Single-sideband modulation/Proofs

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[edit] Signal generation

An alternate method of signal generation has been gaining popularity recently in part due to the availability of low-cost digital signal processor (DSP) systems. To generate an SSB signal with this method, first two versions of the original signal are generated which are mutually 90° out of phase, usually by implementing a Hilbert transformer in a DSP. Each one of these signals are then mixed with carrier waves that are also 90° out of phase with each other. By either adding or subtracting the resulting signals, this can generate a lower or upper sideband signal.

[edit] Mathematical Highlights

Let \hat s(t)\, represent the Hilbert transform of the original signal:  s(t)\,.   Then s_a(t) = s(t)+j \hat s(t)\, is a useful mathematical concept, called the analytic signal.  The Fourier transform of s(t)\, is symmetrical about the 0 Hz axis.  So if we simply translate s(t)\, to a radio transmission frequency, F_c\,, its spectrum will be symmetrical about F_c\,, and the two halves are called sidebands.  The Fourier transform of s_a(t)\, is the same as 2\cdot s(t)\, above 0 Hz, but it has no negative frequency components.  So when we translate it to a radio transmission frequency, it has just a single sideband.  The product of s_a(t)\, with the complex sinusoid:  e^{j2\pi F_c\cdot t}\,, accomplishes that translation. And by Euler's formula, e^{j2\pi F_c\cdot t} = cos(2\pi F_c\cdot t)+j\cdot sin(2\pi F_c\cdot t)\,, which comprises the out-of-phase carrier waves mentioned earlier.

The translated signal also has no negative frequency components.  So it is the analytic representation of the single sideband signal we are trying to generate.  As we have already seen, that means the signal we are trying to generate is just the real part of that product.  I.e,  s_a(t)\cdot e^{j2\pi F_c\cdot t} = s_{ssb}(t) +j\hat s_{ssb}(t) \,.  Therefore we don't need to compute the entire complex product.  (And how would we transmit & receive a complex-valued waveform anyway?)  So the SSB modulator is:  s_{ssb}(t) = Re\big\{s_a(t)\cdot e^{j2\pi F_c\cdot t}\big\} = s(t)\cdot cos(2\pi F_c\cdot t) - \hat s(t)\cdot sin(2\pi F_c\cdot t)\,, as described earlier.

s_a(t)\, represents the signal's upper sideband.  It is also possible, and useful, to transmit the lower sideband instead, which is a mirror image about 0 Hz.  By a general property of the Fourier transform, that symmetry means it is the complex conjugate of s_a(t)\,:  s_a^*(t) = s(t)-j \hat s(t)\,.    Again we multiply by e^{j2\pi F_c\cdot t}\,. The typical F_c\, is large enough that the translated lower sideband has no negative frequency components... another analytic signal. As before, all we need is the real part:  s_{lsb}(t) = Re\big\{s_a^*(t)\cdot e^{j2\pi F_c\cdot t}\big\} = s(t)\cdot cos(2\pi F_c\cdot t) + \hat s(t)\cdot sin(2\pi F_c\cdot t)\,.

Finally, notice that the sum of the two sideband signals is:  2s(t)\cdot cos(2\pi F_c\cdot t)\,, which is the classic model of suppressed-carrier double sideband AM.

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