Envelope Extraction via Complex Homomorphic Filtering I.A.Rezek and S.J.Roberts June 12, 1998 Keywords
Keywords: Hilbert Transform, Homomorphic Filter, Amplitude Modulation, Envelope, Respiration,
EMG.
Abstract
In many examples in biomedical signal processing the modulation amplitude of a signal (e.g. respiration movement) conveys information. Traditional methods of demodulation typically involve peak- and valley tracing which is computationally expensive and noise-sensitive. In this paper we present results for the well-known, but little-used, technique of complex-domain homomorphic ltering which o ers an alternative approach to such demodulation problems.
1 INTRODUCTION Physiological signals have a varied form, but often the signal of interest is embedded or encoded within in some other signal. One such example is respiratory amplitude, which appears as the envelope of the respiration trace. In conjunction with heart-rate variability, the information contained in the breath amplitude is often used to monitor the cardio-respiratory nervous system. Another example is the electromyographic signal envelope which give information about phasic muscle activity. The most obvious approach to envelope extraction is of course the tracing of extrema in the signal. The approach is not only tedious and computationally intensive but also very sensitive to signal noise. A better way to extract the envelope signals is to view them as the result of some amplitude modulation process. The mathematical formation of the model is then s(t) = A m(t) cos(wc t):
(1)
The cosine term is called the carrier signal oscillating at a frequency of !c. The term A m(t) is the modulating signal, which carries the information. The physiological equivalents of these components are, in the case of respiration, respiratory frequency and breath amplitude. A special situation occurs, if the carrier signal peak-to-peak variation is smaller than the modulating signal DC component (see Figure 1). Department of Electrical and Electronic Engineering; Imperial College of Science, Technology and Medicine, Exhibition Road, London, SW7 2BT, UK; email:
[email protected],
[email protected] .uk
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Amplitude Modulation with sufficient Carrier DC 4 2 0 −2 −4 0
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Figure 1: E ect of Carrier DC Component in Amplitude Modulation: bottom graph shows the inversion of the amplitude modulating signal if the Carrier DC component is too large, compared to the modulating signal amplitude. In this case, the modulated signal envelope can not be uniquely reconstructed. A physiological equivalent is periodic breathing, during which such type of modulation might be present. Given the model, it is now theoretically easy to extract the envelope information. To see this we take the Fourier transformation of s(t), S(!) = Ffs(t)g = A(t) 2 [M(! + !c ) + (! ? !c)]
(2)
where Ffm(t)g is the Fourier transform of M(!). Thus, modulation causes the modulating signal spectrum M(!) to be shifted and replicated such that it appears as sidebands of the carrier frequency. If the carrier frequency, !c , is known, demodulation can be achieved simply by multiplying the modulated signal with the carrier signal. This causes M(!) to be shifted to the low-frequency band from which it can be subsequently extracted by low-pass ltering. This, however, requires that the carrier frequency is known and constant, a condition that is not met in physiological signals. For example, respiratory control requires frequent adjustment of respiration rate and depth. When the carrier frequency is not known or is variable within a certain frequency band, demodulation is better achieved by a method known as homomorphic ltering [Oppenheim and Schafer, 1975]. The principle of the method is to nd a transformation such that the operation by which two signals were combined is mapped to a simple linear space. Since most standard ltering techniques were designed for linear spaces, they can now be employed to lter out any desired component. Finally, the extracted signal is mapped back 2
to its original domain.
2 HOMOMORPHIC FILTERING Homomorphic signal processing is concerned with the mapping of algebraically combined signals to a signal domain in which classical linear time invariant signal processing tools can be applied. The method is ideal for the separation of signals that were combined by amplitude modulation or convolution. Because amplitude modulation involves multiplying the message signal with carrier signal, the homomorphic system operating on such signals is referred to as a multiplicative system. In this paper, we focus only on multiplicative systems. (For convolution systems see [Oppenheim and Schafer, 1975]). Suppose the input signals, x1 (t) and x2 (t), have been combined by an operation denoted by 2. Also, let the combination of any one of the signals with a scalar be denoted by . The aim of a homomorphic systems is to nd a transform, D such that the output obeys the superposition principle,
D[x1(t)2x2 (t)] = D[x1(t)] + D[x2(t)] and, D[c x1 (t)] = c D[x1(t)]:
(3) (4)
After the mapping with D, we apply a conventional linear system ( lter), L, to separate the desired signal and map the signal back into its original domain via the inverse transformation D?1 . Here we shish to extract amplitude modulated signals, hence D must satisfy
D[x1(n) x2 (n) ] = D[x1(n)] + D[x2(n)]:
(5)
Clearly, D must be the logarithmic and the inverse transformation D?1 exponentiation. A natural consequence of the transformation is that the frequency spectra of x1(t) and x2 (t) are superimposed. This causes ambiguities and prohibits simple ltering to separate the signals (see Figure (2)). There is no problem if the respective frequency bands are well separated, i.e. x1 (t) and x2(t) reside in non-overlapping frequency bands. If, however, the condition is not met (for instance due to noise or trends), prior band-pass ltering is required. The scene is theoretically set to separate two amplitude-modulated signals. It turns out, however, that the singularity of the logarithm at zero causes major concerns for, say, zero-mean signals. Another concern is the un-symmetrical weighting of signals values in the range of 0 1 and 1 1. The singularity can be 3
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