# Calculating the signal-to-noise ratio for CW (Morse Code) signals?

How is the signal-to-noise ratio for received CW and QRSS CW (and/or other ultra narrow-band DSP digital modes) typically computed? e.g. What noise bandwidth should be used to measure the spectrum power against the narrow CW "on" signal power?

Should one use the DSP detection bandwidth (roughly 1X the WPM), the human decode bandwidth (approximately 3X to 5X the WPM according to an ISO or ITU document), the actual bandwidth used by the transmitter rise time, or some other standard radio bandwidth (SSB LSB filter width?)?

How should the CW S/N be computed to compare its data rate (at a given WPM) against the theoretical Shannon limit for that S/N?

Is the same S/N number meaningful depending on whether the noise is mostly Gaussian noise, white noise, or 1/f RF noise?

• I don't think there's a "typical" calculation, but only application-specific ones: Why would you care about some arbitrary SNR definition when it doesn't describe the problem your receiver is facing? May I ask the reason for your question? – Marcus Müller Jan 7 at 15:30
• What do academic papers use for comparison of actual (ultra narrow band?) RF signals versus the Shannon channel limit for S/N? Are any numbers for CW comparable? – hotpaw2 Jan 7 at 16:51
• I'd assume the most sensible papers, they'd use the effective bandwidth of the receiver when describing an S/N from a receiver's perspective. Since ultra-narrow band kind of suggests the channel is flat, it's fair to assume that every practical transceiver system would use matched filtering (since that maximizes the SNR on the flat, AWN channel), so that'd be the equivalent noise bandwidth of the matched filter. – Marcus Müller Jan 7 at 17:46
• So one answer for CW might be to use the ENB of a typical (averaged across the specs of many of the most popular HF rigs) CW filter? – hotpaw2 Jan 7 at 18:52
• hm, yes that's certainly something that would be representative. But I think you could honestly also argue very much on the actual bandwidth/WPM basis and define that a realistically fast keyed channel needs so and so many Hz bandwidth and no more. – Marcus Müller Jan 7 at 19:13

The answer can be fond in The ARRL Handbook 2019, Vol 3, although it's spreaded across different chapters. In short, SNR is typically calculated for the noise floor of 2500 Hz SSB signal. Particularly this is how WSJT-X calculates negative SNRs for FT8 and other modes.

Now the trick is that by deviding the bandwidth in half you decrease the noise floor by 3 dB assuming the noise is distributed evenly. When I use 500 Hz wide DSP filter in my Yaesu FT-891 I typically see S2-S3 noise floor on 20m, and S4 when 2500 Hz bandwidth is used. Now keeping in mind that one S-unit is approximately 6 dB we get:

>>> from math import log2
>>> 3*log2(3000/500)/6
1.292481250360578


And we really see 1-2 less S-units, as expected.

Now back to CW. The ARRL Handbook claims with the reference to "The ITU Classification of Emission Standards" that the bandwidth of CW signal can be estimated as:

BW = WPM * 0.8 * 5


For instance, a 17WPM signal has a bandwidth about 68Hz. This means that the signal can be received:

>>> 3*log2(2500/68)
15.600748614897329


... 15.6 dB "under the noise floor".