### Analytic Signals and Hilbert Transform Filters

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

^{4.12}Therefore, in continuous time, every analytic signal can be represented as

^{4.13}For more complicated signals which are expressible as a sum of many sinusoids, a

*filter*can be constructed which shifts each sinusoidal component by a quarter cycle. This is called a

*Hilbert transform filter*. Let denote the output at time of the Hilbert-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 signal*corresponding 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 :

*add in phase*to give a unit impulse (corresponding to ), and at frequency , the two impulses, having opposite sign,

*cancel*in the sum, thus creating an analytic signal , as shown in Fig.4.16d. This sequence of operations illustrates how the negative-frequency component gets

*filtered out*by summing with to produce the analytic signal corresponding to the real signal . As a final example (and application), let , where is a slowly varying amplitude envelope (slow compared with ). This is an example of

*amplitude modulation*applied to a sinusoid at ``carrier frequency'' (which is where you tune your AM radio). The Hilbert transform is very close to (if were constant, this would be exact), and the analytic signal is . Note that AM

*demodulation*

^{4.14}is now nothing more than the

*absolute value*.

*I.e.*, . Due to this simplicity, Hilbert transforms are sometimes used in making

*amplitude envelope followers*for narrowband signals (

*i.e.*, signals with all energy centered about a single ``carrier'' frequency). AM demodulation is one application of a narrowband envelope follower.

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Generalized Complex Sinusoids

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Sinusoidal Frequency Modulation (FM)