$4.$ Calculate the maximum data rate of the channel.

$1)$ A noiseless $10-$kHz channel is sampled to $0,1,2,$ or $3$ every $1$ msec. What is the maximum data rate?

the maximum data rate is $40,000$ bits per second $($bps$).$

$2)$ How does the maximum data rate change if the channel is noisy, with a signal$-$to$-$noise ratio of $20$ dB$?$