Single-sideband modulation

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Single-sideband modulation (SSB) is a refinement of the technique of amplitude modulation designed to be more efficient in its use of electrical power and bandwidth. It is closely related to vestigial sideband modulation (VSB) (see below).

Standard amplitude modulation produces a modulated output signal that has twice the bandwidth of the baseband signal. Single-sideband modulation avoids this bandwidth (and power) doubling, at the cost of device complexity.

SSB was pioneered by telephone companies in the 1930s for use over long-distance lines, as part of a technique known as frequency-division multiplexing (FDM). This enabled many voice channels to be sent down a single physical circuit. The use of SSB meant that the channels could be spaced (usually) just 4,000 Hz apart, while offering a speech bandwidth of nominally 300 – 3,400 Hz.

Radio amateurs began to experiment with the method seriously after World War II. It has become a de facto standard for long-distance voice radio transmissions since then.

1 Signal generation
- 1.1 Mathematical highlights
- 1.2 Lower sideband
2 Demodulation
3 Suppressed carrier SSB and VSB
4 Vestigial sideband
5 See also
6 References

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Signal generation

Consider an amplitude-modulated signal, which will have two frequency-shifted copies of the modulating signal (the lower one is frequency-inverted) on either side of the remaining carrier wave. These are known as sidebands.

One method of producing an SSB signal is to remove one of the sidebands via filtering, leaving only either the upper sideband (USB) or less commonly the lower sideband (LSB). Most often, the carrier is reduced (suppressed) or removed entirely. Assuming both sidebands are symmetric, no information is lost in the process. Since the final RF amplification is now concentrated in a single sideband, the effective power output is greater than in normal AM (the carrier and redundant sideband account for well over half of the power output of an AM transmitter). Though SSB uses substantially less bandwidth and power, it cannot be demodulated by a simple envelope detector like standard AM.

An alternate method of generation uses phasing to suppress the unwanted sideband. To generate an SSB signal with this method, two versions of the original signal are generated which are mutually 90° out of phase. Each one of these signals is then mixed with carrier waves that are also 90° out of phase with each other. By either adding or subtracting the resulting signals, a lower or upper sideband signal results. This method, utilizing the Hilbert transform, can be done at cost with digital circuitry, but can also be done easily using analog circuitry. The method was popular in the days of valve radios, later gained a bad reputation due to poorly adjusted commercial implementations, and is again gaining popularity in the homebrew and DSP fields.

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Mathematical highlights

Let <math>s(t)\,</math> be the baseband waveform to be transmitted. Its Fourier transform, <math>S(f)\,</math>, is symmetrical about the <math>f=0\,</math> axis, because <math>s(t)\,</math> is real-valued. Double sideband modulation of <math>s(t)\,</math> to a radio transmission frequency, <math>F_c\,</math>, moves the axis of symmetry to <math>f=\pm F_c</math>, and the two sides of each axis are called sidebands.

Let <math>\widehat s(t)\,</math> represent the Hilbert transform of <math>s(t)\,</math>. Then

<math>s_a(t) = s(t)+j\cdot \widehat s(t)\,</math>

is a useful mathematical concept, called an analytic signal. The Fourier transform of <math>s_a(t)\,</math> equals <math>2\cdot S(f)\,</math>, for <math>f > 0\,</math>, but it has no negative-frequency components. So it can be modulated to a radio frequency and produce just a single sideband.

The analytic representation of <math>\cos(2\pi F_c\cdot t)\,</math> is:

<math>\cos(2\pi F_c\cdot t)+j\cdot \sin(2\pi F_c\cdot t) = e^{j2\pi F_c\cdot t}</math> (the equality is Euler's formula)

whose Fourier transform is <math>\delta(f-F_c)\,</math>.

When <math>s_a(t)\,</math> is modulated (i.e. multiplied) by <math>e^{j2\pi F_c\cdot t}\,</math>, all frequency components are shifted by <math>+F_c\,</math>, so there are still no negative-frequency components. Therefore, the complex product is an analytic representation of the single sideband signal:

<math>s_a(t)\cdot e^{j2\pi F_c\cdot t} = s_{ssb}(t) +j\cdot \widehat s_{ssb}(t) \,</math>

where <math>s_{ssb}(t)\,</math> is the real-valued, single sideband waveform. Therefore:

<math>s_{ssb}(t)\,</math>	<math>= Re\big\{s_a(t)\cdot e^{j2\pi F_c\cdot t}\big\} </math>
	<math>= Re\left\{\ [s(t)+j\cdot \widehat s(t)]\cdot [\cos(2\pi F_c\cdot t)+j\cdot \sin(2\pi F_c\cdot t)]\ \right\} </math>
	<math>= s(t)\cdot \cos(2\pi F_c\cdot t) - \widehat s(t)\cdot \sin(2\pi F_c\cdot t)\,</math>

And the "out-of-phase carrier waves" mentioned earlier are evident.

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Lower sideband

<math>s_a(t)\,</math> represents the baseband signal's upper sideband, <math>s_{+}(t)\,</math>. It is also possible, and useful, to transmit the lower sideband, <math>s_{-}(t)\,</math>, which is a mirror image about f=0 Hz. By a general property of the Fourier transform, that symmetry means it is the complex conjugate of <math>s_{+}(t)\,</math>:

<math>s_{-}(t) = s_{+}^*(t) = s_a^*(t) = s(t)-j\cdot \widehat s(t)\,</math>

Note that:

The gain of 2 is a result of defining the analytic signal (one sideband) to have the same total energy as <math>s(t)\,</math> (both sidebands).

As before, the signal is modulated by <math>e^{j2\pi F_c\cdot t}\,</math>. The typical <math>F_c\,</math> is large enough that the translated lower sideband (LSB) has no negative-frequency components. Then the result is another analytic signal, whose real part is the actual transmission.

<math>s_{lsb}(t)\,</math>	<math>= Re\big\{s_a^*(t)\cdot e^{j2\pi F_c\cdot t}\big\} </math>
	<math>= s(t)\cdot \cos(2\pi F_c\cdot t) + \widehat s(t)\cdot \sin(2\pi F_c\cdot t)\,</math>

Note that the sum of the two sideband signals is

which is the classic model of suppressed-carrier double sideband AM.

SSB and VSB can also be regarded mathematically as special cases of analog quadrature amplitude modulation.

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Demodulation

The front end of an SSB receiver is the same as that of an AM or FM receiver, consisting of a superheterodyne RF front end that produces a frequency-shifted version of the radio frequency (RF) signal within a standard intermediate frequency (IF) band.

To recover the original signal from the IF SSB signal, the single sideband must be frequency-shifted down to its original range of baseband frequencies, by using a product detector which mixes it with the output of a beat frequency oscillator (BFO). In other words, it is just another stage of heterodyning.

For this to work, the BFO frequency must be accurately adjusted. If the BFO is mis-adjusted, the output signal will be frequency-shifted, making speech sound strange and "Donald Duck"-like, or unintelligible.

As an example, consider an IF SSB signal centered at frequency <math>F_{if}\,</math> = 45000 Hz. The baseband frequency it needs to be shifted to is <math>F_b\,</math> = 2000 Hz. The BFO output waveform is <math>cos(2\pi\cdot F_{bfo}\cdot t)\,</math>. When the signal is multiplied by (aka 'heterodyned with') the BFO waveform, it shifts the signal to <math>(F_{if}+F_{bfo})\,</math> and to <math>|F_{if}-F_{bfo}|\,</math>, which is known as the beat frequency or image frequency. The objective is to choose an <math>F_{bfo}\,</math> that results in <math>|F_{if}-F_{bfo}|=F_b\,</math> = 2000 Hz. (The unwanted components at <math>(F_{if}+F_{bfo})\,</math> can be removed by a lowpass filter (such as the human ear).)

Note that there are two choices for <math>F_{bfo}\,</math>: 43000 Hz and 47000 Hz, aka low-side and high-side injection. With high-side injection, the spectral components that were distributed around 45000 Hz will be distributed around 2000 Hz in the reverse order, also known as an inverted spectrum. That is in fact desirable when the IF spectrum is also inverted, because the BFO inversion restores the proper relationships. One reason for that is when the IF spectrum is the output of an inverting stage in the receiver. Another reason is when the SSB signal is actually a lower sideband, instead of an upper sideband. But if both reasons are true, then the IF spectrum in not inverted, and the non-inverting BFO (43000 Hz) should be used.

If <math>F_{bfo}\,</math> is off by a small amount, then the beat frequency is not exactly <math>F_b\,</math>, which can lead to the speech distortion mentioned earlier.

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Suppressed carrier SSB and VSB

Suppressed carrier SSB modulation is used by ATSC. DSL modems implement suppressed carrier SSB modulation as well.

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Vestigial sideband

A vestigial sideband (in radio communication) is a sideband that has been only partly cut off or suppressed. Television broadcasts (in NTSC, PAL, or SECAM analog video format) use this method if the video is transmitted in AM, due to the large bandwidth used. It may also be used in digital transmission, such as the ATSC-standardized 8-VSB. The Milgo 4400/48 modem (circa 1967) used vestigial sideband and phase-shift keying to provide 4800 bit/s transmission over a 1600 Hz channel. ssb means single side band supressed carrier.