The problem is how the phase φ effects the outcome when the input(message signal) is the DSB-SC LSB.
It's :
message: $m(t)=A_{m}cos(ω_{m}t)$
carrier: $c(t)=A_{c}cos(w_{c}t)$
I found that the LSB through the DSB-SC
$DSB(t)=A_{c}A_{m}cos(ω_{m}t)cos(w_{c}t)=(A_{c}A_{m}/2)[cos((ω_{m}-ω_{c})t)+cos((ω_{m}+ω_{c})t)]$
The component $(A_{c}A_{m}/2)[cos((ω_{m}-ω_{c})t)]$ is the expression for the LSB(t)
So the LSB(t) gets mixed with a Local Oscillator
$LO(t)=Acos(w_{c}t+φ)$
and the output d(t) passes through an Ideal LP Filter and the y(t) signal comes out of LPF.
So I've reached that:
y(t)=$(AA_{c}A_{m}/4)[cos((ω_{m}t+φ)]$
I have reached at a dead end trying to understand the effect of phase φ when: φ=(0,π/4,π/2,3π/4,π)