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 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 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 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:
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-spikesadd in phase to give an amplitude 2 up-spike (which is ), and the negative-frequency up-spike in the cosine iscanceled by the down-spike in times sine at frequency. This sequence of operations illustrates how the negative-frequency component gets filtered 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 modulation applied 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 AM demodulation5.9 is now nothing more than the absolute value. I.e., . Due to this simplicity, Hilbert transforms are sometimes used in makingamplitude 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.