Analytic Signals and Hilbert Transform Filters

A signal which has no negative-frequency components is called an analytic signal.^4.7 Therefore, in continuous time, every analytic signal $z(t)$ can be represented as

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

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 sinusoid $A\cos(\omega t + \phi)$ in real life 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'':

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

The phase-quadrature component can be generated from the in-phase component by a simple quarter-cycle time shift.^4.8

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 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

$$x(t)=2\cos(\omega_0 t) = \exp(j\omega_0 t) + \exp(-j\omega_0 t).$$

Applying the ideal phase shifts, the Hilbert transform is

$$\begin{eqnarray*} y(t) &=& \exp(j\omega_0 t-j\pi/2) + \exp(-j\omega_0 t + j\pi/2... ...& -j\exp(j\omega_0 t) + j\exp(-j\omega_0 t) = 2\sin(\omega_0 t). \end{eqnarray*}$$

The analytic signal is then

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

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: 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 $z = x + jy$.

Figure 4.8 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 $\exp(j\omega_0 t)$ will consist of a single ``spike'' at the frequency $\omega=\omega_0$ and zero at all other frequencies. (Just follow things intuitively for now, and revisit Fig. 4.8 after we've developed the Fourier theorems.) From the identity $2\cos(\omega_0 t) = \exp(j\omega_0 t) + \exp(-j\omega_0 t)$, we see that the spectrum contains unit-amplitude ``spikes'' at $\omega=\omega_0$ and $\omega=-\omega_0$. Similarly, the identity $2\sin(\omega_0 t) = [\exp(j\omega_0 t) - \exp(-j\omega_0 t)]/j = -j \exp(j\omega_0 t) + j \exp(-j\omega_0 t)$ says that we have an amplitude $-j$ spike at $\omega=\omega_0$ and an amplitude $+j$ spike at $\omega=-\omega_0$. Multiplying $y(t)$ by $j$ results in $j\sin(\omega_0 t) = \exp(j\omega_0 t) - \exp(-j\omega_0 t)$ which is a unit-amplitude ``up spike'' at $\omega=\omega_0$ and a unit ``down spike'' at $\omega=-\omega_0$. Finally, adding together the first and third plots, corresponding to $z(t) = x(t) + j y(t)$, we see that the two up-spikes add in phase to give an amplitude 2 up-spike (which is $2\exp(j\omega_0 t)$), and the negative-frequency up-spike in the cosine is canceled by the down-spike in $j$ times sine at frequency $-\omega_0$. This sequence of operations illustrates how the negative-frequency component $\exp(-j\omega_0 t)$ gets filtered out by the addition of $2\cos(\omega_0 t)$ and $j 2\sin(\omega_0 t)$.

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 almost exactly $y(t)\approx A(t)\sin(\omega t)$,<sup>4.9</sup> and the analytic signal is $z(t)\approx A(t)e^{j\omega t}$. Note that AM demodulation<sup>4.10</sup> is 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.