# Nyquist Maximum data rate formula for PCM

The Nyquist maximum data rate formula for a binary PCM is given by

$$C = 2B\log_2L$$

I'm not very sure what "$$B$$" is here. Is it the bandwidth of the signal being sampled or is it the channel bandwidth? Also, is "$$L$$" the no. of quantization levels or is it the no. of M-ary symbols used for baseband transmission?

I'm also not sure about what it means when we refer to "Bandwidth of a PCM system"? Are we talking about the channel bandwidth or the bandwidth of the PCM signal itself? I looked up these things on internet and there are different explanations at different places.

Can someone please clarify this or at least point me to a reliable source where it is explained without any ambiguity! Thanks!

• Thanks for the edit, @AlexTP ! – Navin Jun 5 '19 at 10:38
• where did you get this formula from? And don't they have an explanation for the terms there ? – Fat32 Jun 5 '19 at 11:15
• @Fat32 I've come across this formula on various online resources. They all have different definitions of those terms. Some just mention "B" as "bandwidth" and leave it at that. Here are two such places: mason.gmu.edu/~rmorika2/… and witestlab.poly.edu/blog/… – Navin Jun 5 '19 at 11:39
• ok... let me put an explanation – Fat32 Jun 5 '19 at 11:41
• @Fat32 oh! good! Thanks for your time! – Navin Jun 5 '19 at 11:42

The first document (from mason.gmu.edu) gives a confusing explanation of the formulas. The second document is what's the standard way of referring to the channel capacity

Now the two formulas are:

1. $$C = 2B ~\log_2(M) ~~~~~~~~~~~~,~~~\text{Nyquist}$$
2. $$C = B ~\log_2(1 + \text{SNR}) ~~~,~~~\text{Shannon-Hartley}$$

Eventhough the first formula, (referred to as Nyquist in the first document), is assumed to yield channel capacity (of a noiseless! channel which is infinite) it's actually giving the necessary minimum data bit-rate to represent an analog message signal of bandwidth $$B$$ Hz, and quantized to $$M$$ levels ($$\log_2(M)$$ bits) using a PCM representation. According to the Nyquist sampling theorem, the minimum allowed sampling rate would be $$2B$$ Hz, hence there will be $$2B$$ samples per second each quantized to $$M$$ levels, yielding a total of $$D = 2B ~ \log_2(M)$$ bits per second data rate to represent the analog message signal. Hence it's not channel capacity, it's the minimum necessary bitrate to represent an anlog signal using $$M$$-bit PCM technique.

The second formula is what we call: maximum channel (reliable) transmission capacity in bits per second rates, or Channel-Capacity in short, over a more realistic noisy channel (such as telephone wires, all sorts of RF channels, and also the fiber optic or satellite links by modeling them with the inherent noise floors). It gives you the maximum achievable rate of information (or data) transmission through a channel of bandwith $$B$$ Hertz, by using a modulation scheme with the given SNR (ratio of transmitted signal power to the channel noise power).

Note that despite being treated in communications sections, PCM is not a data transmission format but a data representation format. Analog data is converted to digital by samlping and quantizaiton through an ADC of $$N$$-bit quantization at a sampling rate of $$F_s$$ samples per second. The resulting sampled-quantized data is practically represented in binary digital system for processing and transmission. In such a case every $$N$$-bit quantized sample of the message signal is laid out as $$N$$ consecutive binary digits (bits) to represent a single PCM sample of message data. When it comes to transmission, those bits can be transmitted one by one as is done by using BPSK or in groups of M-bits as is done by 8-QAM (where 8 PCM bits are transmitted at once by a single 8-QAM transmission symbol).

• added some explanation... – Fat32 Jun 5 '19 at 13:16
• Thanks! That clears it up! However, I really want to know more how the first formula, the Nyquist formula is derived. Is there any resource you can point me to where a derivation can be found for that? – Navin Jun 5 '19 at 13:29
• I also have some doubts about the Nyquist formula. Let me add some explanation. – Fat32 Jun 5 '19 at 13:36
• sorry but I had to totally change the answer for the first formula please read and reconsider. – Fat32 Jun 5 '19 at 13:45
• @Navin it should mean the bandwidth of the PCM signal when the bits are represented by specific transmission symbol pulses. But you should again reer to the document which talks about the PCM system bandwidth ? – Fat32 Jun 5 '19 at 15:50