n a digital modulation scheme, we encode bits $(0$s and $1$s$)$ into waveforms, as in the QAM examples we

have seen in class and earlier in the project description. Suppose we decide to use the $4-$QAM scheme,

where the transmitted signal is represented as a signal:

s$($t$)=$

r A

$2$ cos$(2$f$0$t$)($u$($t$)$ u$($t $$ T $))$

r A

$2$ sin$(2$f$0$t$)($u$($t$)$ u$($t $$ T $))(30)$

where T $=$ K $1$

f$0$ for some integer K$.$ As we saw before, we can represent the transmission scheme as This is called a digital modulation scheme because you are sending bits instead of an analog waveform.

The idea is that to transmit two bits $(00,01,10,$ or $11,$ we send one of the following $4$ waveforms:

s$11($t$)=+$

r A

$2$ cos$(2$f$0$t$)($u$($t$)$ u$($t $$ T $))+$

r A

$2$ sin$(2$f$0$t$)($u$($t$)$ u$($t $$ T $))(31)$

s$01($t$)=$

r A

$2$ cos$(2$f$0$t$)($u$($t$)$ u$($t $$ T $))+$

r A

$2$ sin$(2$f$0$t$)($u$($t$)$ u$($t $$ T $))(32)$

s$10($t$)=+$

r A

$2$ cos$(2$f$0$t$)($u$($t$)$ u$($t $$ T $))$

r A

$2$ sin$(2$f$0$t$)($u$($t$)$ u$($t $$ T $))(33)$

s$00($t$)=$

r A

$2$ cos$(2$f$0$t$)($u$($t$)$ u$($t $$ T $))$

r A

$2$ sin$(2$f$0$t$)($u$($t$)$ u$($t $$ T $))(34)$

The $4$ points indicated by the circles correspond to the coordinates of the transmitted signal in signal space.

This is one way to send digital information over an analog link: we modulate the digital information

$($bits$)$ into analog $($CT$)$ signals. Here we are using $2$degrees of freedom$($a fancy phrase for using $2$

dimensions, sine and and cosine, at the same time$)$ to send the bits. Note that in this scheme, the $2$ bits

get mapped as $0$

q A

$2$ and $1+$

q A

$2,$ where the first bit modulates the cosine and the second bit

modulates the sine$.$

At the receiver, we can use the following system to demodulate