# Digital Pulse Amplitude Modulation(Part 1)

Note: I am organizing my notes on Telecommunications topics. I will post programming articles too, for those who look for those kinds of articles.

Update:  April 2, 2007 10:34 PM U.S. Central Time: fixed the typo from 2-PAM to 4-PAM

Digital modulation allows us transfer sequences of bits(digital signal) over analog media(like air). Pulse Amplitude Modulation(PAM) consists of mapping a certain number of bits into symbols. For example in 4-PAM(which is named that way because a pair of bits are mapped onto a symbol), if d denotes some scalar value(like 2?), a mapping table might be:

00 -> 3d
01 -> d
11 -> -d
10 -> -3d

where 3d, d, and so on are the amplitudes of a pulse shape we choose(more on choosing the pulse shape later). For now, if we choose a rectangle wave(even though it can never be implemented exactly in a real system), which can be defined $rect(t) = 0$ if $|t| > \frac{1}{2}$ and $rect(t) = 1$ if $t < \frac{1}{2}$, and $rect(t) = \frac{1}{2}$ if $t = \frac{1}{2}$.

The baud rate is just another name for the symbol rate. So, if we have a symbol rate of 36 baud (36 symbols/second), in the 4-PAM scheme above, our bit rate is 72 bits/sec(bps)

Also, the mapping above from bits to symbols is optimal because it is based on the Grey code, which is an ordering of groups of bits such that group differs from the consecutive by just one bit(this is cyclical because 10 is just different in one bit from 00). Using the 2-bit Grey code helps the receiver to perform some error correction(because consecutive levels such as -d and -3d “are only 1 bit away”

References on technical info: Dr. Brian L. Evans’ lectures(Google: Brian L. Evans or Digital Pulse Amplitude Modulation(Dr. Evans’ notes should be the first hit for Digital Pulse Amplitude Modulation), Wikipedia: Grey Code

Check out the comments for the “My Favorite Software Tools Post” for more info on embedding $\LaTeX$ in WordPress posts(info from Mr.Ali’s blog)

Examples:

$%latex \int_{-\infty}^{\infty}f(x) dx = 0$ produces $\int_{-\infty}^{\infty}f(x) dx = 0$

(You shouldn’t put the % in when you do this yourself)