AX.25 Packet Radio AFSK on FM
Bit,
Packet and Data Set Error Rates
Ralph Wallio, WØRPK WØRPK at netINS.net
Error
detection qualities of AX.25 Amateur Packet Radio protect the integrity of data
as presented to data terminal devices but reception of packets before error
detection is not error free. This discussion covers principles which determine
errors per quantities of bits, bytes and packets (error rates) and places
bounds on reasonable expectations.
We will touch on the relationship of Signal to Noise Ratio (SNR also known as Eb/No
which is expressed in dB) as experienced by a data demodulator to Bit Error
Rate (BER). Then the relationship of BER to Bit Probability of Reception (BPR).
Then the relationship of BPR to Packet Probability of Reception (PPR). Finally
we will extend the relationship to Data Set Probability of Reception (DSPR).
Bit Error Rate
Audio frequency shift keying (AFSK) over VHF/UHF FM is easy to implement via
typical inexpensive Packet Radio modems (MODulator/DEModulator) but these
modems require high quality channels. Back in the early days of Packet Radio,
Steve Goode, K9NG, made one of his important contributions to Amateur Radio by
publishing results of exhaustive BER testing [1]. A partial result of Steve's
analysis is an Eb/No performance curve that is reduced to the following table:
BER 

Eb/No (dB) 
BPR 
1.6x10^5 
0.0016% 
24.0 
99.998% 
1.0x10^4 
0.01% 
22.5 
99.99% 
1.0x10^3 
0.1% 
19.5 
99.9% 
1.0x10^2 
1.0% 
17.5 
99% 
1.0x10^1 
10.0% 
15.5 
10% 
Note
that a change of just a couple of dB makes a significant difference in BER.
Above the best, but not perfect, BER the curve is flat. Said another way,
"this is as Goode as it gets". We will never experience continuing
100% error free packet transmission before error detection.
Bit Probability of Reception
Bit Error Rate (BER) is the probability of not receiving a transmitted bit
properly. It is expressed as a percentage or in decimal form, e.g., 0.1% or
1x10^3. In this example, the chance of receiving a transmitted bit incorrectly
is one in every 1000 bits or, stated positively, the chance of receiving a
transmitted bit correctly is 100%0.1% = 99.9% Bit Probability of Reception
(BPR).
Packet Probability of Reception
Packet Probability of Reception (PPR) recognizes that reception of every bit in
a packet is independent of every other bit and that every bit has to be
received correctly. (This assumes use of AX.25 UI frame error detection with
PASSALL turned OFF.) The probability of receiving N consecutive bits is BPR^N
so PPR = (1BER)^N where N is the total number of bits in a packet.
The size of AX.25 packets can vary significantly from just bare U and S frames
with no digipeating addresses and no data bytes (152 bits) to packets with a
maximum number of digipeating addresses and the maximum of 256 data bytes (2656
bits). A typical APRS navigation UI packet might have 2 digipeating addresses
and 65 data bytes (792 bits). We can apply BER values from the above table to
these packet sizes to understand Probability of Packet Reception (PPR):
Packet Size 
1.6x10^05 
1x10^4 
1x10^3 
1x10^2 
1x10^1 
152 
9.97601 
9.84901 
8.58901 
2.17001 
1.10907 
792 
9.87401 
9.23901 
4.52801 
3.49204 
5.75537 
2656 
9.58401 
7.66701 
7.01402 
2.55312 

At a
BER of 1.6x10^5 which results from 24dB Eb/No (better than full quieting
gauged by ear) the probability of receiving a 792bit APRS packet is 9.874 out
of every 10 sent, or 98.74%. A couple of steps down the EB/No ladder at 19.5db
and a BER of 1x10^3, the probability of receiving the same APRS packet drops
to 4.528 out of every 10 sent, or 45.28%. The next step down, 17.5dB Eb/No and
a BER of 1x10^2, produces no copy.
Data Set Probability of Reception
What if we had to transfer a 30Kbyte binary file using only error detection
(not error correction which requires an AX.25 connection)? Every bit, byte and
packet has to be received correctly or the whole file is bad. Assuming no digipeating,
this 30Kbyte file will require 117 256byte packets each containing 2208 bits
for a total of 258,336 bits.
Our PBR formula still applies but it is now used for Data Set Probability of
Reception, DSPR = (1BER)^N, for N = 258,336:
Data Set 
1.6x10^05 
1x10^4 
1x10^3 
1x10^2 
1x10^1 
258,336 
1.60302 
6.02612 



At
the same "this is as Goode as it gets" BER of 1.6x10^5 the
probability of receiving 117 consecutive 256byte packets without an error is
1.603%. Clearly it is difficult if not impossible to transfer files of this
magnitude without error correction.
[1] Steve Goode, K9NG, BER Performance of TAPR TNC Modem, ARRL QEX, August 1983