Editing Single-sideband modulation (section)

==Mathematical formulation==
[[File:Single-sideband derivation.svg|thumb|right|450px|Frequency-domain depiction of the mathematical steps that convert a baseband function into a single-sideband radio signal]]

Single-sideband has the mathematical form of [[IQ Modulation|quadrature amplitude modulation]] (QAM) in the special case where one of the [[baseband]] waveforms is derived from the other, instead of being independent messages''':'''

{{NumBlk|:|<math>s_\text{ssb}(t) = s(t) \cdot \cos\left(2\pi f_0 t\right) - \widehat{s}(t)\cdot \sin\left(2\pi f_0 t\right),\,</math>|{{EquationRef|Eq.1}}}}

where <math>s(t)\,</math> is the message (real-valued), <math>\widehat{s}(t)\,</math> is its [[Hilbert transform]], and <math>f_0\,</math> is the radio [[carrier frequency]].<ref>{{cite book|last1=Tretter|first1=Steven A.|editor1-last=Lucky|editor1-first=R.W.|title=Communication System Design Using DSP Algorithms|date=1995|publisher=Springer|location=New York|isbn=0306450321|page=80|chapter=Chapter 7, Eq 7.9}}</ref>

To understand this formula, we may express <math>s(t)</math> as the real part of a complex-valued function, with no loss of information:

:<math>s(t) = \operatorname{Re}\left\{s_\mathrm{a}(t)\right\} = \operatorname{Re}\left\{s(t) + j \cdot \widehat{s}(t)\right\},</math>

where <math>j</math> represents the [[imaginary unit]].&nbsp; <math>s_\mathrm{a}(t)</math> is the [[analytic representation]] of <math>s(t),</math>&nbsp; which means that it comprises only the positive-frequency components of <math>s(t)</math>:

:<math>
  \frac{1}{2}S_\mathrm{a}(f) =
    \begin{cases}
      S(f), &\text{for}\ f > 0,\\
      0,    &\text{for}\ f < 0,
    \end{cases}
</math>

where <math>S_\mathrm{a}(f)</math> and <math>S(f)</math> are the respective Fourier transforms of <math>s_\mathrm{a}(t)</math> and <math>s(t).</math>&nbsp; Therefore, the frequency-translated function <math>S_\mathrm{a}\left(f - f_0\right)</math> contains only one side of <math>S(f).</math>&nbsp;  Since it also has only positive-frequency components, its inverse Fourier transform is the analytic representation of <math>s_\text{ssb}(t):</math>

:<math>s_\text{ssb}(t) + j \cdot \widehat{s}_\text{ssb}(t) = \mathcal{F}^{-1} \{S_\mathrm{a}\left(f - f_0\right)\} = s_\mathrm{a}(t) \cdot e^{j2\pi f_0 t},\,</math>

and again the real part of this expression causes no loss of information.&nbsp; With [[Euler's formula]] to expand &nbsp;<math>e^{j2\pi f_0 t},\,</math>&nbsp; we obtain {{EquationNote|Eq.1}}:

:<math>\begin{align}
  s_\text{ssb}(t) &= \operatorname{Re}\left\{s_\mathrm{a}(t)\cdot e^{j2\pi f_0 t}\right\} \\
                  &= \operatorname{Re}\left\{\,\left[s(t) + j \cdot \widehat{s}(t)\right] \cdot \left[\cos\left(2\pi f_0 t\right) + j \cdot \sin\left(2\pi f_0 t\right)\right]\,\right\} \\
                  &= s(t) \cdot \cos\left(2\pi f_0 t\right) - \widehat{s}(t) \cdot \sin\left(2\pi f_0 t\right).
\end{align}</math>

Coherent demodulation of <math>s_\text{ssb}(t)</math> to recover <math>s(t)</math> is the same as AM: multiply by <math>\cos\left(2\pi f_0 t\right),</math>&nbsp; and lowpass to remove the "double-frequency" components around frequency <math>2 f_0</math>.  If the demodulating carrier is not in the correct phase (cosine phase here), then the demodulated signal will be some linear combination of <math>s(t)</math> and <math>\widehat s(t)</math>, which is usually acceptable in voice communications (if the demodulation carrier frequency is not quite right, the phase will be drifting cyclically, which again is usually acceptable in voice communications if the frequency error is small enough, and amateur radio operators are sometimes tolerant of even larger frequency errors that cause unnatural-sounding pitch shifting effects).

===Lower sideband===
<math>s(t)</math> can also be recovered as the real part of the complex-conjugate, <math>s_\mathrm{a}^*(t),</math> which represents the negative frequency portion of <math>S(f).</math>  When <math>f_0\,</math> is large enough that <math>S\left(f - f_0\right)</math> has no negative frequencies, the product <math>s_\mathrm{a}^*(t) \cdot e^{j2\pi f_0 t}</math> is another analytic signal, whose real part is the actual ''lower-sideband'' transmission''':'''

:<math>\begin{align}
  s_\mathrm{a}^*(t)\cdot e^{j2\pi f_0 t} &= s_\text{lsb}(t) + j \cdot \widehat s_\text{lsb}(t) \\
             \Rightarrow s_\text{lsb}(t) &= \operatorname{Re}\left\{s_\mathrm{a}^*(t) \cdot e^{j2\pi f_0 t}\right\} \\
                                         &= s(t) \cdot \cos\left(2\pi f_0 t\right) + \widehat{s}(t) \cdot \sin\left(2\pi f_0 t\right).
\end{align}</math>

The sum of the two sideband signals is:

:<math>s_\text{usb}(t) + s_\text{lsb}(t) = 2s(t) \cdot \cos\left(2\pi f_0 t\right),\,</math>

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