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Analytic signal

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Not to be confused with Analytic function.

In signal processing, the analytic signal, or analytic representation, of a real-valued signal <math>x(t)\,</math> is defined by:

<math>x_\mathrm{a}(t) = x(t) + j\cdot \hat{x}(t)\,</math>

where <math>\hat{x}(t)\,</math> is the Hilbert transform of <math>x(t)\,</math> and <math>j\,</math> is the imaginary unit. The analytic representation facilitates mathematical manipulations, such as the derivation of single-sideband modulation and demodulation. And as shown below, it makes certain attributes of <math>x(t)\,</math> more accessible.

1 Introduction
2 Applications
3 Extensions of the analytic signal to signals of multiple variables
- 3.1 Multi-dimensional analytic signal based on an ad-hoc direction
- 3.2 The monogenic signal
4 Not to be confused with
5 See also
6 References
7 External links

Introduction

The basic idea of the analytic representation is that the negative frequency components of the Fourier transform (or spectrum) of a real-valued function are superfluous, due to the Hermitian symmetry of such a spectrum. They can be discarded with no loss of information, if we are willing to deal with a complex-valued function, <math>x_a(t),\,</math> instead. And that turns out to have the advantages mentioned above. As long as the manipulated function has no negative frequency components (that is, it is still analytic), the conversion from complex back to real is just a matter of discarding the imaginary part.

Analytic signals are often shifted in frequency (down-converted) toward 0 Hz, which creates [non-symmetrical] negative frequency components. One motive is to allow lowpass filters with real coefficients to be used to limit the bandwidth of the signal. Another motive is to reduce the highest frequency, which reduces the minimum rate for alias-free sampling. A frequency shift does not undermine the mathematical tractability of the complex signal representation. So in that sense, the down-converted signal is still "analytic". However, restoring the real-valued representation is no longer a simple matter of just extracting the real component. Up-conversion is obviously required, and if the signal has been sampled (discrete-time), interpolation (upsampling) might also be necessary to avoid aliasing.
The complex conjugate of an analytic signal contains only negative frequency components. In that case also, there is no loss of information or reversibility by discarding the imaginary component. Obviously the real component of the complex conjugate is the same as the real component of the analytic signal. But in this case, its extraction restores the suppressed positive frequency components.
Another way to achieve a spectrum of negative frequencies is to frequency-shift the analytic signal sufficiently far in the negative direction. Extracting the real component again restores the positive frequencies. But in this case their order is reversed... the low-frequency component is now the high one. This can be used to demodulate a type of single sideband signal called lower sideband or inverted sideband.

Example 1: <math>x(t) = \cos(\omega_0 t)\,</math>, for some parameter <math>\omega_0 > 0\,</math>

<math>\hat{x}(t) = \cos(\omega_0 t -\begin{matrix} \frac{\pi }{2} \end{matrix}) = \sin(\omega_0 t)\,</math>

<math>x_\mathrm{a}(t) = \cos(\omega_0 t) + j\cdot \sin(\omega_0 t) = e^{j \omega_0 t}\,</math> (The 2^nd equality is Euler's formula.)

This is a complex-valued signal with increasing phase (positive frequency).

It also follows from Euler's formula that <math>\cos(\omega_0 t) = \begin{matrix} \frac{1}{2} \end{matrix}(e^{j \omega_0 t}+e^{-j \omega_0 t})\,</math>. So <math>x(t)\,</math> comprises both positive and negative frequency components. <math>x_\mathrm{a}(t)\,</math> is just the positive portion.

Example 2: <math>x(t) = \cos(\omega _1 t) + \cos(\omega _2 t)\,</math>

<math>\hat{x}(t) = \sin(|\omega _1 |t) + \sin(|\omega _2 |t)\,</math>

<math>x_\mathrm{a}(t) = e^{j |\omega _1 |t}+e^{j |\omega _2 |t}\,</math>

The definition of the analytic signal can also be expressed in terms of the Fourier transforms of <math>x(t)\,</math> and <math>x_\mathrm{a}(t)\,</math>, respectively denoted by <math>X(\omega)\,</math> and <math>X_\mathrm{a}(\omega)\,</math>:

<math>X_\mathrm{a}(\omega)\,</math>	<math>= \mathcal{F}\left\{x_\mathrm{a}(t)\right\}</math>
	<math>= \mathcal{F}\left\{x(t) + j\cdot \hat{x}(t)\right\}</math>
	<math>= \mathcal{F}\left\{x(t)\right\} + j\cdot \mathcal{F} \left\{ \hat{x}(t) \right\} </math>
	<math>= X(\omega) + j\cdot \left[-j \sgn(\omega)X(\omega) \right]</math>
	<math>= X(\omega)\cdot \left[ 1 + \sgn(\omega) \right] </math>
	<math>= \begin{cases}2 X(\omega) & \omega > 0 \\ X(0) & \omega = 0 \\ 0 & \omega < 0 \end{cases}</math>

The general result is that <math>x_\mathrm{a}(t)\,</math> comprises only the nonnegative frequency components of <math>x(t)\,</math> and the positive frequency components of <math>x(t)\,</math> are doubled. Yet the analytic transforms are reversible for real-valued <math>x(t)\,</math>. I.e., the original functions can be recovered as:

<math>X(\omega) = \begin{matrix} \frac{1}{2} \end{matrix} X_\mathrm{a}(\omega) \,</math>, for <math>\omega > 0\,</math>

Because of Hermitian symmetry,

<math>X(-\omega)= X^{*}(\omega) \,</math> for real <math> x(t) \,</math>,

this results in

<math>X(\omega) = \begin{matrix} \frac{1}{2} \end{matrix} X_\mathrm{a}^{*}(-\omega) \,</math>, for <math>\omega < 0\,</math>.

Since <math>X(0) = X_\mathrm{a}(0) \,</math>, it is also true for real <math> x(t) \,</math> that

<math>X(\omega) = \begin{matrix} \frac{1}{2} \end{matrix} \left( X_\mathrm{a}^{*}(-\omega) + X_\mathrm{a}(\omega) \right) \,</math>, for <math>-\infty < \omega < +\infty \,</math>.

More compactly:

<math>X(\omega) = \begin{matrix} \frac{1}{2} \end{matrix} |X_\mathrm{a}(|\omega|)| e^{j \sgn(\omega) \arg\{X_\mathrm{a}(|\omega|) \}}\,</math> for <math>\omega \ne 0\,</math>.

and, in the time domain,

<math>x(t) = \operatorname{Re}\{x_\mathrm{a}(t)\}\,</math>.

For completeness, we note that nothing prevents us from computing <math>x_\mathrm{a}(t)\,</math> for a complex-valued <math>x(t)\,</math>. The negative-frequency components will be eliminated, as usual (see example 3). But it might not be a reversible representation, because the original spectrum is not symmetrical in general. So except for this example, the general discussion assumes real-valued <math>x(t)\,</math>.

Example 3: <math>x(t) = e^{-j \omega_0 t}\,</math>, for some parameter <math>\omega_0 > 0\,</math>

<math>\hat{x}(t) = j\cdot e^{-j \omega_0 t}\,</math>

<math>x_\mathrm{a}(t) = e^{-j \omega_0 t} + j^2\cdot e^{-j \omega_0 t} = 0\,</math>

Applications

The analytic signal can also be expressed in terms of complex polar coordinates, <math>x_\mathrm{a}(t) = A(t)e^{j\phi(t)}\,</math>, where:

<math>A(t) = |x_\mathrm{a}(t)| = \sqrt { x^2(t) + \hat{x}^2(t) }\,</math>

<math>\phi(t) = \arg \left\{ x_\mathrm{a}(t) \right\} \,</math> (see arg function)

A signal in blue and the magnitude of its analytic signal in red, showing the envelope effect

These functions are respectively called the amplitude envelope and instantaneous phase of the signal <math>x(t)\,</math>. In the accompanying diagram, the blue curve depicts <math>x(t)\,</math>, and the red curve depicts the corresponding <math>A(t)\,</math>.
The time derivative of the unwrapped instantaneous phase is called the instantaneous frequency:

<math>\omega(t) \ \stackrel{\mathrm{def}}{=}\ \phi '(t) = \frac{d}{dt} \phi(t)\,</math>

The amplitude function, and the instantaneous phase and frequency are in some applications used to measure and detect local features of the signal. Another application of the analytic representation of a signal relates to demodulation of modulated signals. The polar coordinates conveniently separate the effects of amplitude modulation and phase (or frequency) modulation, and effectively demodulates certain kinds of signals. The analytic signal can also be represented as:

<math>x_\mathrm{a}(t)\,</math>	<math>= \left[ A(t) e^{j\phi(t)} e^{-j \omega_0 t} \right] \ e^{j \omega_0 t}</math>
	<math>= \gamma (t) e^{j \omega_0 t}</math>

where

<math>\gamma (t) = A(t) e^{j(\phi(t) - \omega_0 t)} \,</math>

is the signal's complex envelope. The complex envelope is not unique; on the contrary, it is determined by an arbitrary <math>\omega_0\,</math> assignment. This concept is often used when dealing with passband signals. If <math>x(t)\,</math> is a modulated signal, <math>\omega_0\,</math> is usually assigned to be a carrier frequency. In other cases it is selected to be somewhere in the middle of the frequency band. Sometimes <math>\omega_0\,</math> is chosen to minimize

<math>\int_{0}^{+\infty}(\omega-\omega_0)^2|X_\mathrm{a}(\omega)|^2\, d\omega</math>.

Alternatively[1], <math>\omega_0\,</math> can be chosen to minimize the mean square error in linearly approximating the unwrapped instantaneous phase <math>\phi(t)\,</math>:

<math>\int_{-\infty}^{+\infty}(\omega(t)-\omega_0)^2 |x_\mathrm{a}(t)|^2 \, dt</math>

or another alternative (for some optimum <math>\theta \,</math>)

<math>\int_{-\infty}^{+\infty} \left( \phi(t)-(\omega_0 t + \theta) \right)^2\, dt</math>.

Extensions of the analytic signal to signals of multiple variables

The concept of analytic signal is well-defined for signals of a single variable which typically is time. For signals of two or more variables, an analytic signal can be defined in different ways, and two approaches are presented below

Multi-dimensional analytic signal based on an ad-hoc direction

A straightforward generalization of the analytic signal can be done for a multi-dimensional signal once it is established what is meant by negative frequencies for this case. This can be done by introducing a normalized vector <math>\hat{n}</math> in the Fourier domain and label any frequency vector <math>u</math> as negative if <math>u \cdot \hat{n} < 0</math>. The analytic signal is then produced by removing all negative frequencies and multiply the result by <math>2</math>, in accordance to the procedure described for the case of one-variable signals. However, it should be noted that there is no particular direction for <math>\hat{n}</math> which must be chosen unless there are some additional constraints. Therefore, the choice of <math>\hat{n}</math> is ad-hoc, or application specific.

The monogenic signal

The real and imaginary parts of the analytic signal correspond to the two elements of the vector-valued monogenic signal, as it is defined for one-variable signals. However, the monogenic signal can be extended to arbitrary number of variables in a straightforward manner, producing an <math>n + 1</math> dimensional vector-valued function for the case of <math>n</math> variable signals.

Not to be confused with

In mathematics, particularly functions of a complex variable, an analytic function refers to a function which satisfies certain properties regarding differentiability. The concept of analytic signal should not be confused with analytic functions. See the analytic representation related to the Hilbert transform for a discussion about the relation between these two concepts.