DSB-SC Time domain and Frequency domain Description:

DSBSC modulators make use of the multiplying action in which the modulating signal multiplies the carrier wave. In this system, the carrier component is eliminated and both upper and lower side bands are transmitted. As the carrier component is suppressed, the power required for transmission is less than that of AM.
If 𝑚(𝑡) is the message signal and 𝑐(𝑡) = 𝐴𝑐 cos(2𝜋𝑓𝑐 𝑡) is the carrier signal, then DSBSC modulated wave 𝑠(𝑡) is given by
𝑠(𝑡) = 𝑐(𝑡)𝑚(𝑡)
𝑠(𝑡) = 𝐴𝑐 cos(2𝜋𝑓𝑐 𝑡) 𝑚(𝑡)
Consequently, the modulated signal s(t) under goes a phase reversal , whenever the message signal m(t) crosses zero as shown below.

(a) DSB-SC waveform (b) DSB-SC Frequency Spectrum

 The envelope of a DSBSC modulated signal is therefore different from the message signal and the Fourier transform of s(t) is given by

𝑆(𝑓) =𝐴𝑐[𝑀(𝑓 − 𝑓 ) + 𝑀(𝑓 + 𝑓 )]2 𝑐 𝑐

For the case when base band signal 𝑚(𝑡)is limited to the interval −𝑊 < 𝑓 < 𝑊 as shown in the figure below, we find that the spectrum 𝑆(𝑓) of the DSBSC wave 𝑠(𝑡) is as illustrated below. Except for a change in scaling factor, the modulation process simply translates the spectrum of the base band signal by𝑓𝑐. The transmission  bandwidth required by DSBSC modulation is the same as that for AM

Message and the corresponding DSBSC spectrum