Analytic signal

In signal processing, the analytic representation of signal $s(t)\,$ is defined by: $s_{\mathrm {a} }(t)=s(t)+i\cdot {\hat {s}}(t)\,$

where ${\hat {s}}(t)\,$ is the Hilbert transform of $s(t)\,$ and $i\,$ is the imaginary unit.

Example 1: $s(t)=\cos(\omega _{0}t)\,$, for some parameter $\omega _{0}>0\,$

${\hat {s}}(t)=\cos(\omega _{0}t-{\begin{matrix}{\frac {\pi }{2}}\end{matrix}})=\sin(\omega _{0}t)\,$

$s_{\mathrm {a} }(t)=\cos(\omega _{0}t)+i\cdot \sin(\omega _{0}t)=e^{i\omega _{0}t}\,$ (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 $\cos(\omega t)={\begin{matrix}{\frac {1}{2}}\end{matrix}}(e^{i\omega t}+e^{-i\omega t})\,$. So $s(t)\,$ comprises both positive and negative frequency components. $s_{\mathrm {a} }(t)\,$ is just the positive portion.

Example 2: $s(t)=\cos(\omega _{1}t)+\cos(\omega _{2}t)\,$

${\hat {s}}(t)=\sin(|\omega _{1}|t)+sin(|\omega _{2}|t)\,$

$s_{\mathrm {a} }(t)=e^{i|\omega _{1}|t}+e^{i|\omega _{2}|t}\,$

The definition can also be expressed in terms of the Fourier transforms of $s(t)\,$ and $s_{\mathrm {a} }(t)\,$, respectively denoted by $S(\omega )\,$ and $S_{\mathrm {a} }(\omega )\,$:

$S_{\mathrm {a} }(\omega )\,$	$=2S(\omega )\,$, for $\omega >0\,$
	$=0\,$, for $\omega <0\,$

So it is a general result that $s_{\mathrm {a} }(t)\,$ comprises only the positive frequency components of $s(t)\,$.

Yet the analytic transforms are reversible for real-valued $s(t)\,$. I.e., the original functions can be recovered:

$S(-\omega )={\overline {S(\omega )}}\,$ ( Hermitian property )

and $s(t)=Re\{s_{\mathrm {a} }(t)\}\,$.

An $s_{\mathrm {a} }(t)\,$ can also be formed for a complex-valued $s(t)\,$, but that is not seen in practice.

Example 3: $s(t)=e^{-i\omega _{0}t}\,$, for some parameter $\omega _{0}>0\,$

${\hat {s}}(t)=e^{-i(\omega _{0}t-{\begin{matrix}{\frac {\pi }{2}}\end{matrix}})}=i\cdot e^{-i\omega _{0}t}\,$

$s_{\mathrm {a} }(t)=e^{-i\omega _{0}t}+i^{2}\cdot e^{-i\omega _{0}t}=0\,$

Polar coordinates

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

A complex function can also be expressed in terms of polar coordinates, $s_{\mathrm {a} }(t)=A(t)e^{i\phi (t)}\,$, where:

• $A(t)=|s_{\mathrm {a} }(t)|={\sqrt {s^{2}(t)+{\hat {s}}^{2}(t)}}\,$

• $\phi (t)=\arg(s_{\mathrm {a} }(t))=\arctan({{\hat {s}}(t) \over s(t)})\,$

These functions are respectively called the amplitude envelope and instantaneous phase of the real-valued signal, $s(t)\,$. In the accompanying diagram, the blue curve depicts $s(t)\,$, and the red curve depicts the corresponding $A(t)\,$.
The time derivative of the instantaneous phase is called the instantaneous frequency:

• $\omega _{\mathrm {inst} }(t)=\phi '(t)={d \over dt}\phi (t)\,$

So polar coordinates conveniently separate the effects of amplitude modulation and phase (or frequency) modulation. The polar coordinate transformation effectively demodulates certain kinds of signals. Phase-modulated signals are sometimes referred to as constant envelope modulation.

Practical implementations

The analytic signal can also be written with the convolution operator as:

$s_{\mathrm {a} }(t)=s(t)+i\left({1 \over \pi t}*s(t)\right)=\left(\delta (t)+i{1 \over \pi t}\right)*s(t)$

The filter impulse-response

$\delta (t)+i{1 \over \pi t}$

is non-causal and therefore cannot be implemented in practice if $s(t)\,$ is a time-dependent signal. On the other hand, if $s(t)\,$ is a function of a non-temporal variable, e.g., spatial, the non-causality may not be a problem. The filter is also of infinite support which may be a problem in certain applications. Another issue relates to what happens with the zero frequency (DC), which can be solved by assuring that $s(t)\,$ does not contain any DC-component.

Consequently, a practical computation of the analytic signal in many cases implies that a finite support filter, which in addition is made causal by means of a suitable delay, is used to approximate the computation. The approximation may also imply that only a specific range of negative frequencies is subject to removal. See also quadrature filter.

In the case of discrete signals of finite length, the analytic signal can be computed by taking a discrete frequency DFT, setting all negative frequency components to zero, and finally doing an inverse DFT. However, this operation may not produce the expected result at the boundaries of the signal since the effect of this operation is a cyclic convolution with the desired filter.

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 ${\hat {n}}$ in the Fourier domain and label any frequency vector $u$ as negative if $u\cdot {\hat {n}}<0$. The analytic signal is then produced by removing all negative frequencies and multiply the result by $2$, 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 ${\hat {n}}$ which must be chosen unless there are some additional constraints. Therefore, the choice of ${\hat {n}}$ 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 $n+1$ dimensional vector-valued function for the case of $n$ variable signals.

Not to be confused with

In mathematics an analytic function refers to a function which satisfies certain properties related to differentiability. The concept of analytic signal should not be confused with analytic functions.

See also

• Negative frequency (background information)
• Hilbert transform (background information)

• Single sideband modulation (application)
• Quadrature filter (application)
• Causal filter (application)

• Monogenic signal (extension)

References

• Bracewell, R; The Fourier Transform and Its Applications, 2nd ed, 1986, McGraw-Hill.
• Leon Cohen, "Time-frequency analysis", Prentice-Hall (1995)

External links

• Analytic Signals and Hilbert Transform Filters