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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 $ z(t)$ can be represented as

$\displaystyle z(t) = \frac{1}{2\pi}\int_0^{\infty} Z(\omega)e^{j\omega t}d\omega $

where $ Z(\omega)$ is the complex coefficient (setting the amplitude and phase) of the positive-frequency complex sinusoid $ \exp(j\omega t)$ at frequency $ \omega$. Any real sinusoid $ A\cos(\omega t + \phi)$ may be converted to a positive-frequency complex sinusoid $ A\exp[j(\omega t + \phi)]$ by simply generating a phase-quadrature component $ A\sin(\omega t + \phi)$ to serve as the ``imaginary part'':

$\displaystyle A e^{j(\omega t + \phi)} = A\cos(\omega t + \phi) + j A\sin(\omega t + \phi) $

The phase-quadrature component can be generated from the in-phase component by a simple quarter-cycle time shift.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 $ {\cal H}_t\{x\}$ denote the output at time $ t$ of the Hilbert-transform filter applied to the signal $ x$. Ideally, this filter has magnitude $ 1$ at all frequencies and introduces a phase shift of $ -\pi/2$ at each positive frequency and $ +\pi/2$ at each negative frequency. When a real signal $ x(t)$ and its Hilbert transform $ y(t) = {\cal H}_t\{x\}$ are used to form a new complex signal $ z(t) = x(t) + j y(t)$, the signal $ z(t)$ is the (complex) analytic signal corresponding to the real signal $ x(t)$. In other words, for any real signal $ x(t)$, the corresponding analytic signal $ z(t)=x(t) + j {\cal H}_t\{x\}$ has the property that all ``negative frequencies'' of $ x(t)$ 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 $ \exp(\pm j\pi/2) = \pm j$. Consider the positive and negative frequency components at the particular frequency $ \omega_0$:
\begin{eqnarray*} x_+(t) &\isdef & e^{j\omega_0 t} \\ x_-(t) &\isdef & e^{-j\omega_0 t} \end{eqnarray*}
Now let's apply a $ -90$ degrees phase shift to the positive-frequency component, and a $ +90$ degrees phase shift to the negative-frequency component:
\begin{eqnarray*} y_+(t) &=& e^{-j\pi/2} e^{j\omega_0 t} = -j e^{j\omega_0 t} \\ y_-(t) &=& e^{j\pi/2} e^{-j\omega_0 t} = j e^{-j\omega_0 t} \end{eqnarray*}
Adding them together gives
\begin{eqnarray*} z_+(t) &\isdef & x_+(t) + j y_+(t) = e^{j\omega_0 t} - j^2 e^{... ... x_-(t) + j y_-(t) = e^{-j\omega_0 t} + j^2 e^{-j\omega_0 t} = 0 \end{eqnarray*}
and sure enough, the negative frequency component is filtered out. (There is also a gain of 2 at positive frequencies.) For a concrete example, let's start with the real sinusoid

$\displaystyle x(t)=2\cos(\omega_0 t) = e^{j\omega_0 t} + e^{-j\omega_0 t}. $

Applying the ideal phase shifts, the Hilbert transform is
\begin{eqnarray*} y(t) &=& e^{j\omega_0 t-j\pi/2} + e^{-j\omega_0 t + j\pi/2}\\ &=& -je^{j\omega_0 t} + je^{-j\omega_0 t} = 2\sin(\omega_0 t). \end{eqnarray*}
The analytic signal is then

$\displaystyle z(t) = x(t) + j y(t) = 2\cos(\omega_0 t) + j 2\sin(\omega_0 t) = 2 e^{j\omega_0 t}, $

by Euler's identity. Thus, in the sum $ x(t) + j y(t)$, the negative-frequency components of $ x(t)$ and $ jy(t)$ cancel out, leaving only the positive-frequency component. This happens for any real signal $ x(t)$, not just for sinusoids as in our example.
Figure 4.16: Creation of the analytic signal $ z(t)=e^{j\omega _0 t}$ from the real sinusoid $ x(t) = \cos(\omega_0 t)$ and the derived phase-quadrature sinusoid $ y(t) = \sin(\omega_0 t)$, viewed in the frequency domain. a) Spectrum of $ x$. b) Spectrum of $ y$. c) Spectrum of $ j y$. d) Spectrum of $ z = x + jy$.
\includegraphics[width=2.8in]{eps/sineFD}
Figure 4.16 illustrates what is going on in the frequency domain. At the top is a graph of the spectrum of the sinusoid $ \cos(\omega_0 t)$ consisting of impulses at frequencies $ \omega=\pm\omega_0$ and zero at all other frequencies (since $ \cos(\omega_0 t) = (1/2)\exp(j\omega_0 t) + (1/2)\exp(-j\omega_0 t)$). Each impulse amplitude is equal to $ 1/2$. (The amplitude of an impulse is its algebraic area.) Similarly, since $ \sin(\omega_0 t) = (1/2j)\exp(j\omega_0 t) - (1/2j)\exp(-j\omega_0 t) = -(j/2) \exp(j\omega_0 t) + (j/2)\exp(-j\omega_0 t)$, the spectrum of $ \sin(\omega_0 t)$ is an impulse of amplitude $ -j/2$ at $ \omega=\omega_0$ and amplitude $ +j/2$ at $ \omega=-\omega_0$. Multiplying $ y(t)$ by $ j$ results in $ j\sin(\omega_0 t) = (1/2)\exp(j\omega_0 t) - (1/2)\exp(-j\omega_0 t)$ which is shown in the third plot, Fig.4.16c. Finally, adding together the first and third plots, corresponding to $ z(t) = x(t) + j y(t)$, we see that the two positive-frequency impulses add in phase to give a unit impulse (corresponding to $ \exp(j\omega_0 t)$), and at frequency $ -\omega_0$, the two impulses, having opposite sign, cancel in the sum, thus creating an analytic signal $ z$, as shown in Fig.4.16d. This sequence of operations illustrates how the negative-frequency component $ \exp(-j\omega_0 t)$ gets filtered out by summing $ \cos(\omega_0 t)$ with $ j\sin(\omega_0 t)$ to produce the analytic signal $ \exp(j\omega_0 t)$ corresponding to the real signal $ \cos(\omega_0 t)$. As a final example (and application), let $ x(t) = A(t)\cos(\omega t)$, where $ A(t)$ is a slowly varying amplitude envelope (slow compared with $ \omega$). This is an example of amplitude modulation applied to a sinusoid at ``carrier frequency'' $ \omega$ (which is where you tune your AM radio). The Hilbert transform is very close to $ y(t)\approx A(t)\sin(\omega t)$ (if $ A(t)$ were constant, this would be exact), and the analytic signal is $ z(t)\approx A(t)e^{j\omega t}$. Note that AM demodulation4.14is now nothing more than the absolute value. I.e., $ A(t) = \left\vert z(t)\right\vert$. 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)