[Elecraft] Digital, Smigital...

[Kok Chen]

On Oct 4, 2010, at 12:41 PM, John Ragle wrote:

>  I must be missing something here. How can one expect high fidelity audio 
> (e.g. 20 HZ to 20 kHz) with a receiver with a pass-band of 2.5 or 3.0 
> kHz?
> 
> With those strictures, one is always going to get "carbon mike" or 
> slightly better audio, no?

No.

The data rate through a channel depends not just on the analog bandwidth of a channel, but also the SNR of the channel.

See the section "The Capacity of a Continuous Channel" in Part IV (Continuous Channel) in Shannon's 1948 BSTJ paper:

http://cm.bell-labs.com/cm/ms/what/shannonday/shannon1948.pdf

Channel capacity in bits/sec is equal to W, the bandwidth, multiplied by the log(base 2) of the (S+N)/N ratio.  

With appropriate modulation and error correction coding, [Shannon shows that] you can transmit a digital signal, with *any* arbitrarily small, non-zero error bound that you wish to set, at a data rate up to the channel capacity.

E.g., 

Consider a 3 kHz wide channel with (S+N) to N ratio of 30 dB.  The power ratio is 1000, thus log2 is 9.96 [2 to the power of 10 = 1024, so log2(1000) is just under 10].  With appropriate modulation and coding, from Shannon, you can potentially get almost 30 kbits/sec of digital data through such a channel.

A practical modem from the 1980s can do 28.8 and 33.6 kbits/second through a 3 kHz telephone circuit.  The 56 kbits/sec modems achieve the higher rate by using source coding in addition to channel coding, but they cannot maintain 56 kb/s with completely random binary data.

Today's DSL modems can do even better (much better), but they depend on the landline's capability to send, albeit attenuated, signals beyond the 3 kHz voice band.

Off Topic:

You can perhaps understand why some of us revere Claude Shannon much more than we do Albert Einstein :-).

http://en.wikipedia.org/wiki/Claude_Shannon

The above Wiki article also refers to Shannon's Master's thesis which connected relay circuits with Boolean Algebra -- making it possible to talk about AND gates and OR gates.  David Huffman, also in a Master's thesis, added the memory element (what we call flip-flops today).  The combination is what makes it possible for me to type this and for you to read it :-).

73
Chen, W7AY