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2 The Effects of Phasing Errors on Sideband Suppression

The phasing method generates a single sideband signal, given mathematically as cos(( $\omega_{c}^{}$ $\pm$ $\omega_{a}^{}$ )t) , where the + (-) sign gives the upper (lower) sideband, and $\omega_{c}^{}$ = 2 $\pi$ f_c where f_c is the carrier frequency. Similarly, $\omega_{a}^{}$ = 2 $\pi$ f_a where f_a is the audio modulating frequency. The cosine can be written as

cos(( $\displaystyle\omega_{c}^{}$ $\displaystyle\pm$ $\displaystyle\omega_{a}^{}$ )t) = cos( $\displaystyle\omega_{c}^{}$ t)cos( $\displaystyle\omega_{a}^{}$ t) $\displaystyle\mp$ sin( $\displaystyle\omega_{c}^{}$ t)sin( $\displaystyle\omega_{a}^{}$ t), (1)

the basic equation of the phasing method. The multiplications on the right-hand side are accomplished using balanced modulators, and the two audio frequencies (as well as the two radio frequencies) must be 90^o out of phase and of equal amplitude. I will assume that the radio frequencies are exactly 90^o out of phase, and of equal amplitude. Using the usual complex notation with V_Ae^jt to be one audio signal, and V_Be^jt to be the other, the result of using a nonideal phasing network will be

Re[cos( $\displaystyle\omega_{c}^{}$ t)V_Ae^jt + sin( $\displaystyle\omega_{c}^{}$ t)V_Be^jt] = $\displaystyle{\textstyle\frac{1}{2}}$ Re[e^{j( + )t}(V_A - jV_B) + e^{- j( - )t}(V_A + jV_B)], (2)

and the sideband suppression (or enhancement) is given by

20 $\displaystyle\log_{10}^{}$ $\displaystyle\left \vert \frac{V_A + j V_B}{V_A - j V_B} \right\vert.$ (3)

Notice if |V_A| equals |V_B| , that is if the two signals have equal amplitude then for a phase error of $\delta$ , the suppression in dB is simply,

- 20 $\displaystyle\log_{10}^{}$ |tan( $\displaystyle{\frac{\delta}{2}}$ )|. (4)

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