In signal processing, the analytic signal s_a(t) corresponding to a signal s(t) is defined by

sa(t)

where is the Hilbert transform of s(t), i is the imaginary number. This definition holds for both real and complex signals s(t).

It can also be written with the convolution operator as:

Properties

When s(t) is real, and has a frequency spectrum S(ω) given by its Fourier transform, the analytic signal s_a(t) is complex but its spectrum only contains the positive frequencies of S(ω). In other words, the positive frequencies are retained exactly (possibly multiplied by a constant, depending on implementation), and the negative frequencies are "filtered out". This property is expressed by the following relation:

As a complex signal, the analytic signal may be written in polar coordinates, s_a(t) = A(t)e^iφ(t). The time derivative of its phase is called the instantaneous frequency: ω_i(t) = φ'(t).

The magnitude of the analytic signal, , will return the complex envelope of a real signal, as shown in the diagram. The blue curve is the original signal, and the red curve is the magnitude of the analytic signal of the blue signal. This is useful for AM demodulation, single-sideband modulation, and audio signal processing, as the envelope maintains its shape in spite of phase shifts of the original signal.

In digital signal processing, the analytic signal is sometimes generated by taking the Fourier transform of the signal (using an FFT), setting all negative frequency components to zero, and taking the inverse Fourier transform (IFFT).

References

Leon Cohen, "Time-frequency analysis", Prentice-Hall (1995)

External links

• Analytic Signals and Hilbert Transform Filters