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Instantaneous Frequency and Amplitude of Complex Signals Based on Quaternion Fourier Transform

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Instantaneous Frequency and Amplitude of Complex Signals Based on Quaternion Fourier Transform Instantaneous frequency and amplitude of complex signals based on quaternion Fourier transform Nicolas Le Bihan∗ Stephen J. Sangwiney Todd A. Ellz August 8, 2012 Abstract plex modulation can be represented mathematically by a polar representation of quaternions previously The ideas of instantaneous amplitude and phase are derived by the authors. As in the classical case, there well understood for signals with real-valued samples, is a restriction of non-overlapping frequency content based on the analytic signal which is a complex sig- between the modulating complex signal and the or- nal with one-sided Fourier transform. We extend thonormal complex exponential. We show that, un- these ideas to signals with complex-valued samples, der these conditions, modulation in the time domain using a quaternion-valued equivalent of the analytic is equivalent to a frequency shift in the quaternion signal obtained from a one-sided quaternion Fourier Fourier domain. Examples are presented to demon- transform which we refer to as the hypercomplex rep- strate these concepts. resentation of the complex signal. We present the necessary properties of the quaternion Fourier trans- form, particularly its symmetries in the frequency do- 1 Introduction main and formulae for convolution and the quater- nion Fourier transform of the Hilbert transform. The The instantaneous amplitude and phase of a real- hypercomplex representation may be interpreted as valued signal have been understood since 1948 from an ordered pair of complex signals or as a quater- the work of Ville [1] and Gabor [2] based on the nion signal. We discuss its derivation and proper- analytic signal. A critical discussion of instanta- ties and show that its quaternion Fourier transform neous amplitude and phase was given by Picinbono is one-sided. It is shown how to derive from the hy- in 1997, particularly with reference to amplitude and percomplex representation a complex envelope and a phase modulation [3, V]. The analytic signal and x phase. its associated modulation concepts can be simply de- A classical result in the case of real signals is that scribed, even though a full theoretical treatment is an amplitude modulated signal may be analysed into quite deep. Given a real-valued signal f(t), the cor- its envelope and carrier using the analytic signal pro- responding analytic signal a(t) is a complex signal vided that the modulating signal has frequency con- with real part equal to f(t) and imaginary part or- tent not overlapping with that of the carrier. We thogonal to f(t). The imaginary part is sometimes show that this idea extends to the complex case, known as the quadrature signal { in the case where arXiv:1208.1363v1 [math.NA] 7 Aug 2012 provided that the complex signal modulates an or- f(t) is a sinusoid, the imaginary part of the analytic thonormal complex exponential. Orthonormal com- signal is in quadrature, that is with a phase differ- ence of π=2. The orthogonal signal is related to f(t) − ∗Nicolas Le Bihan is with the CNRS, GIPSA-Lab, by the Hilbert transform [4, 5]. The analytic signal D´epartement Images et Signal, 961 Rue de la Houille Blanche, has the interesting property that its modulus a(t) is Domaine Universitaire, BP 46, 38402 Saint Martin d'H`eres j j cedex, France, email: [email protected] an envelope of the signal f(t). The envelope is also inp.fr known as the instantaneous amplitude. Thus if f(t) is yStephen J. Sangwine is with the School of Computer Sci- an amplitude-modulated sinusoid, the envelope a(t) , ence and Electronic Engineering, University of Essex, Colch- subject to certain conditions on the frequencyj con-j ester, CO4 3SQ, United Kingdom, email: [email protected] tent, is the modulating signal. The argument of the zTodd A. Ell is with Goodrich Sensors & Integrated Sys- tems, 14300 Judicial Road, Burnsville, MN 55306 USA, email: analytic signal, \ a(t) is known as the instantaneous [email protected] phase and its derivative is known as the instanta- 1 neous frequency. The analytic signal has a further a complex signal, for reasons which should be evi- very interesting property: it has a one-sided Fourier dent later in the paper. Note that, in essence, the transform. Thus a simple method for constructing approach proposed in this paper is inspired by the the analytic signal (algebraically or numerically) is work of B¨ulow [8, 9] and Labunets [10] who consid- to compute the Fourier transform of f(t), multiply ered Clifford Fourier transforms (quaternion in the the Fourier transform by a unit step which is zero for case of B¨ulow) to process multidimensional signals. negative frequencies, and then construct the inverse While B¨ulow considered 2D signals (images), here we Fourier transform of the resulting spectrum. consider 1D signals with 2D samples (complex val- ued samples). We are not focusing on any particular In this paper we extend the above ideas to sig- application in this paper, rather we propose a new nals with complex-valued samples. We show that approach to the processing of complex valued signals if a quaternion Fourier transform is computed from (including proper and improper signals, even though a complex-valued signal, and negative frequencies in proper signals are well-described using the classical the frequency domain representation are suppressed, complex Fourier transform | they are a special case.) the inverse quaternion Fourier transform gives an Schreier and Scharf's book gives examples of appli- equivalent of the analytic signal, which we refer to cations [6]. in this paper as a hypercomplex representation of the complex signal. Just as the classical analytic signal The idea of amplitude modulation is extended to represents a real-valued signal by a complex signal the case where the modulating signal is complex, and (that is with a pair of real signals), the hypercomplex the `carrier' is an orthonormal complex exponential, representation presented here represents a complex- that is a complex exponential which is in a com- valued signal by a quaternion signal (that is with a plex plane orthogonal to the complex plane of the pair of complex signals, based on the Cayley-Dickson modulating signal { this concept is easily realised form of a quaternion). using quaternions, since quaternion algebra has a 4-dimensional basis with three mutually orthogonal We consider signals with complex-valued samples imaginary units. Subject to the same restrictions on in a most general sense, without any restrictions frequency content as in the real-valued case, ampli- on the relationship between the real and imaginary tude modulation in the time domain corresponds to parts. However, we exclude the special case of an a frequency shift in the Fourier domain. Further, a analytic signal (where the real and imaginary parts complex envelope of the hypercomplex representation are orthogonal) because it does not possess an in- can be defined which recovers the complex modulat- teresting hypercomplex representation: the two ad- ing signal. This is the complex instantaneous ampli- ditional quaternion components of the hypercomplex tude. Further, the phase of the orthonormal complex representation are simply copies of the two original exponential may be recovered. This is the instanta- components. The case where the real and imaginary neous phase, and its derivative is the instantaneous parts are correlated in some statistical sense is called frequency. In the case of modulation of a complex ex- improper, whereas an analytic signal is proper. It has ponential the instantaneous frequency is that of the been customary to handle such signals by using an complex exponential `carrier'. Just as the classical augmented representation consisting of a vector con- complex analytic signal contains both the original taining the signal and its conjugate [6]. Recently, real signal (in the real part) and a real orthogonal Lilly and Olhede [7] have published a paper on bi- signal (in the imaginary part), our hypercomplex sig- variate analytic signal concepts. Their approach is nal representation contains two complex signals: the linked to a specific signal model, the modulated ellip- original signal and an orthogonal signal. tical signal, which they illustrate with the example of a drifting oceanographic float, and they utilise pairs We have previously published partial results on of analytic signals derived from the pair of real signals this topic [11{13] as we developed the ideas presented representing the time-varying positional coordinates here. The idea of orthonormal complex modulation of a drifting float. In this paper a different approach, has not previously been presented. using hypercomplex representation, is considered be- The sequence of topics in the rest of the paper is cause it is more suited to the analysis of modulation, as follows. Section 2 presents the quaternion algebra and also because we have at our disposal the existing and quaternion Fourier transform concepts used to tools of quaternion Fourier transforms, which provide construct the hypercomplex representation of a com- a natural fit with modulation of a complex signal by plex signal using a one-sided quaternion spectrum. 2 i Section 3 discusses the hypercomplex representation z1 + z2j in the basis 1; i; j; k with z1; z2 C , i.e. f g 2 of a complex signal. Section 4 discusses the instanta- zα = (zα) + i (zα) for α = 1; 2. This is called the neous amplitude and frequency concept for a complex Cayley-Dickson< = form of q. signal through its hypercomplex representation. Sec- Among the possible representations of q, we will tion 5 presents some signal examples to illustrate the make use in this paper of one that was recently in- ideas presented in the paper. troduced: the polar Cayley-Dickson form [16]. Any quaternion q also has a unique polar Cayley-Dickson 2 Quaternion Fourier form given by: Transform q = Aq exp(Bqj) = (a0 + a1i) exp((b0 + b1i)j) (3) where Aq = a0 + a1i is the complex modulus of q and In this section, we present the definition and prop- Bq = b0 +b1i its complex phase.
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