James Miller, G3RUH, has been a prolific writer and very hard worker in the radio amateur
digital communications field. This page presents three of his now classic papers on "modern"
digital communications. James presents his ideas in such a down to earth, practical manner
that I think it is appropriate to reprint these papers here as tutorial information. Modern
digital communication system designers can learn a lot from James' ideas and approach to
practical communications theory.

All of this work is the sole copyright property of James Miller, G3RUH, and this information
can ONLY be used for educational or ham radio related, non-commericial applications. We thank
James a great deal for allowing us to reprint this valuable tutorial here in SSS Online.

ABSTRACT: The Shape of Bits To Come
from: Oscar News (UK), 1991 Apr No. 88 p 29-37

This article is about bits, about bandwidth and about control of both.
What is the "raised cosine modulation" used on microsats AO-16/18/19? And
what is the RSM-8 on Rudak/AO-21? What PSK modulation does Fuji-20 use?
Come to that, what is BPSK, and similar abbreviations? Substantial
tutorial article, copiously illustrated.

Also: Amsat Journal (US) July 1991.
Also: ARRL 10th Computer Networking Conference (US), Oct 1991 pps 112-120.
Also: Amsat-DL Journal Jg.18 Nr.4, December 1991,
Also: Radioaficionados (Spain) Enero 1992.
Also: Amsat-OZ (Denmark) 1992 October, No. 8
Also: Radio Communication (RSGB) 1993 September & October.
Also: SM-Info (Sweden) 1994 Jan - 1995 Dec
Also: "Packet: Speed, More Speed and Applications", ARRL 1995.ISBN 0-87259-495-5

It is available on-line from ftp.amsat.org as the file:

This placement: ftp.amsat.org /amsat/articles/g3ruh/a108.zip
Date: 1995 Dec 19

First placement: Amsat-UK's Oscar News, 1991 Apr No.88 p.29-37
Corrections: Minor edits
Size: 26,000 bytes 4200 words 560 lines

Other placements:

Amsat Journal (US) July 1991.
ARRL 10th Computer Networking Conference (US), Oct 1991 pps 112-120.
Amsat-DL Journal Jg.18 Nr.4, December 1991,
Radioaficionados (Spain) Enero 1992.
Amsat-OZ (Denmark) 1992 October, No. 8
Radio Communication (RSGB) 1993 September & October.
SM-Info (Sweden) 1994 Jan - 1995 Dec
"Packet: Speed, More Speed and Applications", ARRL 1995. ISBN 0-87259-495-5

If I have missed out your magazine, please advise!

THE SHAPE OF BITS TO COME
-------------------------

James Miller BSc. G3RUH
3 Benny's Way
Coton
Cambridge
CB3 7PS
England
E-mail: g3ruh@amsat.org

ABSTRACT

This tutorial article is about bits, about bandwidth and about control of
both. Lately a number of expressions have crept into amateur radio data
transmission, creating both interest and confusion. What, for example, is
the "raised cosine modulation" used on microsats AO-16/18/19? And what is
the RSM-8 on Rudak/AO-21? What PSK modulation does Fuji-20 use? Come to
that, what is PSK anyway? This piece is not just "about satellites", or
"about packet". The principles discussed here form the foundation of all
data transmission, and ought to be as familiar as Ohms law. Read on!

Let's come clean at the outset; "raised cosine", RSM-8, PSK and DPSK are
the same thing. They're all forms of PSK - "phase shift keying".

In the amateur environment we are usually trying to send data as fast as
possible within a limited AF or RF bandwidth. This limit may be set by the
receiver alone, or it may be due to some statutory, aesthetic or technical
requirement. So a very important part of data transmission concerns the
data's spectrum.

UNDERSTANDING PSK

Let's begin with a statement and explain later:

PSK spectral considerations at RF can be analysed by examining the
characteristics of an isolated bit at "DC" or baseband.

I have to start somewhere, so I'll make the assumption that you have in
your head an image that "data" is the sort of regular signal you have
running down a piece of cable. On a 'scope it looks like random highs and
lows spaced at regular intervals. Each of these elements is called a
"bit", and they flow at the "bit rate", described as 9600 bits/sec or
sometimes "9600 baud" depending on the context. See the waveform "Data
In" of fig. 1.

Fig 1. PSK is usually generated by a DSB balanced modulator. So DC
spectrum is simply shifted to RF. Data "1"s give carrier phase 0,
data "0"s give phase 180.

There are many ways to modulate this data onto an RF carrier, for example
ON/OFF keying, carrier FSK, two-tone AFSK and so on. Here I am concerned
with double sideband modulation, DSB. That is, the data signal drives a
balanced modulator whose other input is the RF carrier. The output then
is simply a frequency shifted replica of the input data, much the same
as when the input is speech.

Now binary data has only TWO states. It is either a "1", represented by
+1 volt say, or a "0", represented by -1 volt. And that means that the
modulated RF carrier also takes only TWO states; either it has phase 0
degrees or 180 degrees. This is because compared with the +1 volt level,
the -1 volt inverts the carrier. That means a 180 degrees phase shift.
Voila! Binary Phase-Shift-Keying or BPSK. The term PSK is often used by
default, but is actually imprecise as it embraces other shifts as well as
0/180 degrees. "Antipodal PSK" is correct but long-winded.

It follows then that if we know the spectrum of the "baseband" data signal,
then the RF spectrum is just its replica above and below the carrier
frequency, because it's DSB, double sideband suppressed carrier.

If in addition, the data signal has amplitude variations, then these will
translate into identical amplitude variations at RF.

This is a very important equivalence, and it's worth restating:

"Binary data DSB modulating an RF carrier" and "binary phase shift
keying (BPSK)" are exactly the same thing. If we want to control or
analyse the RF spectrum characteristics, we only need to control or
analyse the source data characteristics.

UNDERSTANDING DATA STREAMS AND ISOLATED BITS

We now make a reasonable assumption: the data stream consists of random
bits. If the data is random, for the purposes of analysis we don't then
need to know anything about the content of the message stream. All we
need to know is the properties of ONE ISOLATED BIT. Whatever properties
we can discover about the isolated bit will also apply to the average
summation of the random assortment of them that make up the stream.

So our problem of spectral considerations at RF can be analysed via the
relatively simpler job of examining the characteristics of an isolated bit
at source level, DC or, as it is appropriately called, "baseband".

As a communications theory aside, it is worth noting that with efficient
communications, that data MUST in fact be random. For if it were not,
that would imply something systematic about it. So the data could
actually be represented or coded more efficiently, resulting in less
transmitted bits, which would then have a random distribution.

UNDERSTANDING BIT, EYE & RF SPECTRUM PLOTS

Now let's examine some representative cases. What happens if we modulate
the carrier directly with unprocessed rectangular bits?

Fig.2 shows three things a) an isolated rectangular bit, b) its RF
spectrum and c) what the data stream looks like on an oscilloscope. Of
course the bit edges are in reality almost invisibly fast, so I have
emphasised them slightly.

Fig 2. Rectangular bit, its eye and spectrum. See text for details.

a) Isolated Bit. The horizontal axis is time, and is marked off in bit
intervals. (Eight are shown). The vertical is voltage.

b) RF Spectrum. The horizontal axis is frequency, and is marked off in
units of the bit rate R bps. For example, if the data rate is 1200
bit/sec (R = 1200), the vertical markers are at 1200 Hz spacing. The mid
point of this spectrum is of course the RF carrier frequency. The
vertical is marked off at 10 db intervals. Before modulation, at "DC" or
baseband, the audio spectrum is just one half of this, the right of Fc,
which therefore represents 0 Hz

c) Oscilloscope diagram.

The horizontal axis is time, and spans a total of two bits; so it's
expanded x4 compared with the isolated bit. The vertical is voltage. You
are to imagine the 'scope is triggered by a local bit-rate clock. The
resultant pile-up of bits is what you see displayed. It is the sum of
many positive and negative isolated bits, each displaced in time by one
bit.

If this display is the received waveform, it's what is sampled by
the data detector to decide if a "1" or a "0" has been received. Usually
this decision is taken at the mid-time point of the received bit by a
sampler. Above the horizontal line is the regime of "1"s, below "0"s.

This display is also universally known as an "eye diagram". It is a
concise representation of the quality of the received signal. The less
confused the trace, the greater the distinction between high and low at
the mid-bit sample point, then the more reliable (less error prone) bit
detection will be.

These traces show no channel noise. That is to say, the signal to
noise ratio is assumed high. If there were noise, the traces would be
blurred. If the noise were momentarily large enough to cause the waveform
accidentally to cross the middle horizontal line, an erroneous detection
decision would be made, leading to a received bit error.

RECTANGULAR BIT

Returning to our rectangular bit shape. Two things are clear from figure
2; first, the extremely high fidelity of the data waveform and second,
the price for this, profligate use of bandwidth. The diagram shows
significant energy spreading well beyond 10 times the data rate. Indeed,
as far away as 30 R (R = bit rate), the sidelobes are still only -40 db
down. As sufferers from computer hash will know, rectangular bits are
effective noise generators.

In some applications (for example spread spectrum communications) this
might be the desired result. In the amateur radio environment where we
want to get as much data through the limited bandwidth of our receivers,
more finesse is required

Obviously we need to filter the data stream before transmission, because
this will attenuate those sidelobes and constrain the bandwidth.
So the key issue, indeed perhaps the central problem of all data
transmission, is "what form should this filter take"?

PERFORMANCE OF A SIMPLE R-C NETWORK DATA FILTER

We tentatively try a simple filter. Let's pass the data stream through an
R-C network with a -3db point equal to half the data rate. The result is
shown in figure 3.

Fig 3. Rectangular bit through R-C filter has classic exponential rise
and fall, and RF sidebands are reduced slightly.

Look at the spectrum; while the main lobe is hardly changed, the sidelobes
are reduced by 10 db or more, which is what we wanted. Look at the
isolated bit; it has become rounded. But note particularly that its
duration now exceeds one bit. In fact it's stretched to roughly two bits
long.

This means that successive bits will overlap. In turn this brings a new
potential design problem; inter-symbol interference, or ISI. Now we find
that not only do we want spectrum control, but we have to do it with the
constraint that successive overlapping bits must somehow not interfere with
each other.

Finally, look at the 'scope trace. The line at the top corresponds to the
sequence 111..., that at the bottom 000.. . The sweeps from top to bottom
are caused by ..10.. and ..01.. transitions and so on. You can also see
that the traces are not confused; highs and lows of successive bits are
separable, so there is no significant inter-bit interference.

INTER-SYMBOL INTERFERENCE REVEALED

In an attempt to attenuate the sidelobes more, let's now double the RC
filter's time constant. That is, the -3db frequency is now 1/4 the bit
rate (R/4). See Figure 4.

Fig 4. But too much ill-designed filtering gives inter symbol
interference (I.S.I.).

Indeed the sidelobes are reduced; but the real casualty can be seen in the
'scope trace. Under certain conditions (for example the sequence ..010..
or ...101... ) the voltage barely gets half way up to the top line. Sure,
bits are still distinct and detectable, but the noise margin is
drastically reduced by some 50%.

Analytically, the reason for this is seen in the isolated bit diagram of
figure 4. The bit starts off at T=0. At T=1 it reaches its peak. At
T=2, it hasn't fallen back to zero again, but has a value approximately
1/3rd peak. This aberration is the sole cause of the inter-symbol
interference apparent in the 'scope trace. At T = 3,4 etc, the voltage
is back to zero again.

From this observation we can formulate the requirement for a bit shape
that guarantees no ISI. There should be a peak of unity at T=0. Then at
all other exact bit points (T= -2, -1, +1, +2, etc) the isolated bit
waveform should be exactly zero. What happens in between doesn't matter, at
least, not from the ISI point of view.

FO-20's DATA FILTER

An example of a bit shape that meets this requirement is that transmitted
by the satellite Fuji-Oscar 20. The data filter is (I believe) a 3rd
order Bessel type with a -3db point at 0.532 times the bit rate (0.532 R).
"Bessel" filters driven by square waves have a nice steady rise, and
negligible overshoot or ringing. In electrical circuit terms, the filter
is merely 3 resistors, 3 capacitors and an op-amp. The filter is shown in
figure 5, responses in figure 6.

Fig 5. Packet satellite FO-20's 3rd Order Data filter is similar to this.
Waveshapes as figure 6.

Fig 6. Characteristics of FO-20 Data. The far-out RF sidelobes are much
more rapidly attenuated than in figure 4, yet there is no confusion
between adjacent bits (no I.S.I.).

The spectrum is quite reasonably controlled; 99.9% of the energy is
contained within a bandwidth of 1.75 R (as compared with 15 R of the
rectangular pulse). Since the FO-20 bit rate is 1200 bps, this means that
most of the signal occupies an effective RF bandwidth of the order of 2.1
kHz, so it stands a fair chance of passing through the SSB receiver
filters without significant distortion.

The isolated bit shape is zero at all T points except at one, and this
leads to a nice 'scope trace. The origin of the term "eye diagram" can be
seen here, as the trace is supposed to resemble a human eye. Notice
that as a consequence of negligible inter-symbol overlap in the isolated
bit, the 'scope trace has perfect bit convergence at the sample point.

The isolated bit has a duration of just 2 bits. This means that there can
be only 2^2 = 4 trajectories on the 'scope trace (00,01,10, and 11). The
01s and 10s show up as a very clean zero crossover. If the RF bandwidth
were narrower then the isolated bit would span more than 2 bits, so the
'scope trace zero crossings would show dispersion.

AND NOW - "RAISED COSINE"

The term "raised cosine" means several things, a fact which causes a lot
of confusion. A raised cosine is just that. A cosine function raised
above its mid point. The formula for a raised cosine is ( 1 + COS x )/2.
Drawn as a graph, it's merely a shape. But it has a number of analytical
properties that make it convenient to employ. These are simplicity,
left-right and upper-lower symmetries. (See fig 7a).

Fig 7a. A raised cosine shape. There's nothing mysterious about a raised
cosine. Just get out your calculator and plot y = (1 + COS x)/2 for x =
-180 to 180 degrees!

A RAISED COSINE BIT

"Raised cosine" can for example, describe the shape of the isolated bit -
see figure 7. Since the isolated bit is cosine shaped, the 'scope trace
is composed entirely of sine waves and straight lines. Indeed, the bit
sequence ..10101010.. actually creates a pure tone at a frequency R/2 at
baseband, or at RF, two frequencies spaced by +/- R/2.

Fig 7b. Raised cosine bit TIME shape used (in principle) on PacSats AO-16/
WO-18/ LO-19.

The isolated bit shape is very similar to FO-20's, occupying 2 bits. Not
surprisingly the spectrum is very similar too. In fact 99.9% of the
energy is contained within 1.69 R (compared with 1.75 R for FO-20). An
important practical point to remember is that the shape of the bit is
completely specified in time. This means that it can be precisely
generated using real finite hardware, such as a look-up table. The
microsats AO-16/WO-18/LO-19 use a filter not unlike this.

THE RAISED COSINE SPECTRUM

Alternatively "raised cosine" can also be used to describe the shape of
the spectrum. This is illustrated in figure 8. It doesn't look much like
a raised cosine spectrum because of the logarithmic scale, but the
response is unity (0db) at f=0, 0.5 (-6db) at f=R/2 and exactly zero at f
= R, the bit rate. 99.9% of the energy is contained within 1.56 R.

Fig 8. Raised Cosine SPECTRUM Shape. There is a family of these shapes,
with the raised cosine part centred on the middle of each side of the
spectrum. This synthesis gives a range of shapes from the gentle one
shown above to the theoretical minimum "brick wall" (Bandwidth = data
rate).

Because the spectrum is absolutely band limited to +/- R, the isolated bit
shape that gives rise to that spectrum has infinite time duration. This
means that it cannot be synthesised with real finite hardware. On the
other hand, as figure 8 shows, it has negligible amplitude beyond T =
+/- 2 bits, so that discarding the tails has little effect other than to
bring in some very minor spectral sidelobes. So synthesis turns out to be
highly practical.

You will also notice that the isolated bit also has precisely the desired
properties for zero inter-symbol interference. Is this fortuitous?
Certainly not! In fact it is a direct consequence of the two symmetries
of the cosine shape mentioned earlier. That's why the raised cosine
spectrum is so widely used in data transmission system design.

9600 BAUD PACKET RADIO & UOSAT/OSCARs 22/23
- A NARROWBAND RAISED COSINE SPECTRUM

As a practical example of "raised cosine" spectrum shaping consider figure
9. Here only the middle 3/8ths of each half of the spectrum is given a
raised cosine shape. From f=0 to 5/16 R the spectrum is flat (0db), from
5/16 R to 11/16 R it follows the cosine shape, and from 11/16 R onwards
the spectrum is zero.

Fig 9. UOSATs 14, 22 and KitSat Oscar-23 have almost "rectangular" raised
cosine spectra (see text). The narrow single main spectrum lobe implies a
bit that is some 8 bit periods long. There is no I.S.I.

This is used in the baseband filtering of UoSat/KitSat/OSCARs 22/23 which
use the G3RUH 9600 baud Packet Radio modem [1]. The isolated bit shape
spans some eight bit periods. That is to say a bit transmitted "now" will
actually have an influence on up to eight other bits, four before and four
after "now". Hence the title of this article! Note also that the isolated
bit has the desired form for minimising inter-symbol interference. A peak
at T=0, zero at all other T points. So the received "eye" shows good
convergence at the sample point.

The spectrum is almost rectangular ("brick-wall") and sidelobe free.
99.9% of the spectral energy is contained within 1.2 R, or +/- 5.6 kHz at
9600 bps.

The zero-crossing dispersion is a direct consequence of the narrow
bandwidth, and great care is needed in the design of the receiver
bit-clock recovery circuits that perform time bit detection. You can see
that sampling only 1/8th bit away from the convergence point means bit
detection half way down to the zero threshold, and so the instantaneous
noise tolerance would be reduced by 6db. Narrow bandwidth bit-clock
recovery circuits are essential. For implementation details see [1].

The actual signals transmitted by Uosat/KitSat are slightly different to
that shown; they are pre-adjusted (in time) so that when they pass through a
real bandwidth limited receiver the final "eye" is as illustrated.

In the G3RUH/PacComm/Kantronics/Symek/Tasco Telereader embodiment of this
modem, transmit bit-shape is looked up from a table of values in an Eprom
and passed to a DAC. This data filter is known as an F.I.R. for Finite
Impulse Response. UO-22/23 use a tapped digital delay line (shift register)
and summing op-amp; Eproms do not always travel well in spacecraft. This
mechanisation of an F.I.R. is called a Transversal Filter.

In Uosat/KitSat the signal also FM modulates the carrier (as opposed to
PSK). FM is simple to generate, much simpler to process on receive in the
presence of mistuning such as doppler shift or receiver drift, and is very
robust. Though FM does require slightly stronger signals, 6-8 db perhaps,
this is not a disadvantagous for amateurs where we are rarely power limited.

Finally note that this precision data shape could trivially generate
BPSK via a DSB balanced modulator. Such PSK would have all the narrow-band
and non-ISI attributes detailed above.

TRANSMITTER AND RECEIVER FILTERING IN SERIES

So far discussion has mostly been about spectra and bit shapes in
association with transmitters. It was implicit that the receiver was
infinitely wideband. For analytic purposes, if there were any RX
filtering, we have assumed it to be merged with the transmit filtering.

In many amateur applications this is indeed the case. For example, the
current flock of PSK packet satellites (Oscar-16 etc) confine their bit
rates to that which can be passed through a typical SSB receiver. RTTY
users assume the same thing.

We now ask the important question; if the overall filtering were to be
deliberately split between transmitter and receiver, what should the form
of the filters be?

We know from all the forgoing what kind of overall response is required.
But how should we partition it?

Obviously, the overall response can be apportioned in any ratio
whatsoever. However one can prove that in the presence of random noise
the optimum split, so as to maximise detector signal-to-noise ratio, is
that with exactly half the frequency response in the transmitter, and the
other half in the receiver. These are known as a "matched filter pair".

Conceptually, one can first imagine a desired overall frequency response
dictated by inter-symbol interference requirements, spectral constraints
and implementation ease. Then as it were, you take the "square root" of
this, and put the resulting filter into both transmitter and receiver.

AO-21/RUDAK-2 "RSM" BIT SHAPING

A very good example of this is the bit shaping scheme used on one of the
RUDAK-2 links of AO-21 [2]. Winningly dubbed "RSM" Rectangular Spectrum
Modulation by its creator, its transmit filter properties are shown in
fig. 11 (left). Both transmitter and receiver have essentially identical
filters. This bit shape is used to phase modulate (BPSK) the RF carrier.
Bit rate is R = 9600 bps.

Fig 11a & b. Performance of RUDAK-2's "RSM" TX filter (a) and signal after
passing through complementary "matched" TX + RX filters (b). The eye is
as wide as possible at the detection point, and regardless of
neighbours, all bits have the same amplitude at this instant. (i.e. no
I.S.I.)

99.9% of the spectral energy is contained within a bandwidth of 1.5 R,
which is consistent with the isolated bit span of about 3 bits. This
means that there are about 2^3 = 8 trajectories in the 'scope trace, which
can be seen quite clearly. It is notable that this isolated bit shape is
created using analogue filtering rather than more accurate and repeatable
digital synthesis.

The transmit filter (fig. 10) has 3 sections, comprising a) a Cauer
Chebyscheff type CC0315/41 which is flat up to a frequency f = R/2, and
then plunges down to the zero at f = R. But above this there is a rise to
only -15 db. So part b) is a 3rd order Butterworth that's smooth to the
-3db point at f = R/2, and then -18db at f= R and -36 db at f = 2R; c) an
"all-pass" network that has a dead flat frequency response, but a tapered
phase characteristic which is used to flatten out the filter delay vs.
frequency curve, called "linear phase". Linear phase filters result
in a transmit bit shape that is left-right symmetric. This filter is not
driven directly by the rectangular bit stream, but by short + or - pulses of
polarity corresponding to the data bits' polarity.

Fig 10. Data filter used in RUDAK-2 "RSM" transmitter. This comprises
three cascaded sections; a 3rd order Cauer, a 3rd order Butterworth and an
all-pass (refer to text). This transmit filter is used in association
with an almost identical filter at the receiver. These filters are thus
"matched". For spectrum and bit shapes see figure 11.

When cascaded with an identical analogue filter in the receiver, the
overall response is as per figure 11b. The overall isolated bit
shape is almost perfect, spanning some 5 bits with remarkable symmetry.
The 'scope trace shows an excellent eye. The sample point convergence has
about +/- 8% scatter due to the slight non-zero values of the isolated bit
shape at T = -2 and T = +2.

THE FUTURE?

The 9600 baud RSM-PSK implementation is near optimal in performance,
probably within 1 db of the theoretical limit.

[ approx. 20 Log (1 - 8/100) ]

It is flying on RUDAK-2/AO-21 so AMSAT experimenters (i.e. YOU) can
evaluate the effectiveness of high speed PSK with or without coding [3]
from the associated on-board Harris RTX2000 RISC processor. This will
help determine what is practical and realistic for amateur spacecraft and
other digital links of the future.

The G3RUH 9600 baud FM system is not only in daily use by many satellite
stations on UO-22/KO-23, but also hundreds of terrestrial packet radio
links world wide. Indeed, the outstanding German packet radio [4] network
thrives on it. The modem represents a design that pushes the fastest
possible binary data through a basic FM radio. 9600 bps throughput is
astonishing to behold, especially on a full duplex satellite link.

Who will predict that one day even our voice repeater links will be entirely
digital. I will!

REFERENCES

Miller J R, "9600 baud Packet Radio Modem Design", Proc. 7th ARRL
Computer Networking Conference, 1988 Oct, pps 135-140

Meinzer K & Haas W, "RUDAK 2 - The Radio Links", Amsat-DL Journal No.
1/17, March 1990 pps 9-12. (German). English translation in Oscar News
No. 83, 1990 June, pps 16-21

Miller J R, "Shannon, Coding and the Radio Amateur", Oscar News No. 81,
1990 Feb, pps 11-15

Rech W & Kneip J, "The German (Central European) Packet Radio Network: An
Overview", Proc. 11th ARRL Computer Networking Conference, 1992 Nov, pps
52-56.