This example shows how to use the discrete Hilbert Transform to implement Single Sideband Modulation. The Hilbert Transform finds applications in modulators and demodulators, speech processing, medical imaging, direction of arrival DOA measurements, essentially anywhere complex-signal quadrature processing simplifies the design. Introduction Single Sideband SSB Modulation is an efficient form of Amplitude Modulation AM that uses half the bandwidth used by AM. This technique is most popular in applications such as telephony, HAM radio, and HF communications, i. This example shows how to implement SSB Modulation using a Hilbert Transformer. To motivate the need to use a Hilbert Transformer in SSB modulation, it's helpful to first quickly review double sideband modulation. Double Sideband Modulation A simple form of AM is the Double Sideband DSB Modulation, which typically consists of two frequency-shifted copies of a modulated signal on either side of a carrier frequency. Let's zoom to read the new power values Our positive frequency components are now at -6, -18, and -12 dB. Now that we've defined DSB modulation, let's take a look at single sideband modulation. Single Sideband Modulation Single Sideband SSB Modulation is similar to DSB modulation, but instead of using the whole spectrum it uses a filter to select either the lower or upper sideband. The selection of the lower or upper sideband results in the lower sideband LSB or upper sideband USB modulation, respectively. There are two approaches to eliminating one of the sidebands, one is the filter method and the other is the phasing method. The process of selective filtering of the upper or lower sideband is difficult due to the stringent filters required, especially if there's signal content close to DC. This example shows how to use the phasing method, which uses a Hilbert Transformer to implement SSB Modulation. SSB modulation requires the shifting of the message signal to another center frequency without creating pairs of frequency components X f-fo and X f+fo as in the case of the DSB modulation, i. This can be done by using a Hilbert Transformer. Let's first review the definition and properties of the ideal Hilbert Transform before we discuss its use in SSB modulation. This will help motivate its use in SSB modulation. Ideal Hilbert Transform The discrete Hilbert Transform is a process by which a signal's negative frequencies are phase-advanced by 90 degrees and the positive frequencies are phase-delayed by 90 degrees. Shifting the results of the Hilbert Transform +j and adding it to the original signal creates a complex signal as we'll see below. This is because if we shift the imaginary part of our analytic complex signal by 90 degrees +j and add it to the real part, the negative frequencies will cancel while the positive frequencies will add. This results in a signal with no negative frequencies. Also, the magnitude of the frequency component in the complex signal is twice the magnitude of the frequency component in the real signal. This is similar to a one-sided spectrum, which contains the total signal power in the positive frequencies. Next we introduce a Spectral Shifter. The Spectral Shifter shifts translates the spectral content of a signal by modulating the analytic signal formed from the signal whose spectrum we want to shift. This concept can be used for SSB modulation as shown later. The scheme is shown in the diagram below. Using this method of spectral shifting will ensure that the power of our signal is shifted to the frequency of interest while maintaining a real-valued signal in the end. As we indicated earlier the analytic signal is made up of the original real-valued signal plus the Hilbert Transform of that real signal. Note: The hilbert function produces the complete analytic complex signal, not just the imaginary part. As shown in the spectrum plot, our analytic signal is complex and only contains positive frequency components. Moreover, if we measure the power, or zoom in our plot further at the positive frequency component we'll see that the power of the frequency components of the analytic signal is twice the total power of the positive or negative frequency component of the real signal, i. See zoomed-in plot below. We see that the power of the analytic complex signal's frequency components 500, 600, and 700 Hz are roughly 0, -6, and 6 dB, respectively, which is the original signal's total power. These values correspond to our original real-valued signal which has three tones with amplitudes of 1, 0. At this point we can modulate the analytic signal to shift the spectral content to another center frequency without producing frequency component pairs and maintain a real-valued signal. To modulate the signal to the carrier frequency fo, we'll multiply the analytic signal by an exponential. If we compare the spectral plot above with that of the DSB modulation we can see that the Spectral Shifter accomplished the SSB modulation. But before we do that we need to point out the fact that ideal Hilbert transformers are not realizable. However, algorithms that approximate the Hilbert Transformer, such as the Parks-McClellan FIR filter design technique, have been developed which can be used. For the FIR Hilbert transformer we will use an odd length filter which is computationally more efficient than an even length filter. Albeit even length filters enjoy smaller passband errors. The savings in odd length filters is a result that these filters have several of the coefficients that are zero. Also, using an odd length filter will require a shift by an integer time delay, as opposed to a fractional time delay that is required by an even length filter. For even length filers the magnitude response doesn't have to be 0 at pi, therefore they have increased bandwidths. So for odd length filters the useful bandwidth is limited to As seen in the plot above we successfully modulated the message signal three tones to the carrier frequency of 3. Summary As we have seen, by using an approximation to the Hilbert Transform we can produce analytic signals, which are useful in many signal applications that require spectral shifting. Specifically we have seen how an approximate Hilbert Transformer can be used to implement Single Sideband Modulation.
