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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? John Ragle -- W1ZI ______________________________________________________________ Elecraft mailing list Home: http://mailman.qth.net/mailman/listinfo/elecraft Help: http://mailman.qth.net/mmfaq.htm Post: mailto:[hidden email] This list hosted by: http://www.qsl.net Please help support this email list: http://www.qsl.net/donate.html |
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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 ______________________________________________________________ Elecraft mailing list Home: http://mailman.qth.net/mailman/listinfo/elecraft Help: http://mailman.qth.net/mmfaq.htm Post: mailto:[hidden email] This list hosted by: http://www.qsl.net Please help support this email list: http://www.qsl.net/donate.html |
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In reply to this post by Kok Chen
The Shannon formula, and in particular the error free condition, is only
valid for gaussian noise. It is not valid for, for example, impulsive noise. It also only strictly applies when the coding delay goes to infinity. (If the "noise" is predictable, it can exceed the Shannon limit. One also has to be careful with the signal bit rate, e.g. contest exchanges tend to have quite a low bit rate, because so much of them is predictable.) Currently I think we are within a few percent of the Shannon limit, but no real system actually reaches it. 56kbps modems don't rely on source coding, except, possibly, in as much as all modems, other than FSK ones, require scrambling. The enabling technology for 56 kbps modems is actually echo cancellation. They work because there isn't really a modem at all at the ISP side, and the link from the local exchange, where the the D/A happens, to the subscriber, is relatively noise free, so the receiver can actually identify all 256 D/A convertor output levels (this is complicated by the levels being non-linear, low ones are closer together than high ones). The ISP end put "PCM" codes directly onto the digital wire. No clever coding is needed for this, but you do need to accurately subtract out the uplink signal, so it requires good echo cancellation. The actual SNR at the input to a subscriber modem is much worse than the Shannon limit allows, but the input "noise" is not random, so not really noise, and can be predicted and subtracted out. Top quoted for policy, not effectiveness. Kok Chen wrote > > 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. > > 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. > -- David Woolley "we do not overly restrict the subject matter on the list, and we encourage postings on a wide range of amateur radio related topics" List Guidelines <http://www.elecraft.com/elecraft_list_guidelines.htm> ______________________________________________________________ Elecraft mailing list Home: http://mailman.qth.net/mailman/listinfo/elecraft Help: http://mailman.qth.net/mmfaq.htm Post: mailto:[hidden email] This list hosted by: http://www.qsl.net Please help support this email list: http://www.qsl.net/donate.html |
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On Oct 5, 2010, at 12:36 AM, David Woolley (E.L) wrote: > The Shannon formula, and in particular the error free condition, is only valid for gaussian noise. Yes, the main Shannon Theorem (Theorem 17) mentions "thermal" noise. (In addition to being Gaussian, Theorem 17 also requires that the noise be white.) However, check out Theorem 18 in the same paper. It establishes upper and lower bounds for the Channel Capacity for non-Gaussian and non-white noise. > 56kbps modems don't rely on source coding, except, possibly, in as much as all modems, other than FSK ones, require scrambling. FSK is of course just a modulation scheme, so it makes no sense to talk about FSK source coding. But if you consider Baudot RTTY, which is FSK+Baudot, then yes, Baudot RTTY does rely on source coding to maximize the number of characters per bit as long as you don't switch often between text and punctuations/numbers. As far as scrambling/interleaving goes (at least with their presence in systems such as MFSK16 and SITOR-B), I prefer to think of them as a form of channel coding (whose primary objective is to minimize channel errors) instead of a form of source coding (whose primary objective is to reduce the average bit rate of data that is sent to the channel coder). 73 Chen, W7AY ______________________________________________________________ Elecraft mailing list Home: http://mailman.qth.net/mailman/listinfo/elecraft Help: http://mailman.qth.net/mmfaq.htm Post: mailto:[hidden email] This list hosted by: http://www.qsl.net Please help support this email list: http://www.qsl.net/donate.html |
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