Instantaneous frequency Another way to describe a signal simultaneously in time and in frequency is to consider its instantaneous frequency. In order to introduce such a function, we must define first the concept of analytic signal. For any real valued signal $x(t)$, we associate a complex valued signal $x_a(t)$ defined as $$x_a(t) = x(t) + j HT(x(t))$$ where $HT(x)$ is the Hilbert transform of $x$ ($x_a$ can be obtained using the M-file hilbert.m of the Signal Processing Toolbox). $x_a(t)$ is called the analytic signal associated to $x(t)$. This definition has a simple interpretation in the frequency domain since $X_a$ is a single-sided Fourier transform where the negative frequency values have been removed, the strictly positive ones have been doubled, and the DC component is kept unchanged : $$\begin{eqnarray*} X_a(\nu) = 0 \ \ \ \ \ \ &\mbox{if}& \nu < 0 \\ X_a(\nu) = ... ...&\mbox{if}& \nu = 0 \\ X_a(\nu) = 2X(\nu) &\mbox{if}& \nu > 0 \end{eqnarray*}$$ ($X$ is the Fourier transform of $x$, and $X_a$ the Fourier transform of $x_a$). Thus, the analytic signal can be obtained from the real signal by forcing to zero its spectrum for the negative frequencies, which do not alter the information content since for a real signal, $X(-\nu)=X^*(\nu)$. From this signal, it is then possible to define in a unique way the concepts of instantaneous amplitude and instantaneous frequency by : $$\begin{eqnarray*} a(t) &=& \vert x_a(t)\vert \ \ \ \ \ \ \ \ \ \ \ \mbox{\it ins... ...i} \frac{d\arg{x_a(t)}}{dt}\ \mbox{\it instantaneous frequency} \end{eqnarray*}$$ An estimation of the instantaneous frequency is given by the M-file instfreq.m of the Time-Frequency toolbox : Example (see fig. 2.3) >> sig=fmlin(256); t=(3:256); >> ifr=instfreq(sig); plotifl(t,ifr); Figure 2.3: Estimation of the instantaneous frequency of a linear chirp As we can see from this plot, the instantaneous frequency shows with success the evolution with time of the frequency content of this signal.
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