# Quantization levels and SNR

Im' trying to build a big picture of telecommunication and i'm stuck in a topic, related to digital communication. But the problem is that i can't go too deep into the topic because it is imensely huge and due to time constraint, so i'm trying to get an informal and intuitive big-picture ( but as accurate as i can ).
From what i could read, we can think informally that the probability error is dependant solely on two things :

1 - SNR
2 - Our channel codec ( coder-decoder )

If we implement a better channel codec, we need to pay less SNR for the same probability error.

And also from not-to-deep research, i could pin that SNR would be solely dependant on 2 things :
1 - Signal power at the receiver. This is mainly dependant on our channel characteristics regarding path loss , on the desired bit rate ( which dictates the bandwidth required ) and on the delivered signal power at the transmitter.
2 - Noise power at the receiver. This is mainly dependant on our channel noise resistance ( how prone the channel is to all kinds of internal noises and external noises ), quantization noise, ISI,etc.

My first question :
I - Apart from the perharps informal and vague treatise, is there some major thing i'm missing on the picture or is there some major thing i have a misconception ?

The second question is :
II - I have read in a text-book that on a digital communication system with A/D step, SNR is directly proportional to the number M of quantization levels : This makes total sense because by increasing M, we end-up decreasing the quantization noise and hence decreasing the noise power at the output. By decreasing the noise power at the output, we would increase SNR.

But i'm thinking here that whenever we increase M, we are actually increasing the bit-rate ( with M=128, each symbol is worth 5 bits but with M=256 each symbol is worth 6 bits,so for the same symbol rate, we need to increase the bit rate ) , and hence decreasing the signal power at the output ( because of a higher bandwidth required ) and hence decreasing SNR.
So, to me, increasing M would not necessarily increase SNR as my text-book says...
Is my reasoning correct ?

So these are my only two questions. Thanks a lot in advance.

## 1 Answer

In a "purely digital" link where you set an output to "high" and an input the other end of a line is read as "high" then the probability error is purely to do with the SNR of the line. What is the probability that a HIGH can be interpreted as a LOW? By introducing a higher level protocol with error detection and correction you effectively negate most of the SNR errors and the question is now "What is the probability that the protocol cannot correct corrupted bits?"

So yes, the CODEC (or protocol) can be used (and is used) to negate the effects of SNR-induced signal corruption.

As for the second part...

If you assume 1 bit of information is transmitted per quantization level, and 1 bit is received per quantization level, then yes, increasing the quantization level will increase the number of bits sent at any one time. However, the SNR of the transmission medium will then have a greater effect on those now smaller quantization steps, so although you reduce the quantization noise, you now increase the SNR noise.

However, if you don't assume 1 bit per quantization level, but have multiple quantization levels per bit, then you can increase the number of quantization levels and keep the overall bitrate the same, but have more detail about each bit, so can make a better informed decision about what value that bit is.

For instance, you can think of a simple digital link with 2 states (HIGH and LOW) as a 1-bit quantized system. For simplicity we'll call it 1V for HIGH and 0V for low.

Now, you could then have it that anything received >= 0.5V is a HIGH and anything < 0.5V is a LOW. That's 1 bit quantization. 0.5V would be HIGH, but 0.499999999999V would be LOW. That's an infinitesimally small margin for noise.

However, increase the receiving quantization to 2 bits, say, would give you more detail. It would give you 4 voltage levels to consider - 0V, 0.33V, 0.66V and 1V.

You could now say that anything > 0.66V is a HIGH, and anything less than 0.33V is a LOW. You have now introduced a "noise margin". Anything that falls between those values is discarded as noise. The bitrate remains the same, but the overall SNR has fallen.

Then of course you can add a "schmitt trigger" to it (or software equivalent), whereby you toggle the value depending on a transition. When the input rises above 0.66V you see the value as HIGH, and keep it as HIGH. Only when it then drops down below 0.33V do you then switch it to LOW.

For systems where you have discrete voltage levels you could sample them at a higher resolution, and the line-induced noise would occupy the least significant bits of that sampled value. Discarding the noisy bits down to the resolution of the sent data can then reduce the noise in the system. Also taking multiple samples and averaging them, which in effect cancels the random noise out, (known as "oversampling") can reduce the noise as well.

None of those techniques affect the bitrate as such since you're not adding any extra information to the sent values.