Double side band modulation is one of the simplest ways of
modulating a signal. It has the advantages of being
mathematically simple and easy to understand, but it can be difficult
to implement because the phase of the modulating carrier must match
that of the demodulating carrier. Basically, DSB modulation modulates
a signal by multiplying it by a carrier signal (of much higher
frequency than the message signal), and then demodulates it by
multiplying by a signal identical to the modulating signal, except for
a scaling factor. A filter is then applied to retrieve the message signal.

Using double sideband modulation, the message signal, m(t), is
simply multiplied by a carrier signal, usually a cosine. Thus,

x_{c}(t) = A_{c}m(t)
cost(ω_{c}t)

where x_{c}(t) is the modulated signal, ω is the angular
frequency, A_{c} is a scaling constant, and the subscript c
denotes a property of the carrier signal.

Once modulated, a signal may then be transmitted through some
medium, such as the atmosphere. Once it reaches a receiver, the signal
must be demodulated, decoded to regain the original message
signal. This is where the difficulty arises for double side band
modulation.

To demodulate the signal, it must be multiplied by a demodulating
carrier. This is easily expressed as

d(t) = 2A_{c}[m(t) cos(ω_{c}t)]
cos(ω_{c}t)

The factor of two on the front is needed because the
cos^{2} identity produces a fact of 1/2. Without it, you would
recover a message signal with 1/2 the amplitude of the input message
signal. If we apply the cos^{2} identity and multiply out, we have:

d(t) = A_{c}m(t) + A_{c}m(t)
cos(2ω_{c};t)

By applying a lowpass filter around the message signal, the high
frequency cos components can be eliminated.

The problem with double sideband modulation is that both the phase
of the modulating carrier and the demodulating carrier must match
exactly for it to work. Even a small phase offset could cause serious
distortion of the received message signal.