Single-sideband has the mathematical form of
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|:|s_\text{usb}(t) = s(t) \cdot \cos\left(2\pi f_0 t\right) - \widehat{s}(t)\cdot \sin\left(2\pi f_0 t\right),\,|}} where s(t)\, is the message (real-valued), \widehat{s}(t)\, is its
Hilbert transform, and f_0\, is the radio
carrier frequency. To understand this formula, we may express s(t) as the real part of a complex-valued function, with no loss of information: :s(t) = \operatorname{Re}\left\{s_\mathrm{a}(t)\right\} = \operatorname{Re}\left\{s(t) + j \cdot \widehat{s}(t)\right\}, where j represents the
imaginary unit. s_\mathrm{a}(t) is the
analytic representation of s(t), which means that it comprises only the positive-frequency components of s(t): : \frac{1}{2}S_\mathrm{a}(f) = \begin{cases} S(f), &\text{for}\ f > 0,\\ 0, &\text{for}\ f where S_\mathrm{a}(f) and S(f) are the respective
Fourier transforms of s_\mathrm{a}(t) and s(t). Therefore, the frequency-translated function S_\mathrm{a}\left(f - f_0\right) contains only one side of S(f). Since it also has only positive-frequency components, its inverse Fourier transform is the analytic representation of s_\text{usb}(t): :s_\text{usb}(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},\, and again the real part of this expression causes no loss of information. With
Euler's formula to expand e^{j2\pi f_0 t},\, we obtain : :\begin{align} s_\text{usb}(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} Coherent demodulation of s_\text{ssb}(t) to recover s(t) is the same as AM: multiply by \cos\left(2\pi f_0 t\right), and lowpass to remove the "double-frequency" components around frequency 2 f_0. If the demodulating carrier is not in the correct phase (cosine phase here), then the demodulated signal will be some linear combination of s(t) and \widehat s(t), 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 s(t) can also be recovered as the real part of the complex-conjugate, s_\mathrm{a}^*(t), which represents the negative frequency portion of S(f). When f_0\, is large enough that S\left(f - f_0\right) has no negative frequencies, the product s_\mathrm{a}^*(t) \cdot e^{j2\pi f_0 t} is another analytic signal, whose real part is the actual
lower-sideband transmission
: :\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} The sum of the two sideband signals is: :s_\text{usb}(t) + s_\text{lsb}(t) = 2s(t) \cdot \cos\left(2\pi f_0 t\right),\, which is the classic model of suppressed-carrier
double sideband AM. ==Practical implementations==