# Why is discrete-time spectral efficiency of 2-PAM stated as 2 bits per 2-dimensions?

In fact, 2-PAM uses one dimension and transmits one bit per symbol. So I see it as 1-bit per 1-dimension. But in communication theory, it is said to be 2-bits in 2-dimensions, and thus is equivalent to 4-QAM in that regard.

You might say that 2-PAM could send two bits if is used two dimensions, but in fact it doesn't.

I suspect this is done to make the math nicer, but I haven't heard that stated.

Edit to add (for anyone finding this question):

First, I found the book by Proakis and Salehi, "Digital Communications" Fifth Edition, explains the concept of dimensionality in Section 4.6-1. It is a much more general concept than just axis on the complex plane.

To restate my original question: I had learned in a course I am watching that in comparing continuous-time and discrete-time systems, the discrete-time parameter "bits per 2-dimensions" maps value-wise to the continuous-time parameter "bits per second per Hz". The example of 2-PAM was used and it was said that 2-PAM could be viewed as 2-bits per 2-dimensions, which is the same as 4-QAM. This seemed wrong since I thought 4-QAM has twice the spectral efficiency as 2-PAM.

You would normally think of 2-PAM as being 1-bit per 1-dimension (which is true), but in order to map it (value-wise) to "bits per second per Hz" it must be expressed as "bits per 2-dimensions".

So in saying that 2-PAM can be viewed as 2-bits in 2-dimensions, the second dimension is not in the complex plane, but is another dimension, namely time. In other words, you can send two 2-PAM symbols in sequence to achieve 2-bits per 2-dimensions. I had previously discounted this idea because it would halve the (continuous-time) spectral efficiency, and thus 2-PAM would be still be half that of 4-QAM.

Fortunately, the Proakis book reminded me that 2-PAM requires half the bandwidth of 4-QAM (using SSB), and so indeed they do have the same spectral efficiency. That key fact is what caused my confusion, but at least it led to me better the understanding concept of dimensionality, which is important in coding.

• What definition of "dimension" are you using? Also: any method of comparison that says 2-PAM and 4-QAM are equivalent is flawed.
– MBaz
Commented May 4 at 19:32
• @MBaz In simple cases, dimension relates to the Re and Im axis of the symbols. But I believe the concept is more general than that. It is used as the discrete-time equivalent of bits per second per Hz. This concept is used in the MIT courses on comm theory which use a lot of mathematics. Commented May 4 at 21:51
• I recommend you to read on signal spaces and orthogonality. The correct definition of dimension is related to the basis of the signal space. Also, this is different from calling a real constellation "1-D" and a quadrature constellation "2-D".
– MBaz
Commented May 6 at 13:07

But in communication theory, it is said to be 2-bits in 2-dimensions

That would be wrong, simple as that. It transports 1 bit per symbol, no matter how many dimensions you have.

If you use complex signalling, it's still just 1 bit per symbol, not 2. So, it's "1 bit in 2 dimensions", even if I've never actually read that as unit.

Us communications engineers typically simply speak in bit/s/Hz, as that's the relevant metric.

You might say that 2-PAM could send two bits if is used two dimensions, but in fact it doesn't.

It doesn't, so you can't act as if it did; full stop.

• I am probably not presenting the concept very well. I am watching courses on YouTube, and maybe things will become more clear as I go (e.g. as I get deeper into coding theory). The instructors are quite famous so I have to assume they know what they are talking about :). Commented May 5 at 19:50
• I honestly do always assume people know what they talk about; but the extrapolation step you're doing here from "1 bit if I just use real signalling, so 2 bits if I use complex" is wrong! If you do that, you end up with something that's simply not called 2-PAM. Commented May 5 at 19:51