**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW
VERSION:** "Mathematics of the Discrete
Fourier Transform (DFT), with Audio Applications --- Second
Edition", by Julius
O. Smith III, W3K
Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright ©
*2017-09-28* by Julius O. Smith III -
Center for Computer Research
in Music and Acoustics (CCRMA), Stanford University

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## The Analytic Signal and Hilbert Transform Filters

A signal which has no negative-frequency components is called an

analytic signal.^{5.6}Therefore, in continuous time, every analytic signal can be represented as

where is the complex coefficient (setting the amplitude and phase) of the positive-freqency complex sinusoid atfrequency .Any sinusoid in real life may be converted to a positive-frequency complex sinusoid by simply generating a phase-quadrature component to serve as the ''imaginary part'':

The phase-quadrature component can be generated from the in-phase componentby a simple quarter-cycle time shift.^{5.7}For more complicated signals which are expressible as a sum of many sinusoids, a

filtercan be constructed which shifts each sinusoidal component by a quarter cycle. This is called aHilbert transform filter. Let denote the output at time of theHilbert-transform filter applied to the signal . Ideally, this filter has magnitude at all frequencies and introduces a phase shift of at each positive frequency and at each negative frequency. When a real signal and its Hilbert transform are used to form a new complex signal , the signal is the (complex)analytic signalcorresponding to the real signal . In other words, for any real signal , the corresponding analytic signal has the property that all ''negative frequencies'' of have been ''filtered out.''To see how this works, recall that these phase shifts can be impressed on a complex sinusoid by multiplying it by . Consider the positive and negative frequency components at the particular frequency:

Now let's apply a degrees phase shift to the positive-frequency component, and a degrees phase shift to the negative-frequency component:

Adding them together gives

and sure enough, the negative frequency component is filtered out. (There is also a gain of 2 at positive frequencies which we can remove by defining the Hilbert transform filter to have magnitude 1/2 at all frequencies.)For a concrete example, let's start with the real sinusoid

Applying the ideal phase shifts, the Hilbert transform is

The analytic signal is then

by Euler's identity. Thus, in the sum , the negative-frequency components of and cancel out in the sum, leaving only the positive-frequency component. This happens for any real signal , not just for sinusoids as in our example.

Figure 5.8:Creation of the analytic signal from the real sinusoid and the derived phase-quadrature sinusoid , viewed in the frequency domain. a) Spectrum of . b) Spectrum of . c) Spectrum of . d) Spectrum of .Figure 5.8 illustrates what is going on in the frequency domain. While we haven't ''had'' Fourier analysis yet, it should come as no surprise that the spectrum of a complex sinusoid will consist of a single ''spike'' at the frequency and zero at all other frequencies. (Just follow things intuitively for now, and revisit Fig. 5.8 after we've developed the Fourier theorems.) From the identity , we see that the spectrum contains unit-amplitude ''spikes'' at and . Similarly, the identity says that we have an amplitude spike at and an amplitude spike at . Multiplying by results in which is a unit-amplitude ''up spike'' at and a unit ''down spike'' at . Finally, adding together the first and third plots, corresponding to , we see that the two up-spikes

add in phaseto give an amplitude 2 up-spike (which is ), and the negative-frequency up-spike in the cosine iscanceledby the down-spike in times sine at frequency. This sequence of operations illustrates how the negative-frequency component getsfiltered outby the addition of and .As a final example (and application), let , where is a slowly varying amplitude envelope (slow compared with). This is an example of

amplitude modulationapplied to a sinusoid at ''carrier frequency'' (which is where you tune your AM radio). The Hilbert transform is almost exactly^{5.8}, and the analytic signal is . Note that AMdemodulation^{5.9}is now nothing more than theabsolute value. I.e., . Due to this simplicity, Hilbert transforms are sometimes used in makingamplitude envelope followersfor narrowband signals (i.e., signals with all energy centered about a single ''carrier'' frequency). AM demodulation is one application of a narrowband envelope follower.