnition, each pole of AP should be paired with conjugate reciprocal zero, as shown in OSB Figure 5.21. rational all-pass system has the general form given below: AP (1 =1 If pole is at then zero is at /a i.e. pole at re j is paired with zero at j If h[n] is real, then OSB Figure 5.24 shows the frequency response for an all-pass system with the pole-zero plot in OSB Figure 5.21. Note that Figure 5.24(b) shows the wrapped phase. The group delay in (c) is largest when π/ and the points on the unit circle that are closest to the poles and zeros. The phase change is greatest around these points. An all-pass system is always stable, since when frequency response characteristics (such as all- pass) are discussed, it is naturally assumed that the Fourier transform exists, thus stability is implied. Example: ) = has pole at the origin, and zero at thus it is an all-pass system. In general, any rational function will have an equal number of poles and zeros (some at ).

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Minimum-Phase Systems The basic deﬁnition of minimum-phase system is as follows: Stable causal and has stable causal inverse Stable and causal All poles of are inside the unit circle. Stable and causal inverse All poles of /H are inside the unit circle or equivalently, all zeros of are inside the unit circle. Thus, to have minimum-phase system, all poles and zeros of must be inside the unit circle (no pole or zero at ). Since the number of poles is always equal to the number of zeros, you have the same number of poles and zeros inside the unit circle. The system in OSB Figure 5.30(a) has four poles and four zeros inside the unit circle, and thus is minimum-phase. Example: NOT minimum-phase (a pole at From the deﬁnitions, it is clear that an all-pass system cannot be minimum-phase. Spectral Factorization Generally, several diﬀerent systems can have diﬀerent phase responses and yet have the same magnitude response. However, for minimum-phase signal ], the frequency response can be uniquely recovered (to within sign change) from the magnitude alone. This also means that you cannot specify both magnitude and phase independently for minimum-phase system. For real rational system: j j j j j (1 /z j j The following example demonstrates the process of recovering from the given Example: 17 16 17 16 j j cos j j j cos

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with (1 /z 17 16 Let z H (1 /z ). z z z z −1 −2 −1.5 −1 −0.5 0.5 1.5 1/4 1/2 We will just replace j (analytic continuation). j j ), then we need to factor Zeros of Poles of Since is minimum-phase, we know that must have all its poles and zeros inside the unit circle, thus it has pole at 1/2 and zero at 1/4. (1 /z (1 /z =1( =0) (1 (1 j From the given 17 1 j 16 16 | | =0 This problem of recovering frequency response from the magnitude response is commonly called spectral factorization.

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um-Phase Systems deﬁnition of a maximum-phase system is as follows: anti-causal with a stable and anti-causal inverse. oles and zeros are outside the unit circle and ROC includes the unit circle. 5.30(b) shows the pole/zero plot of a delayed maximum-phase system. The system but causal due to the 4th order pole at = 0. The corresponding sequences associated Figure 5.30 are shown in OSB Figure 5.31. Note that ] is the ﬂipped and delayed ] as ) = (1 /z ). system ) can always be expressed as ) = MIN MAX integer M. The factor allows us to compensate for poles and zeros at the origin of the factorization is shown below. MIN ) takes the the pole inside the unit MAX ) takes the zero outside the unit circle. In order to make the number of zeros equal, we place a zero at the origin for MIN , and a pole at for MAX . The compensates these zero and pole. ) = Maxim factor −1 −0.5 0.5 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 −1 −0.5 0.5 1.5 −1 −0.5 0.5 1.5 INF −1 −0.5 0.5 1.5 −1 −0.5 0.5 1.5 INF The basic Stable and All OSB Figure is stable with OSB version of stable for some or at One example circle and poles and MIN MAX

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can also be expressed as ) = MIN AP ) = −1 −0.5 0.5 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 Real Part Imaginary Part 1/b −1 −0.5 0.5 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1/b stable system MIN /b AP /b OSB problem 5.64 illustrates the importance of this concept in compensating the magnitude response of nonminimum-phase system. MIN AP Since MIN has stable and causal inverse, we can deﬁne stable and causal system as follows: MIN With this compensating system, the magnitude of the overall frequency response is unity. j ) = j j j ) = AP j j j | | | | | | | |

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There are diﬀerent ways of deﬁning minimum-phase systems. Now, we will talk about what’s “minimum about minimum-phase systems. 1. Minimum-phase systems have minimum group delay. 2. Minimum-phase systems have minimum energy delay. Minimum Group Delay As shown in OSB section 5.5, for causal all-pass system, AP ω. The equality holds for all when AP is just constant. Consider single factor AP Let re j then re j j j Now, consider all casual systems with the same frequency response magnitude j MIN AP If AP is not constant, then it increases the group delay, i.e. among all causal systems with the same frequency response magnitude, the minimum-phase one has the smallest group delay at all frequencies. MIN and the equality holds i MIN ). Minimum Energy Delay Closely related is the minimum energy delay property. =0 MIN =0 i.e. the minimum-phase impulse response is the most “front-loaded of all the causal ones with the same frequency response magnitude. The energy accumulates faster for the minimum-phase impulse response as shown in OSB Figure 5.32. OSB Problem 5.66 proves this result.

341 DiscreteTime Signal Processing OpenCourseWare 2006 Lecture MinimumPhase and AllPass Systems Reading Sections 55 and 56 in Oppenheim Schafer Buck OSB AllPass Systems De64257nition of an allpass system AP is as follows AP j The gain of an allpass s ID: 22137

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Page 1

Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.341: Discrete-Time Signal Processing OpenCourseWare 2006 Lecture Minimum-Phase and All-Pass Systems Reading: Sections 5.5 and 5.6 in Oppenheim, Schafer Buck (OSB). All-Pass Systems Deﬁnition of an all-pass system AP is as follows: AP j The gain of an all-pass system is real constant (A doesn’t necessarily need to be 1). In order to satisfy the above deﬁnition, each pole of AP should be paired with conjugate reciprocal zero, as shown in OSB Figure 5.21. rational all-pass system has the general form given below: AP (1 =1 If pole is at then zero is at /a i.e. pole at re j is paired with zero at j If h[n] is real, then OSB Figure 5.24 shows the frequency response for an all-pass system with the pole-zero plot in OSB Figure 5.21. Note that Figure 5.24(b) shows the wrapped phase. The group delay in (c) is largest when π/ and the points on the unit circle that are closest to the poles and zeros. The phase change is greatest around these points. An all-pass system is always stable, since when frequency response characteristics (such as all- pass) are discussed, it is naturally assumed that the Fourier transform exists, thus stability is implied. Example: ) = has pole at the origin, and zero at thus it is an all-pass system. In general, any rational function will have an equal number of poles and zeros (some at ).

Page 2

Minimum-Phase Systems The basic deﬁnition of minimum-phase system is as follows: Stable causal and has stable causal inverse Stable and causal All poles of are inside the unit circle. Stable and causal inverse All poles of /H are inside the unit circle or equivalently, all zeros of are inside the unit circle. Thus, to have minimum-phase system, all poles and zeros of must be inside the unit circle (no pole or zero at ). Since the number of poles is always equal to the number of zeros, you have the same number of poles and zeros inside the unit circle. The system in OSB Figure 5.30(a) has four poles and four zeros inside the unit circle, and thus is minimum-phase. Example: NOT minimum-phase (a pole at From the deﬁnitions, it is clear that an all-pass system cannot be minimum-phase. Spectral Factorization Generally, several diﬀerent systems can have diﬀerent phase responses and yet have the same magnitude response. However, for minimum-phase signal ], the frequency response can be uniquely recovered (to within sign change) from the magnitude alone. This also means that you cannot specify both magnitude and phase independently for minimum-phase system. For real rational system: j j j j j (1 /z j j The following example demonstrates the process of recovering from the given Example: 17 16 17 16 j j cos j j j cos

Page 3

with (1 /z 17 16 Let z H (1 /z ). z z z z −1 −2 −1.5 −1 −0.5 0.5 1.5 1/4 1/2 We will just replace j (analytic continuation). j j ), then we need to factor Zeros of Poles of Since is minimum-phase, we know that must have all its poles and zeros inside the unit circle, thus it has pole at 1/2 and zero at 1/4. (1 /z (1 /z =1( =0) (1 (1 j From the given 17 1 j 16 16 | | =0 This problem of recovering frequency response from the magnitude response is commonly called spectral factorization.

Page 4

um-Phase Systems deﬁnition of a maximum-phase system is as follows: anti-causal with a stable and anti-causal inverse. oles and zeros are outside the unit circle and ROC includes the unit circle. 5.30(b) shows the pole/zero plot of a delayed maximum-phase system. The system but causal due to the 4th order pole at = 0. The corresponding sequences associated Figure 5.30 are shown in OSB Figure 5.31. Note that ] is the ﬂipped and delayed ] as ) = (1 /z ). system ) can always be expressed as ) = MIN MAX integer M. The factor allows us to compensate for poles and zeros at the origin of the factorization is shown below. MIN ) takes the the pole inside the unit MAX ) takes the zero outside the unit circle. In order to make the number of zeros equal, we place a zero at the origin for MIN , and a pole at for MAX . The compensates these zero and pole. ) = Maxim factor −1 −0.5 0.5 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 −1 −0.5 0.5 1.5 −1 −0.5 0.5 1.5 INF −1 −0.5 0.5 1.5 −1 −0.5 0.5 1.5 INF The basic Stable and All OSB Figure is stable with OSB version of stable for some or at One example circle and poles and MIN MAX

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can also be expressed as ) = MIN AP ) = −1 −0.5 0.5 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 Real Part Imaginary Part 1/b −1 −0.5 0.5 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1/b stable system MIN /b AP /b OSB problem 5.64 illustrates the importance of this concept in compensating the magnitude response of nonminimum-phase system. MIN AP Since MIN has stable and causal inverse, we can deﬁne stable and causal system as follows: MIN With this compensating system, the magnitude of the overall frequency response is unity. j ) = j j j ) = AP j j j | | | | | | | |

Page 6

There are diﬀerent ways of deﬁning minimum-phase systems. Now, we will talk about what’s “minimum about minimum-phase systems. 1. Minimum-phase systems have minimum group delay. 2. Minimum-phase systems have minimum energy delay. Minimum Group Delay As shown in OSB section 5.5, for causal all-pass system, AP ω. The equality holds for all when AP is just constant. Consider single factor AP Let re j then re j j j Now, consider all casual systems with the same frequency response magnitude j MIN AP If AP is not constant, then it increases the group delay, i.e. among all causal systems with the same frequency response magnitude, the minimum-phase one has the smallest group delay at all frequencies. MIN and the equality holds i MIN ). Minimum Energy Delay Closely related is the minimum energy delay property. =0 MIN =0 i.e. the minimum-phase impulse response is the most “front-loaded of all the causal ones with the same frequency response magnitude. The energy accumulates faster for the minimum-phase impulse response as shown in OSB Figure 5.32. OSB Problem 5.66 proves this result.

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