Analytic signal#Complex envelope

File:Analytisches-signal-uebertragungsfunktion-en.svg

If $s(t)$ is a real-valued function with Fourier transform $S(f)$ (where $f$ is the real value denoting frequency), then the transform has Hermitian symmetry about the $f\; =\; 0$ axis:

:$S(-f)\; =\; S(f)^*,$

where $S(f)^*$ is the complex conjugate of $S(f)$.

The function:

:$$

\begin{align}

S_\mathrm{a}(f) &\triangleq

\begin{cases}

2S(f), &\text{for}\ f > 0,\\

S(f), &\text{for}\ f = 0,\\

0, &\text{for}\ f < 0

\end{cases}\\

&= \underbrace{2 \operatorname{u}(f)}_{1 + \sgn(f)}S(f) = S(f) + \sgn(f)S(f),

\end{align}

where

• $\backslash operatorname\{u\}(f)$ is the Heaviside step function,
• $\backslash sgn(f)$ is the sign function,

contains only the non-negative frequency components of $S(f)$. And the operation is reversible, due to the Hermitian symmetry of $S(f)$:

:$$

\begin{align}

S(f) &=

\begin{cases}

\frac{1}{2}S_\mathrm{a}(f), &\text{for}\ f > 0,\\

S_\mathrm{a}(f), &\text{for}\ f = 0,\\

\frac{1}{2}S_\mathrm{a}(-f)^*, &\text{for}\ f < 0\ \text{(Hermitian symmetry)}

\end{cases}\\

&= \frac{1}{2}[S_\mathrm{a}(f) + S_\mathrm{a}(-f)^*].

\end{align}

The analytic signal of $s(t)$ is the inverse Fourier transform of $S\_\backslash mathrm\{a\}(f)$:

:$\backslash begin\{align\}$

s_\mathrm{a}(t) &\triangleq \mathcal{F}^{-1}[S_\mathrm{a}(f)]\\

&= \mathcal{F}^{-1}[S (f)+ \sgn(f) \cdot S(f)]\\

&= \underbrace{\mathcal{F}^{-1}\{S(f)\}}_{s(t)} + \overbrace{

\underbrace{\mathcal{F}^{-1}\{\sgn(f)\}}_{j\frac{1}{\pi t}} * \underbrace{\mathcal{F}^{-1}\{S(f)\}}_{s(t)}

}^\text{convolution}\\

&= s(t) + j\underbrace{\left[{1 \over \pi t} * s(t)\right]}_{\operatorname{\mathcal{H}}[s(t)]}\\

&= s(t) + j\hat{s}(t),

\end{align}

where

• $\backslash hat\{s\}(t)\; \backslash triangleq\; \backslash operatorname\{\backslash mathcal\{H\}\}[s(t)]$ is the Hilbert transform of $s(t)$;
• $*$ is the binary convolution operator;
• $j$ is the imaginary unit.

Noting that $s(t)=\; s(t)*\backslash delta(t),$ this can also be expressed as a filtering operation that directly removes negative frequency components:

:$s\_\backslash mathrm\{a\}(t)\; =\; s(t)*\backslash underbrace\{\backslash left[\backslash delta(t)+\; j\{1\; \backslash over\; \backslash pi\; t\}\backslash right]\}\_\{\backslash mathcal\{F\}^\{-1\}\backslash \{2u(f)\backslash \}\}.$

Since $s(t)\; =\; \backslash operatorname\{Re\}[s\_\backslash mathrm\{a\}(t)]$, restoring the negative frequency components is a simple matter of discarding $\backslash operatorname\{Im\}[s\_\backslash mathrm\{a\}(t)]$ which may seem counter-intuitive. The complex conjugate $s\_\backslash mathrm\{a\}^*(t)$ comprises only the negative frequency components. And therefore $s(t)\; =\; \backslash operatorname\{Re\}[s\_\backslash mathrm\{a\}^*(t)]$ restores the suppressed positive frequency components. Another viewpoint is that the imaginary component in either case is a term that subtracts frequency components from $s(t).$ The $\backslash operatorname\{Re\}$ operator removes the subtraction, giving the appearance of adding new components.

:$s(t)\; =\; \backslash cos(\backslash omega\; t),$   where  $\backslash omega\; >\; 0.$

Then:

:$\backslash begin\{align\}$

\hat{s}(t) &= \cos\left(\omega t - \frac{\pi}{2}\right) = \sin(\omega t), \\

s_\mathrm{a}(t) &= s(t) + j\hat{s}(t) = \cos(\omega t) + j\sin(\omega t) = e^{j\omega t}.

\end{align}

The last equality is Euler's formula, of which a corollary is $\backslash cos(\backslash omega\; t)\; =\; \backslash frac\{1\}\{2\}\; \backslash left(e^\{j\backslash omega\; t\}\; +\; e^\{j\; (-\backslash omega)\; t\}\backslash right).$ In general, the analytic representation of a simple sinusoid is obtained by expressing it in terms of complex-exponentials, discarding the negative frequency component, and doubling the positive frequency component. And the analytic representation of a sum of sinusoids is the sum of the analytic representations of the individual sinusoids.

Here we use Euler's formula to identify and discard the negative frequency.

:$s(t)\; =\; \backslash cos(\backslash omega\; t\; +\; \backslash theta)\; =\; \backslash frac\{1\}\{2\}\; \backslash left(e^\{j\; (\backslash omega\; t+\backslash theta)\}\; +\; e^\{-j\; (\backslash omega\; t+\backslash theta)\}\backslash right)$

Then:

:$s\_\backslash mathrm\{a\}(t)\; =$

\begin{cases}

e^{j(\omega t + \theta)} \ \ = \ e^{j |\omega| t}\cdot e^{j\theta} , & \text{if} \ \omega > 0, \\

e^{-j(\omega t + \theta)} = \ e^{j |\omega| t}\cdot e^{-j\theta} , & \text{if} \ \omega < 0.

\end{cases}

This is another example of using the Hilbert transform method to remove negative frequency components. Nothing prevents us from computing $s\_\backslash mathrm\{a\}(t)$ for a complex-valued $s(t)$. 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 $s(t)$.

:$s(t)\; =\; e^\{-j\backslash omega\; t\}$, where $\backslash omega\; >\; 0$.

Then:

:$\backslash begin\{align\}$

\hat{s}(t) &= je^{-j\omega t}, \\

s_\mathrm{a}(t) &= e^{-j\omega t} + j^2 e^{-j\omega t} = e^{-j\omega t} - e^{-j\omega t} = 0.

\end{align}

Image:analytic.svg

An analytic signal can also be expressed in polar coordinates:

:$s\_\backslash mathrm\{a\}(t)\; =\; s\_\backslash mathrm\{m\}(t)e^\{j\backslash phi(t)\},$

where the following time-variant quantities are introduced:

• $s\_\backslash mathrm\{m\}(t)\; \backslash triangleq\; |s\_\backslash mathrm\{a\}(t)|$ is called the instantaneous amplitude or the envelope;
• $\backslash phi(t)\; \backslash triangleq\; \backslash arg\backslash !\backslash left[s\_\backslash mathrm\{a\}(t)\backslash right]$ is called the instantaneous phase or phase angle.

In the accompanying diagram, the blue curve depicts $s(t)$ and the red curve depicts the corresponding $s\_\backslash mathrm\{m\}(t)$.

The time derivative of the unwrapped instantaneous phase has units of radians/second, and is called the instantaneous angular frequency:

:$\backslash omega(t)\; \backslash triangleq\; \backslash frac\{d\backslash phi\}\{dt\}(t).$

The instantaneous frequency (in hertz) is therefore:

:$f(t)\backslash triangleq\; \backslash frac\{1\}\{2\backslash pi\}\backslash omega(t).$  B. Boashash, "Estimating and Interpreting the Instantaneous Frequency of a Signal-Part I: Fundamentals", Proceedings of the IEEE, Vol. 80, No. 4, pp. 519–538, April 1992

The instantaneous amplitude, 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.

Analytic signals are often shifted in frequency (down-converted) toward 0 Hz, possibly creating [non-symmetrical] negative frequency components:

$$\{s\_\backslash mathrm\{a\}\}\_\{\backslash downarrow\}(t)\; \backslash triangleq\; s\_\backslash mathrm\{a\}(t)e^\{-j\backslash omega\_0\; t\}\; =\; s\_\backslash mathrm\{m\}(t)e^\{j(\backslash phi(t)\; -\; \backslash omega\_0\; t)\},$$

where $\backslash omega\_0$ is an arbitrary reference angular frequency.

This function goes by various names, such as complex envelope and complex baseband. The complex envelope is not unique; it is determined by the choice of $\backslash omega\_0$. This concept is often used when dealing with passband signals. If $s(t)$ is a modulated signal, $\backslash omega\_0$ might be equated to its carrier frequency.

In other cases, $\backslash omega\_0$ is selected to be somewhere in the middle of the desired passband. Then a simple low-pass filter with real coefficients can excise the portion of interest. 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 may be required, and if the signal has been sampled (discrete-time), interpolation (upsampling) might also be necessary to avoid aliasing.

If $\backslash omega\_0$ is chosen larger than the highest frequency of $s\_\backslash mathrm\{a\}(t),$ then $\{s\_\backslash mathrm\{a\}\}\_\{\backslash downarrow\}(t)$ has no positive frequencies. In that case, extracting the real component restores them, but in reverse order; the low-frequency components are now high ones and vice versa. This can be used to demodulate a type of single-sideband signal called lower sideband or inverted sideband.

Other choices of reference frequency are sometimes considered:

• Sometimes $\backslash omega\_0$ is chosen to minimize $$\backslash int\_0^\{+\backslash infty\}(\backslash omega\; -\; \backslash omega\_0)^2|S\_\backslash mathrm\{a\}(\backslash omega)|^2\backslash ,\; d\backslash omega.$$
• Alternatively,{{Cite journal|last=Justice|first=J.|date=1979-12-01|title=Analytic signal processing in music computation| journal=IEEE Transactions on Acoustics, Speech, and Signal Processing|volume=27|issue=6|pages=670–684| doi=10.1109/TASSP.1979.1163321| issn=0096-3518}} $\backslash omega\_0$ can be chosen to minimize the mean square error in linearly approximating the unwrapped instantaneous phase $\backslash phi(t)$: $$\backslash int\_\{-\backslash infty\}^\{+\backslash infty\}[\backslash omega(t)\; -\; \backslash omega\_0]^2\; |s\_\backslash mathrm\{a\}(t)|^2\backslash ,\; dt$$
• or another alternative (for some optimum $\backslash theta$): $$\backslash int\_\{-\backslash infty\}^\{+\backslash infty\}[\backslash phi(t)\; -\; (\backslash omega\_0\; t\; +\; \backslash theta)]^2\backslash ,\; dt.$$

In the field of time-frequency signal processing, it was shown that the analytic signal was needed in the definition of the Wigner–Ville distribution so that the method can have the desirable properties needed for practical applications.B. Boashash, "Notes on the use of the Wigner distribution for time frequency signal analysis", IEEE Trans. on Acoustics, Speech, and Signal Processing, vol. 26, no. 9, 1987

Sometimes the phrase "complex envelope" is given the simpler meaning of the complex amplitude of a (constant-frequency) phasor;{{efn|"the complex envelope (or complex amplitude)"{{Cite book|url=https://books.google.com/books?id=tOeeJyP95IQC |title=Time-Frequency Analysis|last1=Hlawatsch|first1=Franz|last2=Auger|first2=François|date=2013-03-01|publisher=John Wiley & Sons |isbn=9781118623831|language=en}}}}{{efn|"the complex envelope (or complex amplitude)", p. 586 {{Cite book |url=https://books.google.com/books?id=kxICp6t-CDAC&q=%2522complex%2520amplitude%2522%2520%2522complex%2520envelope%2522&pg=RA1-PA586 |title=Encyclopedia of Optical Engineering: Abe-Las, pages 1-1024|last=Driggers|first=Ronald G.|date=2003-01-01 |publisher=CRC Press|isbn=9780824742508|language=en}}}}

other times the complex envelope $s\_m(t)$ as defined above is interpreted as a time-dependent generalization of the complex amplitude.{{efn|"Complex envelope is an extended interpretation of complex amplitude as a function of time." p. 85{{Cite book|url=https://books.google.com/books?id=tXQy5JdQyZoC|title=Global Environment Remote Sensing| last=Okamoto|first=Kenʼichi|date=2001-01-01|publisher=IOS Press|isbn=9781586031015|language=en}}}} Their relationship is not unlike that in the real-valued case: varying envelope generalizing constant amplitude.

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.

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 unit vector $\backslash boldsymbol\; \backslash hat\{u\}$ in the Fourier domain and label any frequency vector $\backslash boldsymbol\; \backslash xi$ as negative if . 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, there is no particular direction for $\backslash boldsymbol\; \backslash hat\{u\}$ which must be chosen unless there are some additional constraints. Therefore, the choice of $\backslash boldsymbol\; \backslash hat\{u\}$ is ad hoc, or application specific.

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 {{nowrap|(n + 1)}}-dimensional vector-valued function for the case of n-variable signals.

• Practical considerations for computing Hilbert transforms
• I/Q data
• Negative frequency

• Single-sideband modulation
• Quadrature filter
• Causal filter

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{{further reading cleanup|date=October 2014}}

• Leon Cohen, Time-frequency analysis, Prentice Hall, Upper Saddle River, 1995.
• Frederick W. King, Hilbert Transforms, vol. II, Cambridge University Press, Cambridge, 2009.
• B. Boashash, Time-Frequency Signal Analysis and Processing: A Comprehensive Reference, Elsevier Science, Oxford, 2003.

• [http://www.dsprelated.com/freebooks/mdft/Analytic_Signals_Hilbert_Transform.html Analytic Signals and Hilbert Transform Filters]

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Category:Signal processing

Category:Time–frequency analysis

Category:Fourier analysis