expressing it in terms of complex-exponentials, discarding the negative frequency component, and doubling the positive frequency component. And the analytic representation of a sum of sinusoids is the sum of the analytic representations of the individual sinusoids. Example 2 Here we use Euler's formula to identify and discard the negative frequency. Then: Example 3 This is another example of using the Hilbert transform method to remove negative frequency components. We note that nothing prevents us from computing for a complex-valued . But it might not be a reversible representation, because the original spectrum is not symmetrical in general. So except for this example, the general discussion assumes real-valued . , where . Then: Properties Instantaneous amplitude and phase An analytic signal can also be expressed in polar coordinates: where the following time-variant quantities are introduced: is called the instantaneous amplitude or the envelope; is called the instantaneous phase or phase angle. In the accompanying diagram, the blue curve depicts and the red curve depicts the corresponding . The time derivative of the unwrapped instantaneous phase has units of radians/second, and is called the instantaneous angular frequency: The instantaneous frequency (in hertz) is therefore: The instantaneous amplitude, and the instantaneous phase and frequency are in some applications used to measure and detect local features of the signal. Another application of the analytic representation of a signal relates to demodulation of modulated signals. The polar coordinates conveniently separate the effects of amplitude modulation and phase (or frequency) modulation, and effectively demodulates certain kinds of signals. Complex envelope/baseband Analytic signals are often shifted in frequency (down-converted) toward 0 Hz, possibly creating [non-symmetrical] negative frequency components: where is an arbitrary reference angular frequency. This function goes by various names, such as complex envelope and complex baseband. The complex envelope is not unique; it is determined by the choice of . This concept is often used when dealing with passband signals. If is a modulated signal, might be equated to its carrier frequency. In other cases, is selected to be somewhere in the middle of the desired passband. Then a simple low-pass filter with real coefficients can excise the portion of interest. Another motive is to reduce the highest frequency, which reduces the minimum rate for alias-free sampling. A frequency shift does not undermine the mathematical tractability of the complex signal representation. So in that sense, the down-converted signal is still analytic. However, restoring the