Analytic signal

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 where • \operatorname{u}(f) is the Heaviside step function, • \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 The analytic signal of s(t) is the inverse Fourier transform of S_\mathrm{a}(f): :\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 • \hat{s}(t) \triangleq \operatorname{\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)*\delta(t), this can also be expressed as a filtering operation that directly removes negative frequency components: :s_\mathrm{a}(t) = s(t)*\underbrace{\left[\delta(t)+ j{1 \over \pi t}\right]}_{\mathcal{F}^{-1}\{2u(f)\}}. Negative frequency components Since s(t) = \operatorname{Re}[s_\mathrm{a}(t)], restoring the negative frequency components is a simple matter of discarding \operatorname{Im}[s_\mathrm{a}(t)] which may seem counter-intuitive. The complex conjugate s_\mathrm{a}^*(t) comprises only the negative frequency components. And therefore s(t) = \operatorname{Re}[s_\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 \operatorname{Re} operator removes the subtraction, giving the appearance of adding new components. ==Examples==

Example 1 :s(t) = \cos(\omega t),   where  \omega > 0. Then: :\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 \cos(\omega t) = \frac{1}{2} \left(e^{j\omega t} + e^{j (-\omega) t}\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. Example 2 Here we use Euler's formula to identify and discard the negative frequency. :s(t) = \cos(\omega t + \theta) = \frac{1}{2} \left(e^{j (\omega t+\theta)} + e^{-j (\omega t+\theta)}\right) Then: :s_\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 Example 3 This is another example of using the Hilbert transform method to remove negative frequency components. Nothing prevents us from computing s_\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\omega t}, where \omega > 0. Then: :\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} ==Properties==

Instantaneous amplitude and phase An analytic signal can also be expressed in polar coordinates: :s_\mathrm{a}(t) = s_\mathrm{m}(t)e^{j\phi(t)}, where the following time-variant quantities are introduced: • s_\mathrm{m}(t) \triangleq |s_\mathrm{a}(t)| is called the instantaneous amplitude or the envelope; • \phi(t) \triangleq \arg\!\left[s_\mathrm{a}(t)\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_\mathrm{m}(t). The time derivative of the unwrapped instantaneous phase has units of radians/second, and is called the instantaneous angular frequency: :\omega(t) \triangleq \frac{d\phi}{dt}(t). The instantaneous frequency (in hertz) is therefore: :f(t)\triangleq \frac{1}{2\pi}\omega(t).   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. Complex envelope/baseband Analytic signals are often shifted in frequency (down-converted) toward 0 Hz, possibly creating [non-symmetrical] negative frequency components: {s_\mathrm{a}}_{\downarrow}(t) \triangleq s_\mathrm{a}(t)e^{-j\omega_0 t} = s_\mathrm{m}(t)e^{j(\phi(t) - \omega_0 t)}, where \omega_0 is an arbitrary reference angular frequency. \omega_0 can be chosen to minimize the mean square error in linearly approximating the unwrapped instantaneous phase \phi(t): \int_{-\infty}^{+\infty}[\omega(t) - \omega_0]^2 |s_\mathrm{a}(t)|^2\, dt • or another alternative (for some optimum \theta): \int_{-\infty}^{+\infty}[\phi(t) - (\omega_0 t + \theta)]^2\, 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. Sometimes the phrase "complex envelope" is given the simpler meaning of the complex amplitude of a (constant-frequency) phasor; other times the complex envelope s_m(t) as defined above is interpreted as a time-dependent generalization of the complex amplitude. Their relationship is not unlike that in the real-valued case: varying envelope generalizing constant amplitude. ==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 unit vector \boldsymbol \hat{u} in the Fourier domain and label any frequency vector \boldsymbol \xi as negative if \boldsymbol \xi \cdot \boldsymbol \hat{u} . 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 \boldsymbol \hat{u} which must be chosen unless there are some additional constraints. Therefore, the choice of \boldsymbol \hat{u} 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 -dimensional vector-valued function for the case of n-variable signals. ==See also==