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On nontrivial analytic signals with positive instantaneous frequency
Signal Processing 83 (2003) 655 – 658 www.elsevier.com/locate/sigpro
Short communication
On nontrivial analytic signals with positive instantaneous frequency Milo$s I. Doroslova$cki∗ Department of Electrical and Computer Engineering, The George Washington University, 801 22nd Street, N.W., Washington, DC 20052, USA Received 18 March 2002; received in revised form 12 November 2002
Abstract This communication recalls that the design of analytic signals with positive instantaneous frequency is a dual problem to the design of causal linear time-invariant systems (e.g., 1lters) with monotonically decreasing phase response. Recent publications that specify the structure of analytic signals with instantaneous positive frequency for the aperiodic and periodic case, are referenced. The design of a 1nite-energy analytic signal with positive periodic instantaneous frequency is illustrated. ? 2002 Elsevier Science B.V. All rights reserved. Keywords: Analytic signal; Transfer function; Amplitude response; Phase response; Causal system; Instantaneous frequency; Group delay response; Duality
1. Introduction The analytic signal f+ (t) of a real signal f(t) is de1ned as [1]: ˆ f+ (t) = f(t) + j f(t);
(1)
ˆ where f(t) is the Hilbert transform of f(t). The Fourier transform of f+ (t) vanishes at negative frequencies. The analytic signal can be written in the polar form: f+ (t) = af (t)e
j’f (t)
;
(2)
where the analytic amplitude af (t) and analytic phase ’f (t) are real functions. The derivative of the analytic phase d’f (t)=dt is called the analytic instantaneous ∗
Tel.: +1-202-994-6916; fax: +1-202-994-0227. E-mail address: [email protected] (M.I. Doroslova$cki).
frequency (!f (t)). In this way, the af (t), ’f (t), and !f (t) are uniquely de1ned and moreover, they are the only choice if we want to satisfy plausible physical conditions listed in [8]. Recently, questions have been raised about the existence of complex signals with prescribed structure that are analytic signals with positive instantaneous frequency. It is shown in [9] that a 1nite-energy signal g(t) in the form (2), i.e. g(t) = a(t)ej’(t)
(3)
with polynomial phase and band-limited amplitude of bandwidth B, is analytic, if and only if d’(t) = !0 ¿ B dt
(4)
for all t, but possibly isolated points. That is, the 1nite-energy analytic signals with band-limited
0165-1684/03/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0165-1684(02)00483-8
656
M.I. Doroslova+cki / Signal Processing 83 (2003) 655 – 658
amplitude a(t) and polynomial phase must be in the trivial form a(t)ej(!0 t+ 0 ) ;
(5)
where !0 ¿ B. These trivial analytic signals have constant positive instantaneous frequency. Can we 1nd nontrivial families of complex functions which are analytic signals with positive instantaneous frequency? One direction to search for that, as also indicated in [9], is to consider the case when the phase ’(t) is not a polynomial. Now, we will recall the fact that the design of analytic signals is dual to the design of causal linear time-invariant (LTI) systems. The duality enable us to directly use transfer functions of LTI systems as analytic signals with desired properties. 2. Duality between analytic signals and causal LTI systems Let s( ) be the impulse response of a causal system and S() be the transfer function of the causal system, i.e. +∞ S() = F{s( )}() = s( )e−j2 d : (6)
(F(− ); f()) is a Fourier transform pair as well, is called the duality [1] or symmetry property [5]. In that sense, we can say that there is duality between f( ) and F(− ), or more speci1cally in our case, between the impulse response of a causal system s( ) and the analytic signal S(− ).
3. Analytic signals with positive aperiodic instantaneous frequency In this case, we are going to consider transfer functions of causal continuous-time systems S() and take g(t) = S(−t). If we restrict ourselves to the rational transfer functions, the monotonically decreasing phase response (i.e., the positiveness of analytic signal instantaneous frequency) can be controlled by an appropriate placement of poles and zeros. If we consider the causal all-pass systems, the dual counterpart are the analytical signals of the phase signals [6]. These analytic signals have a monotonically increasing instantaneous phase ’(t) that is a sum of a constant term, of a linear term with nonnegative slope 2p0 , and of arctangent terms:
−∞
The imaginary part of S(−) is the Hilbert transform of the real part of S(−) [5, p. 251]. S(−) is an analytic signal in the -domain. Let us de1ne g(t) = S(−t):
(7)
The complex signal g(t) is an analytic signal in the t-domain. It can be easily checked that F{g(t)}(p) = F{S(−t)}(p) +∞ S(−t)e−j2pt dt = s(p): = −∞
(8)
Moreover, if S() has a monotonically decreasing phase, g(t) = S(−t) will have a positive instantaneous frequency. Note that in the cases when the Fourier integral does not exist in the strict sense, the existence of the integral in the distribution sense is assumed [7]. Particularly, that means that discrete-time signals are represented using the delta functions. The general property of the Fourier transform that if (f( ); F()) is a Fourier transform pair, then
’(t) = + 2p0 t + 2
N k=1
Arctg
bk ; ak − t
(9)
where ∈ [ − ; ), p0 ∈ R+ ∪ {0}, ak ∈ R, bk ∈ R+ , and N is the number of poles (zeros) in a causal all-pass system. and p0 correspond to the phase and frequency of a modulating complex exponential signal, respectively. Consequently, the instantaneous frequency is a rational function of t, it is always positive and moreover, it is always larger than 2p0 . The instantaneous frequency tends to 2p0 when t → ±∞. That is, a complex signal with constant amplitude can be analytic if its phase derivative is a rational function of t (not a polynomial of t and, of course, not any rational function of t). By choosing poles of the analytic signal of a phase signal (i.e., by choosing poles of the all-pass 1lter), we can approximate functions of interest. Also, by multiplying the analytic signal of a phase signal with a real positive window W (t), whose spectrum w(p) = 0 for |p| ¿ p0 , we can obtain a 1nite-energy analytic signal with positive instantaneous frequency.
M.I. Doroslova+cki / Signal Processing 83 (2003) 655 – 658
4. Analytic signals with positive periodic instantaneous frequency
657
The sequence {s(np1 )} is chosen to be the impulse response of the all pass 1lter
Periodic analytic signals can be obtained from transfer functions of causal discrete-time systems. A transfer function can be factored as a product of a minimum phase transfer function and an all-pass transfer function. As it is well known, the all-pass transfer function has the phase response monotonically decreasing [4]. Consequently, the analytic signal obtained from the all-pass part has a positive instantaneous frequency. The minimum phase part gives an analytic signal that is the complex envelope of the total signal. This approach has been recently used in [2,3] for modeling of signals over a 1nite interval and applied to speech characterization [2]. The positive instantaneous frequency d’(t)=dt of an “all-pass” analytic signal is in this case a sum of a constant nonnegative term 2p0 and periodic positive terms: d’(t) = 2p0 dt
0:9 − z −1 (14) 1 − 0:9z −1 and it is plotted in Fig. 1. The obtained nonlinear part m(t) of the instantaneous phase is depicted in Fig. 2. The corresponding instantaneous frequency 2p0 + dm(t)=dt is in Fig. 3. The amplitude spectrum of the signal is in Fig. 4. As can be seen from the 1gures, the complex signal is an analytic signal with positive instantaneous frequency. 1.2
1
0.8
0.6
0.4
N + 2q0
2k − 1 ; 2 + 1 − 2k cos(2q0 t + k ) k=1 k
0.2
0
(10)
where p0 ∈ R+ ∪ {0}, q0 ∈ R+ , k ∈ (1; +∞), k ∈ [ − ; +), and N is the number of poles (zeros) in a causal all-pass discrete-time system. p0 is the frequency of a modulating complex exponential signal. By choosing poles of the “all-pass” analytic signal, we can approximate functions of interest. Of course, by windowing the “all-pass” periodic analytic signal by a positive real window W (t), whose spectrum is w(p) = 0 for |p| ¿ p0 , we obtain a 1nite-energy analytic signal with positive instantaneous frequency. As an example consider the complex signal g(t) = S(−t) = a(t)ej2p0 t+jm(t) ;
(11)
where ejm(t) =
+∞
-0.2 -10
0
10
20
30
40
50
n
Fig. 1. Causal sequence for generating an “all pass” analytic signal.
60
50
40
30
20
10
s(np1 )ej2np1 t ;
(12)
0
n=0
p0 = 1=500, p1 = 1=1000, and t − 5000 2 : a(t) = sinc 625
-10 0
1000
2000
3000
4000
5000
6000
7000
8000
9000 10000
TIME [s]
(13)
Fig. 2. Nonlinear part m(t) of instantaneous phase.
658
M.I. Doroslova+cki / Signal Processing 83 (2003) 655 – 658
0.14
5. Conclusion
0.12
Considered as functions of time, transfer functions of causal LTI systems are analytical signals reversed in time. We can use the well-developed art of designing causal LTI systems to design the analytic signals with positive instantaneous frequency. As it is well known, by appropriately placing poles and zeros, a transfer function with monotonically decreasing phase response (i.e., with positive group delay response) can be obtained. Equivalently, that means that an analytical signal with positive instantaneous frequency is obtained as well.
0.1
0.08
0.06
0.04
0.02
0 0
1000
2000
3000
4000
5000
6000
7000
8000
9000 10000
TIME [s]
References
Fig. 3. Instantaneous frequency of g(t).
600
500
400
300
200
100
0 0.05
0.04
0.03
0.02
0.01
0
0.01
0.02
0.03
FREQUENCY [Hz]
Fig. 4. Amplitude spectrum of g(t).
0.04
0.05
[1] S. Haykin, Communication Systems, 3rd Edition, Wiley, New York, 1994. [2] R. Kumaresan, A. Rao, Model-based approach to envelope and positive instantaneous frequency estimation of signals with speech applications, J. Acoust. Soc. Amer. 105 (3) (March 1999) 1912–1924. [3] R. Kumaresan, A. Rao, On minimum/maximum/all-pass decompositions in time and frequency domains, IEEE Trans. Signal Process. 48 (10) (October 2000) 2973–2976. [4] A.V. Oppenheim, R.W. Schafer, J.R. Buck, Discrete-Time Signal Processing, 2nd Edition, Prentice-Hall, Upper Saddle River, NJ, 1998. [5] A. Papoulis, Signal Analysis, McGraw-Hill, New York, 1977. [6] B. Picinbono, On instantaneous amplitude and phase of signals, IEEE Trans. Signal Process. 45 (3) (March 1997) 552–560. [7] L. Schwartz, Mathematics for the Physical Sciences, Addison-Wesley, London, 1966. [8] D. Vakman, On the analytic signal, the Teager–Kaiser energy algorithm, and other methods for de1ning amplitude and frequency, IEEE Trans. Signal Process. 47 (7) (July 1999) 2066–2068. [9] X.-G. Xia, L. Cohen, On analytic signals with nonnegative instantaneous frequency, Proceedings of the ICASSP-99, Phoenix, March 1999, Paper 1483.
Short communication
On nontrivial analytic signals with positive instantaneous frequency Milo$s I. Doroslova$cki∗ Department of Electrical and Computer Engineering, The George Washington University, 801 22nd Street, N.W., Washington, DC 20052, USA Received 18 March 2002; received in revised form 12 November 2002
Abstract This communication recalls that the design of analytic signals with positive instantaneous frequency is a dual problem to the design of causal linear time-invariant systems (e.g., 1lters) with monotonically decreasing phase response. Recent publications that specify the structure of analytic signals with instantaneous positive frequency for the aperiodic and periodic case, are referenced. The design of a 1nite-energy analytic signal with positive periodic instantaneous frequency is illustrated. ? 2002 Elsevier Science B.V. All rights reserved. Keywords: Analytic signal; Transfer function; Amplitude response; Phase response; Causal system; Instantaneous frequency; Group delay response; Duality
1. Introduction The analytic signal f+ (t) of a real signal f(t) is de1ned as [1]: ˆ f+ (t) = f(t) + j f(t);
(1)
ˆ where f(t) is the Hilbert transform of f(t). The Fourier transform of f+ (t) vanishes at negative frequencies. The analytic signal can be written in the polar form: f+ (t) = af (t)e
j’f (t)
;
(2)
where the analytic amplitude af (t) and analytic phase ’f (t) are real functions. The derivative of the analytic phase d’f (t)=dt is called the analytic instantaneous ∗
Tel.: +1-202-994-6916; fax: +1-202-994-0227. E-mail address: [email protected] (M.I. Doroslova$cki).
frequency (!f (t)). In this way, the af (t), ’f (t), and !f (t) are uniquely de1ned and moreover, they are the only choice if we want to satisfy plausible physical conditions listed in [8]. Recently, questions have been raised about the existence of complex signals with prescribed structure that are analytic signals with positive instantaneous frequency. It is shown in [9] that a 1nite-energy signal g(t) in the form (2), i.e. g(t) = a(t)ej’(t)
(3)
with polynomial phase and band-limited amplitude of bandwidth B, is analytic, if and only if d’(t) = !0 ¿ B dt
(4)
for all t, but possibly isolated points. That is, the 1nite-energy analytic signals with band-limited
0165-1684/03/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0165-1684(02)00483-8
656
M.I. Doroslova+cki / Signal Processing 83 (2003) 655 – 658
amplitude a(t) and polynomial phase must be in the trivial form a(t)ej(!0 t+ 0 ) ;
(5)
where !0 ¿ B. These trivial analytic signals have constant positive instantaneous frequency. Can we 1nd nontrivial families of complex functions which are analytic signals with positive instantaneous frequency? One direction to search for that, as also indicated in [9], is to consider the case when the phase ’(t) is not a polynomial. Now, we will recall the fact that the design of analytic signals is dual to the design of causal linear time-invariant (LTI) systems. The duality enable us to directly use transfer functions of LTI systems as analytic signals with desired properties. 2. Duality between analytic signals and causal LTI systems Let s( ) be the impulse response of a causal system and S() be the transfer function of the causal system, i.e. +∞ S() = F{s( )}() = s( )e−j2 d : (6)
(F(− ); f()) is a Fourier transform pair as well, is called the duality [1] or symmetry property [5]. In that sense, we can say that there is duality between f( ) and F(− ), or more speci1cally in our case, between the impulse response of a causal system s( ) and the analytic signal S(− ).
3. Analytic signals with positive aperiodic instantaneous frequency In this case, we are going to consider transfer functions of causal continuous-time systems S() and take g(t) = S(−t). If we restrict ourselves to the rational transfer functions, the monotonically decreasing phase response (i.e., the positiveness of analytic signal instantaneous frequency) can be controlled by an appropriate placement of poles and zeros. If we consider the causal all-pass systems, the dual counterpart are the analytical signals of the phase signals [6]. These analytic signals have a monotonically increasing instantaneous phase ’(t) that is a sum of a constant term, of a linear term with nonnegative slope 2p0 , and of arctangent terms:
−∞
The imaginary part of S(−) is the Hilbert transform of the real part of S(−) [5, p. 251]. S(−) is an analytic signal in the -domain. Let us de1ne g(t) = S(−t):
(7)
The complex signal g(t) is an analytic signal in the t-domain. It can be easily checked that F{g(t)}(p) = F{S(−t)}(p) +∞ S(−t)e−j2pt dt = s(p): = −∞
(8)
Moreover, if S() has a monotonically decreasing phase, g(t) = S(−t) will have a positive instantaneous frequency. Note that in the cases when the Fourier integral does not exist in the strict sense, the existence of the integral in the distribution sense is assumed [7]. Particularly, that means that discrete-time signals are represented using the delta functions. The general property of the Fourier transform that if (f( ); F()) is a Fourier transform pair, then
’(t) = + 2p0 t + 2
N k=1
Arctg
bk ; ak − t
(9)
where ∈ [ − ; ), p0 ∈ R+ ∪ {0}, ak ∈ R, bk ∈ R+ , and N is the number of poles (zeros) in a causal all-pass system. and p0 correspond to the phase and frequency of a modulating complex exponential signal, respectively. Consequently, the instantaneous frequency is a rational function of t, it is always positive and moreover, it is always larger than 2p0 . The instantaneous frequency tends to 2p0 when t → ±∞. That is, a complex signal with constant amplitude can be analytic if its phase derivative is a rational function of t (not a polynomial of t and, of course, not any rational function of t). By choosing poles of the analytic signal of a phase signal (i.e., by choosing poles of the all-pass 1lter), we can approximate functions of interest. Also, by multiplying the analytic signal of a phase signal with a real positive window W (t), whose spectrum w(p) = 0 for |p| ¿ p0 , we can obtain a 1nite-energy analytic signal with positive instantaneous frequency.
M.I. Doroslova+cki / Signal Processing 83 (2003) 655 – 658
4. Analytic signals with positive periodic instantaneous frequency
657
The sequence {s(np1 )} is chosen to be the impulse response of the all pass 1lter
Periodic analytic signals can be obtained from transfer functions of causal discrete-time systems. A transfer function can be factored as a product of a minimum phase transfer function and an all-pass transfer function. As it is well known, the all-pass transfer function has the phase response monotonically decreasing [4]. Consequently, the analytic signal obtained from the all-pass part has a positive instantaneous frequency. The minimum phase part gives an analytic signal that is the complex envelope of the total signal. This approach has been recently used in [2,3] for modeling of signals over a 1nite interval and applied to speech characterization [2]. The positive instantaneous frequency d’(t)=dt of an “all-pass” analytic signal is in this case a sum of a constant nonnegative term 2p0 and periodic positive terms: d’(t) = 2p0 dt
0:9 − z −1 (14) 1 − 0:9z −1 and it is plotted in Fig. 1. The obtained nonlinear part m(t) of the instantaneous phase is depicted in Fig. 2. The corresponding instantaneous frequency 2p0 + dm(t)=dt is in Fig. 3. The amplitude spectrum of the signal is in Fig. 4. As can be seen from the 1gures, the complex signal is an analytic signal with positive instantaneous frequency. 1.2
1
0.8
0.6
0.4
N + 2q0
2k − 1 ; 2 + 1 − 2k cos(2q0 t + k ) k=1 k
0.2
0
(10)
where p0 ∈ R+ ∪ {0}, q0 ∈ R+ , k ∈ (1; +∞), k ∈ [ − ; +), and N is the number of poles (zeros) in a causal all-pass discrete-time system. p0 is the frequency of a modulating complex exponential signal. By choosing poles of the “all-pass” analytic signal, we can approximate functions of interest. Of course, by windowing the “all-pass” periodic analytic signal by a positive real window W (t), whose spectrum is w(p) = 0 for |p| ¿ p0 , we obtain a 1nite-energy analytic signal with positive instantaneous frequency. As an example consider the complex signal g(t) = S(−t) = a(t)ej2p0 t+jm(t) ;
(11)
where ejm(t) =
+∞
-0.2 -10
0
10
20
30
40
50
n
Fig. 1. Causal sequence for generating an “all pass” analytic signal.
60
50
40
30
20
10
s(np1 )ej2np1 t ;
(12)
0
n=0
p0 = 1=500, p1 = 1=1000, and t − 5000 2 : a(t) = sinc 625
-10 0
1000
2000
3000
4000
5000
6000
7000
8000
9000 10000
TIME [s]
(13)
Fig. 2. Nonlinear part m(t) of instantaneous phase.
658
M.I. Doroslova+cki / Signal Processing 83 (2003) 655 – 658
0.14
5. Conclusion
0.12
Considered as functions of time, transfer functions of causal LTI systems are analytical signals reversed in time. We can use the well-developed art of designing causal LTI systems to design the analytic signals with positive instantaneous frequency. As it is well known, by appropriately placing poles and zeros, a transfer function with monotonically decreasing phase response (i.e., with positive group delay response) can be obtained. Equivalently, that means that an analytical signal with positive instantaneous frequency is obtained as well.
0.1
0.08
0.06
0.04
0.02
0 0
1000
2000
3000
4000
5000
6000
7000
8000
9000 10000
TIME [s]
References
Fig. 3. Instantaneous frequency of g(t).
600
500
400
300
200
100
0 0.05
0.04
0.03
0.02
0.01
0
0.01
0.02
0.03
FREQUENCY [Hz]
Fig. 4. Amplitude spectrum of g(t).
0.04
0.05
[1] S. Haykin, Communication Systems, 3rd Edition, Wiley, New York, 1994. [2] R. Kumaresan, A. Rao, Model-based approach to envelope and positive instantaneous frequency estimation of signals with speech applications, J. Acoust. Soc. Amer. 105 (3) (March 1999) 1912–1924. [3] R. Kumaresan, A. Rao, On minimum/maximum/all-pass decompositions in time and frequency domains, IEEE Trans. Signal Process. 48 (10) (October 2000) 2973–2976. [4] A.V. Oppenheim, R.W. Schafer, J.R. Buck, Discrete-Time Signal Processing, 2nd Edition, Prentice-Hall, Upper Saddle River, NJ, 1998. [5] A. Papoulis, Signal Analysis, McGraw-Hill, New York, 1977. [6] B. Picinbono, On instantaneous amplitude and phase of signals, IEEE Trans. Signal Process. 45 (3) (March 1997) 552–560. [7] L. Schwartz, Mathematics for the Physical Sciences, Addison-Wesley, London, 1966. [8] D. Vakman, On the analytic signal, the Teager–Kaiser energy algorithm, and other methods for de1ning amplitude and frequency, IEEE Trans. Signal Process. 47 (7) (July 1999) 2066–2068. [9] X.-G. Xia, L. Cohen, On analytic signals with nonnegative instantaneous frequency, Proceedings of the ICASSP-99, Phoenix, March 1999, Paper 1483.