The previous method of transmission was known as \emph{Coherent Detection}, or \emph{Coherent Demodulation}, or \emph{Synchronous Detection/Demodulation}. The transmitter and receiver may be even hundreds of miles and in the case of short wave radio, thousands of miles away. What you transmit is not what you receive. The atmosphere degrades the signal in various ways---all kinds of interference, climate issues can affect the integrity of the signal. This major assumption that we made in the last discussion in regards to transmitting signals is that we send $y(t)$ and receive $y(t)$. In particular, in the city, the signal you transmit gets reflected off buildings. At the receiver you not only get the original signal but echoes of that signal with attenuated amplitudes. This level of generalization by allowing frequency jitter and the phase difference is just a small level of generalization. There are all kinds of things that can go wrong. Can we use a better method? Lets look at \emph{Quadrature Multiplexing}, otherwise known as \emph{Quadrature Amplitude Modulation}:

%\img{images/modulators/quadmult/quad.ps}

\psset{unit=1cm}
\begin{center}
\begin{pspicture}(-1,-1)(6,4)
    
%text
\rput(0.8,3){$x_1(t)$}

%text
\rput(0.8,1.5){$x_2(t)$}

%text
\rput(5.5,2.25){$y(t)$}

%text
\rput(2.75,0.5){$sin(\omega_0t)$}

%text
\rput(2.75,2){$cos(\omega_0t)$}


% wire with arrow 
\psline{->}(1.2,3)(2.5,3)

% wire with arrow 
\psline{->}(1.2,1.5)(2.5,1.5)

% wire with arrow 
\psline{}(3,3)(4.5,3)

% wire with arrow 
\psline{}(3,1.5)(4.5,1.5)

% wire with arrow 
\psline{->}(4.5,3)(4.5,2.5)

% wire with arrow 
\psline{->}(4.5,1.5)(4.5,2)

% wire with arrow 
\psline{->}(4.75,2.25)(5.25,2.25)

% wire with arrow 
\psline{->}(2.75,2.25)(2.75,2.75)

% wire with arrow 
\psline{->}(2.75,0.75)(2.75,1.25)

% multiplier
\pscircle(2.75,3){0.25}
\psline(2.575,3.175)(2.925,2.825)
\psline(2.925,3.175)(2.575,2.825)


% multiplier
\pscircle(2.75,1.5){0.25}
\psline(2.575,1.675)(2.925,1.325)
\psline(2.925,1.675)(2.575,1.325)


% adder
\pscircle(4.5,2.25){0.25}
\psline(4.25,2.25)(4.75,2.25)
\psline(4.5,2.5)(4.5,2)

\end{pspicture}
\end{center}

This is used to transmit color signals in the standard television system before HDTV.  It works in the following way: The system transmits two signals, carrying them on to the same frequency and at the receiver separates the signals:

% \img{images/modulators/quadmult/rec.ps}

\psset{unit=1cm}
\begin{center}
\begin{pspicture}(-1,-1)(6,4)
    

%frame
\rput(4.5,3){$\mbox{LPF}$}
\psframe(5,2.75)(4,3.25)


%frame
\rput(4.5,1.5){$\mbox{LPF}$}
\psframe(5,1.25)(4,1.75)

%text
\rput(6,3){$g_1(t)$}

%text
\rput(6,1.5){$g_2(t)$}

%text
\rput(0.8,2.25){$y(t)$}

%text
\rput(2.75,0.5){$sin(\omega_0t)$}

%text
\rput(2.75,2){$cos(\omega_0t)$}


% wire with arrow 
\psline{->}(1.2,2.25)(1.5,2.25)

% wire with arrow 
\psline{->}(1.5,3)(2.5,3)

% wire with arrow 
\psline{->}(1.5,1.5)(2.5,1.5)

% wire with arrow 
\psline{->}(5,3)(5.5,3)

% wire with arrow 
\psline{->}(5,1.5)(5.5,1.5)

% wire with arrow 
\psline{}(3,3)(4,3)

% wire with arrow 
\psline{}(3,1.5)(4,1.5)

% wire with arrow 
\psline{}(1.5,3)(1.5,1.5)

% wire with arrow 
\psline{->}(2.75,2.25)(2.75,2.75)

% wire with arrow 
\psline{->}(2.75,0.75)(2.75,1.25)

% multiplier
\pscircle(2.75,3){0.25}
\psline(2.575,3.175)(2.925,2.825)
\psline(2.925,3.175)(2.575,2.825)


% multiplier
\pscircle(2.75,1.5){0.25}
\psline(2.575,1.675)(2.925,1.325)
\psline(2.925,1.675)(2.575,1.325)

\end{pspicture}
\end{center}

Hopefully $g_1(t),g_2(t)$ are equal to $x_1(t),x_2(t)$ respectively. Lets do the math for a received signal $y(t) = x_1(t)\cos(\omega_0t) + x_2(t)\sin(\omega_0t)$, by passing it through the cosine portion of the receiving end system:

\begin{align*}
y(t) &= x_1(t)\cos(\omega_0t) + x_2(t)\sin(\omega_0t) \\
y(t)\cos(\omega_0t) &= x_1(t)\cos^2(\omega_0t) + x_2(t)\sin(\omega_0t)\cos(\omega_0t) \\
r_1(t) &= x_1(t)\lr{\frac{1+\cos(2\omega_0t)}{2}} + x_2(t)\frac{1}{2}\sin(2\omega_0t) \\
r_1(t) &= \frac{x_1(t)}{2} + \frac{x_1(t)}{2}\cos(2\omega_0t) + \frac{x_2(t)}{2}\sin(2\omega_0t) \\
\end{align*}

Once we pass $r_1(t)$ through a low-pass filter with a gain of 2 to get $g_1(t)$, we can recover the signal $x_1(t)$ because all terms with frequency $2\omega_0$ will be cut out.
