processing gain in the receiver

Question

I am always confused about the processing gain in the receiver. Assume my receiver had the following: $$\boxed{\text{LNA}}{\longrightarrow}\boxed{\text{Mixer}}{\longrightarrow}\boxed{\text{Filter}}{\longrightarrow}\boxed{\text{ADC}}$$

1. Does the filter in the analog section has any role to play with the processing gain?
2. I know if the ADC is 12 bit and 100 msps sampling its SNR is $$12\cdot6.02+1.76=74\rm \ dB$$ if I sample at 200 msps does processing gain increase?
3. If I am using FFT analysis the processing gain increases, but if there is no FFT how to increase processing gain in digital domain (by filtering)?
4. What is the significance of processing gain?

Answer 1

"processing gain" isn't something that a specific step in your signal processing brings you, for every possible signal, in every possible situation.

Instead: If you have knowledge about properties of your signal, you can use that to somehow discriminate signal from noise and thus increase the SNR. The increase in SNR is called a processing gain.

Notice how you didn't mention any properties? Especially none you could exploit for this?

Typical properties include narrow bandwidth (where a simple filter could reduce the amount of noise power and thus increase SNR), or being shaped with a specific filter to which you can use a matched filter (where you specifically design a filter so that it maximizes SNR in AWN), or you could know your signal is spread with a specific spreading sequence (where you can despread and thus exploit that you have power growing quadratically with the number of accumulated correlated signal samples, but only linearly with the number of uncorrelated noise samples).

But really, these aren't all properties one can exploit. There's things like known temporal behaviour, channel state information, and many nonlinear noise operations and much more that can have a processing gain for any specific system- and noise-model. So, you need to write that down and see how your processing step affects the SNR.

if i am using fft analysis the processing gain increases

No! Or, only for signals that match the implicit assumption you're making (which I don't know, but which seems to assume your signal is very narrowband and fits exactly into one of your FFT bins; don't know how often that is the case in an unsynchronized system).

but if there is no fft how to increase processing gain in digital domain

That depends on your signal model and your noise model, and we can't tell you that.

Answer 2

The processing gain is usually defined as the ratio of signal-to-noise ratios at the output and input of a signal processing system. Let's denote the processing gain $PG$, and $SNR_{\rm I}$ and $SNR_{\rm O}$ the signal-to-noise ratios at the input and output respectively, this means that

$$PG \triangleq\frac{SNR_{\rm O}}{SNR_{\rm I}}\tag{1}$$

In other words, the improvement in SNR. In a way, $PG$ in $(1)$ can be used as a measure of performance.