It is important for the users and designers of RFICs to know the theoretical limit of these receivers so that they can determine if their design improvements are fully successful.
This application note describes a step-by-step method to predict the sensitivity of an ASK receiver, given a system noise figure, IF bandwidth, and Baseband bandwidth.
The results show that the logarithmic amplitude detection in the Received Signal Strength Indicator (RSSI) amplifier decreases the output Signal to Noise Ratio (SNR) for low input SNRs (threshold effect) and that the sensitivity increases as the square root of the IF to Baseband bandwidth ratio.\t
Because the RSSI detector is a nonlinear detector, it changes the Signal to Noise Ratio (SNR) of the signal that goes into it. The key to the ASK sensitivity calculation is the SNROUT vs SNRIN curve of the RSSI detector.
Once we know the SNROUT vs. SNRIN relationship, the steps to finding the ASK sensitivity for a given Noise Figure, IF Bandwidth, and Data Rate are given below.
Where kT is the noise spectral density at 290K (-174dBm/Hz) BIF is the IF (pre-detection) BW, and FS is the system (not just the front-end) noise figure of the receiver.
Because the RSSI detector is a logarithmic detector, the SNR input-output relationship can be expressed in a closed-form expression, albeit a messy one. An old paper published in the IEEE Transactions on Aerospace and Electronic Systems derived the expression and plotted the SNROUT vs SNRIN curve. The curve in the article is small and doesn't have enough gridlines, but it is possible to evaluate the expression in an Excel spreadsheet and plot it in better detail. The curve appears below, plotted along with a simple SNROUT = SNRIN curve (linear detection) for comparison. Notice the threshold effect. Below the "crossover point" SNR of 3.7dB, the SNR gets worse going through the detector. Above this point, it improves.
Another Excel spreadsheet incorporates Steps 1 through 4 above with the SNROUT vs SNRIN curve to produce the sensitivity calculations shown in the next graph. They are plotted as Sensitivity vs. Data Rate for three IF bandwidths, using 7dB for a Noise Figure. Notice that sensitivity improves roughly as the SQUARE ROOT of either the IF BW or the Data Rate. This is because at sensitivity we are working in the range of the RSSI SNR curve where the slope of SNROUT to SNRIN is roughly 2 (a square-law relationship) in log scale.
The curves are consistent with practical experience for carefully designed ASK receivers. For instance, at a 3 kbps data rate and 280kHz IF BW, the sensitivity is -114dBm. An 11dB Eb/No, corresponding to a 10-3 BER for ASK, is used in this calculation, which leads to about 12dB SNR for a steady CW signal ("peak" Eb/No is 14dB because data is 50% duty cycle on average, less about 2dB for the ratio of 1.5:1 of BBW to Data Rate).
It is important to point out two assumptions made here: (1) That the noise bandwidth at the output of the RSSI detector is the same as the IF bandwidth, and (2) that the noise distribution at the output of the RSSI detector is Gaussian. In fact, the noise bandwidth of the RSSI detector might be much larger than the IF bandwidth. This can be taken into account by increasing the effective system noise figure. The output noise distribution is not Gaussian, so a complete analysis would require calculating probabilities of error for the exact noise distribution at the RSSI output. We believe that the difference in Eb/No for a given BER is small, and that it will not change the fundamental results of this paper listed below.
Bales, C. W., "A Comparison of Logarithmic and K-th Law Detectors", IEEE Trans. AES, July 1978, pp. 693-696