tcomplex - SIGNAL PROCESSING SIMULATION NEWSLETTER Note...

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Note: This is not a particularly interesting topic for anyone other than those who are involved in simulation. So if you have difficulty with this issue, you may safely stop reading after page 5 without feeling guilty. Hilbert Transform, Analytic Signal and the Complex Envelope In Digital Signal Processing we often need to look at relationships between real and imaginary parts of a complex signal. These relationships are generally described by Hilbert transforms. Hilbert transform not only helps us relate the I and Q components but it is also used to create a special class of causal signals called analytic which are especially important in simulation. The analytic signals help us to represent bandpass signals as complex signals which have specially attractive properties for signal processing. Hilbert Transform is not a particularly complex concept and can be much better understood if we take an intuitive approach first before delving into its formula which is related to convolution and is hard to grasp. The following diagram that is often seen in text books describing modulation gives us a clue as to what a Hilbert Transform does.
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Figure 1 - Role of Hilbert Transform in modulation The role of Hilbert transform as we can guess here is to take the carrier which is a cosine wave and create a sine wave out of it. So let’s take a closer look at a cosine wave to see how this is done by the Hilbert transformer. Figure 2a shows the amplitude and the phase spectrum of a cosine wave. Now recall that the Fourier Series is written as where and and A n and B n are the spectral amplitudes of cosine and sine waves. Now take a look at the phase spectrum. The phase spectrum is computed by Cosine wave has no sine spectral content, so B n is zero. The phase calculated is 90 ° for both positive and negative frequency from above formula. The wave has two spectral components each of magnitude 1/2A, both positive and lying in the real plane. (the real plane is described as that passing vertically (R-V plane) and the Imaginary plane as one horizontally (R-I plane) through the Imaginary axis)
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Figure 2b shows the same two spectrums for a sine wave. The sine wave phase is not symmetric because the amplitude spectrum is not symmetric. The quantity A n is zero and B n has either a positive or negative value. The phase is +90 ° for the positive frequency and -90 ° for the negative frequency. Now we wish to convert the cosine wave to a sine wave. There are two ways of doing that, one in time domain and the other in frequency domain. Hilbert Transform in Frequency Domain Now compare Figure 2a and 2b, in particular the spectral amplitudes. The cosine spectral amplitudes are both positive and lie in the real plane. The sine wave has spectral components that lie in the Imaginary plane and are of opposite sign. To turn cosine into sine, as shown in Figure 3 below, we need to rotate the negative frequency component of
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tcomplex - SIGNAL PROCESSING SIMULATION NEWSLETTER Note...

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