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Analytic signal
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<h2 id="definition">Definition</h2> <p>If s ( t ) {\displaystyle s(t)} is a <i>real-valued</i> function with Fourier transform S ( f ) {\displaystyle S(f)} (where f {\displaystyle f} is the real value denoting frequency), then the transform has <a href="/facts/Hermitian_function/3U40nuaK">Hermitian</a> symmetry about the f = 0 {\displaystyle f=0} axis: </p> S ( − f ) = S ( f ) ∗ , {\displaystyle S(-f)=S(f)^{*},} <p>where S ( f ) ∗ {\displaystyle S(f)^{*}} is the <a href="/facts/Complex_conjugate/7TEaoQDe">complex conjugate</a> of S ( f ) {\displaystyle S(f)} . The function: </p> S a ( f ) ≜ { 2 S ( f ) , for f > 0 , S ( f ) , for f = 0 , 0 , for f < 0 = 2 u ( f ) ⏟ 1 + sgn ( f ) S ( f ) = S ( f ) + sgn ( f ) S ( f ) , {\displaystyle {\begin{aligned}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+\operatorname {sgn}(f)}S(f)=S(f)+\operatorname {sgn}(f)S(f),\end{aligned}}} <p>where </p> <ul><li> u ( f ) {\displaystyle \operatorname {u} (f)} is the <a href="/facts/Heaviside_step_function/bgsUtxgP">Heaviside step function</a>,</li> <li> sgn ( f ) {\displaystyle \operatorname {sgn}(f)} is the <a href="/facts/Sign_function/VADy9zQq">sign function</a>,</li></ul> <p>contains only the <i>non-negative frequency</i> components of S ( f ) {\displaystyle S(f)} . And the operation is reversible, due to the Hermitian symmetry of S ( f ) {\displaystyle S(f)} : </p> S ( f ) = { 1 2 S a ( f ) , for f > 0 , S a ( f ) , for f = 0 , 1 2 S a ( − f ) ∗ , for f < 0 (Hermitian symmetry) = 1 2 [ S a ( f ) + S a ( − f ) ∗ ] . {\displaystyle {\begin{aligned}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{aligned}}} <p>The analytic signal of s ( t ) {\displaystyle s(t)} is the inverse Fourier transform of S a ( f ) {\displaystyle S_{\mathrm {a} }(f)} : </p> s a ( t ) ≜ F − 1 [ S a ( f ) ] = F − 1 [ S ( f ) + sgn ( f ) ⋅ S ( f ) ] = F − 1 { S ( f ) } ⏟ s ( t ) + F − 1 { sgn ( f ) } ⏟ j 1 π t ∗ F − 1 { S ( f ) } ⏟ s ( t ) ⏞ convolution = s ( t ) + j [ 1 π t ∗ s ( t ) ] ⏟ H [ s ( t ) ] = s ( t ) + j s ^ ( t ) , {\displaystyle {\begin{aligned}s_{\mathrm {a} }(t)&\triangleq {\mathcal {F}}^{-1}[S_{\mathrm {a} }(f)]\\&={\mathcal {F}}^{-1}[S(f)+\operatorname {sgn}(f)\cdot S(f)]\\&=\underbrace {{\mathcal {F}}^{-1}\{S(f)\}} _{s(t)}+\overbrace {\underbrace {{\mathcal {F}}^{-1}\{\operatorname {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{aligned}}} <p>where </p> <ul><li> s ^ ( t ) ≜ H [ s ( t ) ] {\displaystyle {\hat {s}}(t)\triangleq \operatorname {\mathcal {H}} [s(t)]} is the <a href="/facts/Hilbert_transform/RoQaC8MS">Hilbert transform</a> of s ( t ) {\displaystyle s(t)} ;</li> <li> ∗ {\displaystyle *} is the binary <a href="/facts/Convolution/PVPrdz9J">convolution</a> operator;</li> <li> j {\displaystyle j} is the <a href="/facts/Imaginary_unit/1SCllQlU">imaginary unit</a>.</li></ul> <p>Noting that s ( t ) = s ( t ) ∗ δ ( t ) , {\displaystyle s(t)=s(t)*\delta (t),} this can also be expressed as a filtering operation that directly removes negative frequency components: </p> s a ( t ) = s ( t ) ∗ [ δ ( t ) + j 1 π t ] ⏟ F − 1 { 2 u ( f ) } . {\displaystyle s_{\mathrm {a} }(t)=s(t)*\underbrace {\left[\delta (t)+j{1 \over \pi t}\right]} _{{\mathcal {F}}^{-1}\{2u(f)\}}.} <h3>Negative frequency components</h3> <p>Since s ( t ) = Re [ s a ( t ) ] {\displaystyle s(t)=\operatorname {Re} [s_{\mathrm {a} }(t)]} , restoring the negative frequency components is a simple matter of discarding Im [ s a ( t ) ] {\displaystyle \operatorname {Im} [s_{\mathrm {a} }(t)]} which may seem counter-intuitive. The complex conjugate s a ∗ ( t ) {\displaystyle s_{\mathrm {a} }^{*}(t)} comprises <i>only</i> the negative frequency components. And therefore s ( t ) = Re [ s a ∗ ( t ) ] {\displaystyle 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 ) . {\displaystyle s(t).} The Re {\displaystyle \operatorname {Re} } operator removes the subtraction, giving the appearance of adding new components. </p> <h2 id="examples">Examples</h2> <h3>Example 1</h3> s ( t ) = cos ( ω t ) , {\displaystyle s(t)=\cos(\omega t),} where ω > 0. {\displaystyle \omega >0.} <p>Then: </p> s ^ ( t ) = cos ( ω t − π 2 ) = sin ( ω t ) , s a ( t ) = s ( t ) + j s ^ ( t ) = cos ( ω t ) + j sin ( ω t ) = e j ω t . {\displaystyle {\begin{aligned}{\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{aligned}}} <p>The last equality is <a href="/facts/Euler%2527s_formula/Wq7VXIV0">Euler's formula</a>, of which a <a href="/facts/Corollary/ntQ8zR9N">corollary</a> is cos ( ω t ) = 1 2 ( e j ω t + e j ( − ω ) t ) . {\textstyle \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 <a href="/facts/Negative_frequency/YasYp0L0">negative frequency</a> 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. </p> <h3>Example 2</h3> <p>Here we use Euler's formula to identify and discard the negative frequency. </p> s ( t ) = cos ( ω t + θ ) = 1 2 ( e j ( ω t + θ ) + e − j ( ω t + θ ) ) {\displaystyle s(t)=\cos(\omega t+\theta )={\frac {1}{2}}\left(e^{j(\omega t+\theta )}+e^{-j(\omega t+\theta )}\right)} <p>Then: </p> s a ( t ) = { e j ( ω t + θ ) = e j | ω | t ⋅ e j θ , if ω > 0 , e − j ( ω t + θ ) = e j | ω | t ⋅ e − j θ , if ω < 0. {\displaystyle 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 <0.\end{cases}}} <h3>Example 3</h3> <p>This is another example of using the Hilbert transform method to remove negative frequency components. Nothing prevents us from computing s a ( t ) {\displaystyle s_{\mathrm {a} }(t)} for a complex-valued s ( t ) {\displaystyle 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 ) {\displaystyle s(t)} . </p> s ( t ) = e − j ω t {\displaystyle s(t)=e^{-j\omega t}} , where ω > 0 {\displaystyle \omega >0} . <p>Then: </p> s ^ ( t ) = j e − j ω t , s a ( t ) = e − j ω t + j 2 e − j ω t = e − j ω t − e − j ω t = 0. {\displaystyle {\begin{aligned}{\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{aligned}}} <h2 id="properties">Properties</h2> <h3>Instantaneous amplitude and phase</h3> <p>An analytic signal can also be expressed in <a href="/facts/Polar_coordinates/HBko4YoN">polar coordinates</a>: </p> s a ( t ) = s m ( t ) e j ϕ ( t ) , {\displaystyle s_{\mathrm {a} }(t)=s_{\mathrm {m} }(t)e^{j\phi (t)},} <p>where the following time-variant quantities are introduced: </p> <ul><li> s m ( t ) ≜ | s a ( t ) | {\displaystyle s_{\mathrm {m} }(t)\triangleq |s_{\mathrm {a} }(t)|} is called the <i>instantaneous amplitude</i> or the <i><a href="/facts/Envelope_(waves)/3JsLqPoh">envelope</a></i>;</li> <li> ϕ ( t ) ≜ arg [ s a ( t ) ] {\displaystyle \phi (t)\triangleq \arg \!\left[s_{\mathrm {a} }(t)\right]} is called the <i><a href="/facts/Instantaneous_phase/fJ17zuan">instantaneous phase</a></i> or <i>phase angle</i>.</li></ul> <p>In the accompanying diagram, the blue curve depicts s ( t ) {\displaystyle s(t)} and the red curve depicts the corresponding s m ( t ) {\displaystyle s_{\mathrm {m} }(t)} . </p><p>The time derivative of the <a href="/facts/Phase_wrapping/fJ17zuan">unwrapped</a> instantaneous phase has units of <i>radians/second</i>, and is called the <i>instantaneous angular frequency</i>: </p> ω ( t ) ≜ d ϕ d t ( t ) . {\displaystyle \omega (t)\triangleq {\frac {d\phi }{dt}}(t).} <p>The <i><a href="/facts/Instantaneous_phase/fJ17zuan">instantaneous frequency</a></i> (in <a href="/facts/Hertz/6Wshj5qm">hertz</a>) is therefore: </p> f ( t ) ≜ 1 2 π ω ( t ) . {\displaystyle f(t)\triangleq {\frac {1}{2\pi }}\omega (t).} <a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> <p>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 <a href="/facts/Modulation/aAXPjINv">modulated signals</a>. The polar coordinates conveniently separate the effects of <a href="/facts/Amplitude_modulation/3sTxsXk6">amplitude modulation</a> and phase (or frequency) modulation, and effectively demodulates certain kinds of signals. </p> <h3> Complex envelope/baseband</h3> <p>Analytic signals are often shifted in frequency (down-converted) toward 0 Hz, possibly creating [non-symmetrical] negative frequency components: s a ↓ ( t ) ≜ s a ( t ) e − j ω 0 t = s m ( t ) e j ( ϕ ( t ) − ω 0 t ) , {\displaystyle {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 ω 0 {\displaystyle \omega _{0}} is an arbitrary reference angular frequency.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p><p>This function goes by various names, such as <i>complex envelope</i> and <i>complex <a href="/facts/Baseband/yC0jJwbL">baseband</a></i>. The complex envelope is not unique; it is determined by the choice of ω 0 {\displaystyle \omega _{0}} . This concept is often used when dealing with <a href="/facts/Passband_signal/VebMrazm">passband signals</a>. If s ( t ) {\displaystyle s(t)} is a modulated signal, ω 0 {\displaystyle \omega _{0}} might be equated to its <a href="/facts/Carrier_frequency/ae98pBlY">carrier frequency</a>. </p><p>In other cases, ω 0 {\displaystyle \omega _{0}} is selected to be somewhere in the middle of the desired passband. Then a simple <a href="/facts/Low-pass_filter/QFsUjw7h">low-pass filter</a> 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 <i>analytic</i>. 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 <a href="/facts/Sampling_(signal_processing)/xxck7aLN">sampled</a> (discrete-time), <a href="/facts/Interpolation/8PyetX4f">interpolation</a> (<a href="/facts/Upsampling/ndAk3IxG">upsampling</a>) might also be necessary to avoid <a href="/facts/Aliasing/Pzg84HL0">aliasing</a>. </p><p>If ω 0 {\displaystyle \omega _{0}} is chosen larger than the highest frequency of s a ( t ) , {\displaystyle s_{\mathrm {a} }(t),} then s a ↓ ( t ) {\displaystyle {s_{\mathrm {a} }}_{\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 <a href="/facts/Single-sideband_modulation/Cfuv9KX1">single-sideband</a> signal called <i>lower sideband</i> or <i>inverted sideband</i>. </p><p>Other choices of reference frequency are sometimes considered: </p> <ul><li>Sometimes ω 0 {\displaystyle \omega _{0}} is chosen to minimize ∫ 0 + ∞ ( ω − ω 0 ) 2 | S a ( ω ) | 2 d ω . {\displaystyle \int _{0}^{+\infty }(\omega -\omega _{0})^{2}|S_{\mathrm {a} }(\omega )|^{2}\,d\omega .} </li> <li>Alternatively,<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> ω 0 {\displaystyle \omega _{0}} can be chosen to minimize the mean square error in linearly approximating the <i>unwrapped</i> instantaneous phase ϕ ( t ) {\displaystyle \phi (t)} : ∫ − ∞ + ∞ [ ω ( t ) − ω 0 ] 2 | s a ( t ) | 2 d t {\displaystyle \int _{-\infty }^{+\infty }[\omega (t)-\omega _{0}]^{2}|s_{\mathrm {a} }(t)|^{2}\,dt} </li> <li>or another alternative (for some optimum θ {\displaystyle \theta } ): ∫ − ∞ + ∞ [ ϕ ( t ) − ( ω 0 t + θ ) ] 2 d t . {\displaystyle \int _{-\infty }^{+\infty }[\phi (t)-(\omega _{0}t+\theta )]^{2}\,dt.} </li></ul> <p>In the field of time-frequency signal processing, it was shown that the analytic signal was needed in the definition of the <a href="/facts/Wigner%25E2%2580%2593Ville_distribution/5tyVfdFF">Wigner–Ville distribution</a> so that the method can have the desirable properties needed for practical applications.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p><p>Sometimes the phrase "complex envelope" is given the simpler meaning of the <a href="/facts/Complex_amplitude/LyjSSUDo">complex amplitude</a> of a (constant-frequency) phasor;<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a><a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> other times the complex envelope s m ( t ) {\displaystyle s_{m}(t)} as defined above is interpreted as a time-dependent generalization of the complex amplitude.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> Their relationship is not unlike that in the real-valued case: varying <a href="/facts/Envelope_(waves)/3JsLqPoh">envelope</a> generalizing constant <a href="/facts/Amplitude/aEckuXJM">amplitude</a>. </p> <h2 id="extensions-of-the-analytic-signal-to-signals-of-multiple-variables">Extensions of the analytic signal to signals of multiple variables</h2> <p>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. </p> <h3>Multi-dimensional analytic signal based on an ad hoc direction</h3> <p>A straightforward generalization of the analytic signal can be done for a multi-dimensional signal once it is established what is meant by <i>negative frequencies</i> for this case. This can be done by introducing a <a href="/facts/Unit_vector/GyaXneDn">unit vector</a> u ^ {\displaystyle {\boldsymbol {\hat {u}}}} in the Fourier domain and label any frequency vector ξ {\displaystyle {\boldsymbol {\xi }}} as negative if ξ ⋅ u ^ < 0 {\displaystyle {\boldsymbol {\xi }}\cdot {\boldsymbol {\hat {u}}}<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, there is no particular direction for u ^ {\displaystyle {\boldsymbol {\hat {u}}}} which must be chosen unless there are some additional constraints. Therefore, the choice of u ^ {\displaystyle {\boldsymbol {\hat {u}}}} is ad hoc, or application specific. </p> <h3>The monogenic signal</h3> <p>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 (<i>n</i> + 1)-dimensional vector-valued function for the case of <i>n</i>-variable signals. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Hilbert_transform/RoQaC8MS">Practical considerations for computing Hilbert transforms</a></li> <li><a href="/facts/I%2fQ_data/w2WgxUqg">I/Q data</a></li> <li><a href="/facts/Negative_frequency/YasYp0L0">Negative frequency</a></li></ul> <h3>Applications</h3> <ul><li><a href="/facts/Single-sideband_modulation/Cfuv9KX1">Single-sideband modulation</a></li> <li><a href="/facts/Quadrature_filter/u6wr94Mg">Quadrature filter</a></li> <li><a href="/facts/Causal_filter/vBWknoay">Causal filter</a></li></ul> <h2 id="notes">Notes</h2> <h2 id="further-reading">Further reading</h2> <ul><li>Leon Cohen, <i>Time-frequency analysis</i>, Prentice Hall, Upper Saddle River, 1995.</li> <li>Frederick W. King, <i>Hilbert Transforms</i>, vol. II, Cambridge University Press, Cambridge, 2009.</li> <li>B. Boashash, <i>Time-Frequency Signal Analysis and Processing: A Comprehensive Reference</i>, Elsevier Science, Oxford, 2003.</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="http://www.dsprelated.com/freebooks/mdft/Analytic_Signals_Hilbert_Transform.html">Analytic Signals and Hilbert Transform Filters</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Smith, J.O. "Analytic Signals and Hilbert Transform Filters", in Mathematics of the Discrete Fourier Transform (DFT) with Audio Applications, Second Edition, https://ccrma.stanford.edu/~jos/r320/Analytic_Signals_Hilbert_Transform.html, or https://www.dsprelated.com/freebooks/mdft/Analytic_Signals_Hilbert_Transform.html, online book, 2007 edition, accessed 2021-04-29. <a href="https://ccrma.stanford.edu/~jos/r320/Analytic_Signals_Hilbert_Transform.html" target="_blank">https://ccrma.stanford.edu/~jos/r320/Analytic_Signals_Hilbert_Transform.html</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Bracewell, Ron. The Fourier Transform and Its Applications. McGraw-Hill, 2000. pp. 361-362 <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>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 <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Bracewell, Ron. The Fourier Transform and Its Applications. McGraw-Hill, 2000. pp. 361-362 <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Justice, J. (1979-12-01). "Analytic signal processing in music computation". IEEE Transactions on Acoustics, Speech, and Signal Processing. 27 (6): 670–684. doi:10.1109/TASSP.1979.1163321. ISSN 0096-3518. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>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 <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>"the complex envelope (or complex amplitude)"[6] <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>"the complex envelope (or complex amplitude)", p. 586 [7] <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>"Complex envelope is an extended interpretation of complex amplitude as a function of time." p. 85[8] <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> </ol>