Extreme narrow bandwidth techniques

Extreme narrow bandwidth techniques ON7YD
Extreme narrow bandwidth techniques

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Any comments and/or suggestions are welcome at : on7yd@qsl.net

last updated on 11 December 2001

Index

Introduction
Bandwidth
Digital Signal Processing (DSP)
1. DSP basics
2. Fast Fourier Transform (FFT)
Extreme narrowband modes
Available software
Operating practice
Literature
Acknowledgements

1. Introduction

Due to several reasons the signal to noise ratio (SNR) of ham signals on 136kHz is often very low :

A wavelength of about 2.2km (1.35 mile) makes any ham antenna small and inefficient.
1 Watt ERP power limit
High noiselevels
Strong commercial stations (100kW and more) very close to the ham band

One way to improve the SNR is to reduce the receiver bandwidth and thus have less unwanted signals and noise while leaving the level of the wanted signal unchanged. But the reception of any signal requires a minimal receiver bandwidth depending on the type of modulation. SSB has a typical bandwidth of 2.4kHz, the bandwidth of a CW signal is dependent on the speed but in any (practical) case less than 100Hz. The use of a filter with a narrower bandwidth than that of the transmitted signal will distort the signal.
Here I will describe a technique that allows to communicate with signals far below the noise level. It can be used to make a basic QSO and I think that in some way the 'value' of such QSO's can be compared to Meteor Scatter QSO's on VHF.

2. Bandwidth

The dominant and most efficient mode on 136kHz is CW. As mentioned before the minimal bandwidth at the receiver side is determined by spectrum of the transmitted signal. In case of CW it is the (keying) speed that puts a limit to the minimal bandwidth.
An accepted method to measure the speed of CW is the PARIS system. The word Paris has a length of exactlty 50 'dots', word spacing included. Based on this system a CW signal of 12 words per minute (WPM) means 600 'dotlengths' per minute or 10 'dotlengths' per second. But as each dot is separated by space of the same length the actual length of the 'dot-cycle' is the double. If a continious series of dots is given at 12WPM this results in a 5Hz square wave. If an RF signal is keyed with this series of dots you will get a carrier with 2 sidebands at 5Hz, resulting in a 10Hz wide signal. Depending on how 'hard' the keying is more sideband further away from the carrier will be created but these do not contain additional information and can be considered as a waste of energy (and a source of QRM to others). So basicaly the minimum bandwidth that is required to receive a CW signal undistorted is :

B = 0.833 * WPM (Hz)

Assuming that the only noise source is a frequency independent (white) noise, the total receiver noise will be directly proportional to the receiver bandwidth. Taking a 12WPM CW signal as a reference and assuming that the receiver bandwidth is optimized to the transmission speed the table below shows the SNR improvement that can be achieved by reducing the CW speed :

Speed	optimum bandwidth	SNR vs. 12WPM
12WPM	10Hz	0dB
8WPM	6.67Hz	+1.8dB
4WPM	3.33Hz	+4.8dB
1 sec./dot	1Hz	+10dB
3 sec./dot	0.33Hz	+14.8dB
10 sec./dot	0.1Hz	+20dB

It is clear that a significant SNR improvement can be achieved by reducing the CW speed. On 136kHz a dotlength of 3 seconds has become a kind of standard and this mode is called QRSS (from the Q-code QRS : please reduce your speed). At these very slow CW speeds it becomes rather difficult to copy the signal by ear as you would almost need a chronometer to time the dots and dashes. Furthermore the frequency of the signal needs to be very stable as smaller bandwidths are used. Fortunately this not a big problem on 136kHz where a frequency stability of 0.1Hz is rather easy to achieve.
Another problem is that filters become more and more complicated to built as the bandwidth becomes smaller. And also tuning into a signal can be a rather tricky thing at bandwidths below 1Hz.
So reducing bandwidth not only has benefits in the way of an improved SNR but creates also a lot of additional problems. One way to overcome many of these problems is the use of Digital Signal Processing (DSP).

3. Digital Signal Processing (DSP)

3.1. DSP basics
Digital Signal Processing is one of these magic expressions that you hear once in a while, but seems the domain of specialized electronic engineers and maybe some 'happy few' hams. Until very recentely special (and rather expensive) hardware was needed to perform DSP. But now all the special hardware can be replaced by a pentium PC with soundcard and the software you need is available for free. Practical details can be found here. As the expression Digital Signal Processing says, the analog (input) signal is converted to digital, then processed and eventually converted back to an analog (output) signal.

The conversion of the analog signal to a digital form is done by analog-to-digital conversion (ADC). The most basic version of ADC is often done by ourselves when we use a voltmeter to determine the value of a voltage. With DSP this 'reading of voltages' is done automatically at a known time interval, this is called sampling. The result is a series of measurements, where we know the measured voltage and the time when it was measured.

These data are processed digitally, what in practice means that they undergo a series of more or less complicated calcualtions. The result can be interpreted as digital data or eventually converted back to an analog signal. Using DSP all kind of things can be done, not only filtering but also reducing bandwidth, time multiplexing of several signals etc...
Here we will only discuss the filtering of a signal.

3.2. Fast Fourier Transform (FFT)
Althought there are several methods to filter a signal digitally the major technique is by using Fast Fourier Transform (FFT). The mathematical background on this transform was developed by Joseph Fourier, about 200 years ago. The basic idea behind this transform is that any signal can be seen as the sum of a series of sinusoidal signals, where each sine can have a different amplitude and phase.

In the above picture the complex red signal is equal to the sum of the green, blue, yellow and white sines. The mathematical equations of the Fourier transform are rather complicated, those who are interested can have a look at following webpages :

Fourier and the frequency domain (Strathclyde University, Glasgow, Scotland) : very complete, but it takes some math
The Fourier series and The fast Fourier transform (VisLab, Sydney, Australia) : idem
The basic tutorial on frequency analysis (Steema Software, Girona, Spain) : explains Fourier transform almost without math

Fortunately it is not necessary to go deep into these mathematics to understand how FFT works and therefore the maths will be kept to a minimum. But as there is a lot of calculating involved Fourier transforms will take a lot of 'computing time'. To reduce this a special algoritm was developed to enhance the speed of the Fourier transforms, this algoritm is called the Fast Fourier Transform (FFT).
When we take the Fourier transform of any signal we actually split the signal up in a number of sines and for each of these sines the amplitude and phase is calculated. Each of these sines represents a certain frequency (or better frequency band) and from these sum of sines (and their amplitudes) we can reconstruct the frequency spectrum of the measured signal.

The 'quality' of the reconstructed frequency spectrum depends on 3 things :

the sample rate (interval between 2 A-D conversions)
the sampling time for one transform
the number of bits of the A-D convertor

The sample rate determines the maximum frequency of the spectrum : the maximum frequency that can be reconstructed is 50% of the sampling frequency.
eg. : If we take a sample every 0.2ms (equals a sampling frequency of 5kHz) the maximum frequency that can be reconstructed is 2.5kHz.
The sampling time determines the frequency resolution (or the bandwidth of each 'channel') : the frequency resolution is equal to one over the sampling time.
eg. : If we take a sampling time of 0.1 seconds the frequency resolution (or channel bandwidth) will be 10Hz, this means that in the series of sines of the Fourier transform each sine will represent a 10Hz wide channel.
The number of samples in a Fourier transform has to be a power of 2 (2, 4, 8, 16, ... , 256, ... 65536, ...). Although you can take any number of samples and just add a series of 'zeros' until you get a power of 2 it is more practical to choose the correct ratio between sample rate and sampling time in order to get the right number of samples.
eg. : if we have a sample rate of 0.2ms we will not take a sampling time of 0.1 seconds, what would result in 500 samples, but a sample time of 0.1024 seconds in order to get 512 samples (= 2⁹). The result of the Fourier transform will be a series of 256 sines where each sine represents a 9.766Hz wide channel between 0Hz and 2.5kHz.
The picture below shows a simple example, the Fourier transform of 16 samples at a rate of 1ms results in a series of 8 sines that each represent a 62.5Hz wide channel between 0 and 500Hz :

The number of bits of the A-D convertor determines the dynamic range of the spectrum. In practice (using the PC soundcard) we can choose between a 8-bit or 16-bit A-D conversion.
eg. : For a 8-bit A-D conversion we have 2⁸ = 256 levels and the dynamic range will be 20.Log(256) = 48dB. For a 16-bit A-D conversion we have 2¹⁶ = 65536 levels and the dynamic range will be 20.Log(65536) = 96dB.

4. Extreme narrowband modes

4.1. QRSS
QRSS is extreme slow speed CW, the name is derived from the Q-code QRS (reduce your speed). To take advantage of the very narrow bandwidth of the transmitted signal an appropriate filter at the receiver end is needed. Making a 'software filter' using FFT has some advantages over the oldfashioned hardware filter. One of the main advantages, when using it for reception of slow CW signals, is that FFT does not give you one single filter but you get a series of filters with which you can monitor a complete spectrum at once. This means that you do not have to tune exactly into the signal, what can be very delicate at sub-Herz bandwidths. Also it is possible to monitor more than one QRSS signal at the same time. At first glance it looks as if it is complicated to do this, even if FFT presents you this nice multi-channel filter it might be difficult to monitor all these channels. Further the long duration of the dots and dashes is unfavourable for aural monitoring.
A solution to the above problems is to show the outcome of the FFT on screen rather than making it audible. The result is a graphic where one axis represents time, the other axis represents frequency and the color represents the signal strength. If the vertical axis represents time we call it a waterfall display while it is called a curtain display if the horizontal axis represents time.
All this may sound complicated but it is easy to understand when you see an example (curtain display) :

The picture above show the signal of HB9ASB who was not audible due to very strong QRN, the vertical lines are the result of S9++ static crashes.
Some nice collections of screen captures can be found at the webpages of DK8KW, G3XDV, OK1FIG and NL9222

In April 2000 Geri Kinzel (DK8KW) did some measurements to compare QRSS with normal (aural) CW :
This morning I made some laboratory tests to get some indication about the ability to communicate with signals below noise level using QRSS. I used a calibrated frequency synthesizer (Adret 2230), an 0-120 dB attenuator in 1 dB steps (Schlumberger BMD500) and my Praecitronic MV61 Selective Level Meter. With a BNC T-connector I fed the normal band noise including LORAN lines on 137.500 kHz (+/- 50 Hz) to one side and the output of the frequency synthesizer to the other side. With the attenuator I made sure that a 0 dBm (50 Ohm) signal with the synthesizer corresponds to a -80dBu ( 0dBu = 0.775V into 75 Ohm = +9dBm, -80dBu = -71dBm) signal at the MV62 (plus/minus 1 dB). The band was quite, with a background noise around -110dBu (S4, -101dBm) and LORAN lines clearly visible. Using the 100 Hz bandwidth of the MV62 and the cascaded 250 Hz/500 Hz CW filters of the IC-746 I checked the signal by ear as well as with the Spectrogram software with the normal parameters I use for "3-5 second-dot-length" QRSS (5.5k sample rate, 16bit mono, 16384 points FFT = 0.3 Hz resolution, 60 dB scale, 300 ms time scale, 10 x average) and obtained the following results:

Signal strength at RX input	Comment
-100dBu / -91dBm	good audible CW (S6)
-110dBu / -101dBm	CW signal equal to noiselevel (S4), can just be copied
-115dBu / -106dBm	boundary for aural CW, signal just detectable by ear
-125dBu / -116dBm	perfect readable QRSS signal ('O' report)
-130dBu / -121dBm	good readable QRSS signal ('M' report)
-135dBu / -126dBm	just detectable QRSS signal ('T' report)
-140dBu / -131dBm	signal not detectable

Conclusions:
QRSS has a 20 dB signal level advantage over normal (aural CW), which means that the minimum detectable and/or readable QRSS signal that might just allow communication lies 20 dB below the signal, that can just be detected and/or decoded by a trained CW-operator's ear. If I consider the "CW-operator's ear/brain bandwidth" to be 30 Hz, this roughly corresponds to the bandwidths used (0.3 vs 30 Hz).

John Andrews (W1TAG) did also some measurements. He was using an Icom R75 receiver and 6 foot loop to receive the signal transmitted by a 10 mw exciter, feeding a small loop antenna through a variable attenuator. The receiver was picking up whatever noise was present at the time of the test. The signal was decoded using the ARGO software.
The transmitted signal was attenuated to a level that just allowed a solid copy :

Dotlength	Measured level (vs. 6WPM)	Theoretical level (vs. 6WPM)
0.2 sec. (6 WPM)	reference	reference
3 sec.	-10dB	-11.8dB
10 sec.	-15dB	-17dB
30 sec.	-19dB	-21.8dB
60 sec.	-23dB	-24.8dB

For all measurements there is a +/- 2dB difference between the measured and theoretical result, but I believe that this can be explained by the fact that the 6WPM reference screenshot has a clearly lower SNR than the other screenshots. The full report on this measurement can be found here

4.2. Dual Frequency CW (DFCW)
At a speed of 3 seconds per dot a very basic QSO will take about 30 minutes. Changing QRN levels and/or propagation during this period can have a vast effect on a QSO. Therefore a new transmission mode has been developed that enhances the average speed by a factor of 2.5 to 3.
When the nature of CW is analyzed it seems a digital mode where 'key down' respresents a logic '1' and 'key up' a logic '0'. But another approach it is to see it as a mode with 3 'logical states' : the 'dash' (3 periods of key down + 1 period of key up or '1110'), the 'dot' (1 period of key down + 1 period of key up or '10') and the 'character space' (2 periods of key up or '00'). The spacing between words is 3 character spaces. So there are 2 elements that play a role : the presence/absense of a signal and the duration of the signal. As CW was intended to be received by ear the different duration of the signals is essential, but it lengthens the time needed to transmit a text.
In Dual Frequency CW (DFCW) the element 'duration' is replaced by the element 'frequency'. So dots and dashes no longer have a different length but they are transmitted on a different frequency. Due to this frequency shift there is no 'space' needed between the dots / dashes and the character space can be reduced to the same (dot)length.
When the idea of DFCW first was introduced there was a lot of scepticism about the readablility of there frequency shifted signals but in practice it seems rather easy to read it from the screen. To make it even more easy to read, especially during a sequence of dots ot dashes, a short space (typicaly 1/3 of a dotlength) is added between the dots and dashes. This reduces the average speed a bit, but is improves the readability and also reduces the duty cycle (what is better for the PA).
The example below show the text 'CQ ON7YD K' in QRSS and DFCW, at the same speed :

At a speed of 3 seconds per dot this CQ will take 5'30" in QRSS while it will take only 1'54" in DFCW. The speed advantage of DFCW over QRSS can be taken in 2 ways, either by reducing the duration of a QSO or by increasing the dot length and working at a narrower bandwidth. The last means that, for the same duration of a QSO, the dot length in DFCW can be 2.5 to 3 times longer and as a result of this get a 4 to 5dB better SNR.

The screenshot below shows a 'real-world' picture of a DFCW signal. I hope this proves how easy it is to decode a DFCW signal by eye.

4.3. Future developments of extreme narrowband modes
Over the past year a dot length of 3 seconds has become a kind of informal standard for QRSS and DFCW, as practical test have shown best results for this speed. Most hams use the Spectogram software at the receiving end with a sample rate of 11kHz and blocks of 16384 samples, this means a sample time of about 1.5 seconds. At first sight it is not so obvious why the sample time is only half the dot length, wouldn't it be better either to make the sample time longer or the dot length shorter ?
But there is a good reason why the sample time is so much shorter than the dot length, it is because so far the transmitter and receiver do not work 'synchronised'. This means that a sample block (at the RX end) begins somewhere in the middle of a dot (at the TX end) and vice versa. What happens when dot length and sample block have the same duration is shown in the pictures below :

To ensure that at least 1 sample block falls completely within each dot (or dash, space) a sample block can not be longer than half the dot length.
If we can create some kind of 'synchronisation' between TX and RX it would be possible to (almost) double the sample block duration without increasing the dot length and thus achieve a 3dB gain. When DFCW is used and the dot length is increased to 10 seconds a QSO will take about the same time as a 3 sec./dot QRSS QSO. Assuming that it should be not so difficult to 'synchronise' transmitter and receiver software with a 1 second accuracy a sample block of 8 seconds (with a 2 seconds interleave) and a dot length of 10 seconds seems possible to me :

Compared to the 'traditional' QRSS a gain of over 7dB can be achieved while the duration of the QSO is about the same. Timing errors (between TX and RX) upto 1 second will not affect the SNR.

An alternative solution is given by the programmers of Spectran. Instead of using a complete new set of samples for each FFT they take only partly new data and 'shift' the existing data up in the data-array used to perform the FFT:

eg. : Assume that 4096 datapoints are used to perform a FFT. Instead of using a complete new set of data for the next FFT you remove the 128 'oldest' points (that are in position 3969 to 4096 in the data-array), shift the datapoints in position 1 to 3968 upward to the end of the data-array and fill positions 1 to 128 with new data. This procedure is repeated for every FFT.
This method has the advantage that the 'duration' of the FFT array can be almost as long as the duration of a dot, but there are also some disadvantages. First of all the workload for the computer increases significantly, in case of the above example the computer has to perform 32 FFT's in the time that with the 'traditional' method only 1 FFT calculation is needed. On screen this method also causes some 'blur' at the beginning and end of the dots :

Another way to improve the SNR is averaging. It is based on the fact that noise is random and cancels itself out over a number of measurements while the signal is consistent. Therefore the results of several FFT's are added and the average is taken. While the advantage is an improved SNR the disadvantage is that the results on screen appear slower, as you have only 1 output to screen for several FFT's. Fortunately there is a way arround this, but it has also some drawbacks. Alberto di Bene (I2PHD), one of the programmers of Spectran, sent me some interesting comments on how averaging can be done :
In Spectran there are two averaging mechanisms :

A) the averaging that you can activate with the AVG button on the main panel. This has two sub-choices : the sliding window and the integration radiobuttons. With the sliding window a moving average is performed on the computed spectrum, and a result is output for each input spectrum, no matter what the averaging factor is. This factor has only the effect of adjusting the width of the moving window. On the contrary, when doing integration, if the the averaging factor is N, an output spectrum is generated every N input spectra, simply taking their average.
B) the averaging which is produced as a side effect of the overlapping done on the input time series. The purpose of this overlapping is to produce an output in a time shorter of what would otherwise be needed, should overlapping not be used. To clarify a little : suppose that the sampling rate is 8000 Hz, and the resolution set to 0.12207 (rounded in the display to 0.12) Hz. This implies that the Nyquist frequency is 4000 Hz, and you have 4000 / 0.12207 magnitude values, obtained with 8000 / 0.12207 complex values (the real and the imaginary parts). So you have to compute 8000 / 0.12207 = 65536 complex values, corresponding to an equal amount of real time samples. The time needed for this is 65536 / 8000 (the number of samples you get each second) = 8.192 seconds. And, actually, if you set Spectran with the speed slider at its minum (the slider colour changes to green), i.e. an overlapping factor of 1, you will have a refresh of the screen about every 8 seconds. Things go worse with increasing resolutions, the wait time increases proportionally. If a faster refresh rate is desired, the overlapping can be of help. With this method, you don't wait that a complete new set of input values is ready, but a part of the old data is reused, together with some new ones. Of course there is a compromise (no free lunch...) : in reusing the old data, you loose some time resolution, and this appears on the screen as a smearing, or blurring, of the spectral lines. This effect increases with the overlapping factor. With factors up to about 8, it is not very noticeable, but things go worse with higher values. It's the price to be paid for having a quicker refresh rate. As with almost any choice in life, a suitable compromise must be found, and this has also implications on the optimal dot/dash length.

5. Available software

5.1. Spectogram

Function	:	DSP receiving software
Operating system	:	Windows 95 and up
Author(s)	:	R. Horne
Description	:	Very versatile program. Version 4.2.6.5. is tailored for weak signal reception. The later version 5.1.6. has some extra features but some experience with weak signal reception is required to use them optimal.
Recommended settings	:
Status	:	Freeware
Web page	:	here
Download	:	version 5.1.6. or version 4.2.6.5.

5.2. Spectran

Function	:	DSP receiving software
Operating system	:	Windows 95/98
Author(s)	:	A. di Bene (I2PHD) and V. De Tomasi (IK2CZL)
Description	:	This program allows extreme narrowband reception of weak signals. A lot of option, but they require some experience to use them proper.
Recommended settings	:
Status	:	Freeware
Web page	:	here
Download	:	beta 4, build 127a or beta 3, build 295

5.3. EasyGram

Function	:	DSP receiving software
Operating system	:	Windows 95 and up
Author(s)	:	P. Maly (OK1FIG)
Description	:	This program extends possibilities of well-known Spectrogram with some new useful features. It enables to define the scrolling area to any size, it can save screen shots in defined time periods (useful overnight), it enables to browse the saved pictures easily. All the settings can be saved to named profiles.
Recommended settings	:
Status	:	Freeware
Web page	:	here (warning : slow link)
Download	:	lastest version

5.4. Argo

Function	:	DSP receiving software
Operating system	:	Windows 95 and up
Author(s)	:	A. di Bene (I2PHD) and V. De Tomasi (IK2CZL)
Description	:	Especially developed for reception of QRSS signals at 3 sec./dot and 10 sec./dot. Easy to use, since only very few parameters have to be set.
Recommended settings	:
Status	:	Freeware
Web page	:	here
Download	:	beta 1, build 113

5.5. Spectrum Lab

Function	:	DSP receiving software
Operating system	:	Windows 95 and up
Author(s)	:	W. B�scher (DL4YHF)
Description	:	Spectrum Lab incorporates a amazing number of features in a single program. Besides the 'standard' DSP rceiving window you can use it as a terminal for PSK, RTYY etc... and it will even allow you to decode the DCF77 signal and set your PC clock to it.
Recommended settings	:
Status	:	Freeware
Web page	:	here
Download	:	here

5.6. Crunch

Function	:	DSP receiving software
Operating system	:	DOS
Author(s)	:	B. de Carle (VE2IQ)
Description	:	A completely different approach to 'decode' QRSS transmissions is made by VE2IQ. With crunch the incoming audio signal is recorded as a WAV-file, either using the soundcard or VE2IQ's Sigma-Delta DSP board. Afterward the file is filtered and 'speeded up' to bring the QRSS signal to normal speed CW, audible via the soundcard.
Recommended settings	:
Status	:	Freeware
Web page	:
Download	:	here

5.7. QRS

Function	:	QRSS and DFCW transmitting software
Operating system	:	Windows 3.11 and up
Author(s)	:	R. Strobbe (ON7YD)
Description	:	This program is primarily intended to be used for QRSS and DFCW but has also some normal CW facilities. Keying of the transmitter and frequency shifting (for DFCW) is done via the serial or parallel port, using a very simple interface. Various options are available, including QSK and fast CW identification in QRSS and DFCW modes. In 'QSO mode' the program is optimized to share the screen with receiving software.
Recommended settings	:	speed - dot length = 3 seconds DFCW - key gap = 1 second
Status	:	Freeware
Web page	:	here
Download	:	UK site or USA site

5.7. Alternative downloading (mirror side)

Most of the above software (and a lot of other stuff) can also be downloaded from the NL9222 side.

6. Operating practice

Operating QRSS and DFCW is rather simple, just a few 'specialities' :

The (unofficial) QRSS/DFCW segment is 137600Hz - 137800Hz, most activity is between 137700Hz and 137750Hz
You need a stable TX : long term stability of 5Hz is an absolute minimum, 1Hz or better is recommended
Keep your CQ's short : eg. CQ G3XXX K not CQ CQ CQ G3XXX G3XXX G3XXX PSE K
The report system is the TMO system (similar to EME) :
- T = signal traces seen but not good enough for a QSO
- M = weak signal but good enough for a QSO
- O = perfect copy
A dot length of 3 seconds is recommended, eventually 5 seconds if very weak signals are expected
For DFCW a shift of 5Hz (with 'dash' being the higher frequency) and a key gap of 1 second is recommended
During a QSO (once you are sure that both stations have the calls OK) you can use the suffixes instead of complete calls.
If you see the end of a QSO and you want to contact one of the stations you can start calling this station while the previous QSO is going on, just call on another frequency.
If replying to a CQ it is recommended to call NOT exactly on the frequency of the other station, this will avoid QRM in case more than 1 station is responding.
Due to the slow 'transmission rate' it is recommended to limit the QSO to exchange of callsigns and reports, especially if you are located close to another radioamateur that is active in this mode (one strong signal can 'block' the entire QRSS segment).

A basic QRSS (or DFCW) QSO could look like this :

CQ ON7YD K
ON7YD G3XDV K
G3XDV YD OOO K
YD XDV OOO K
XDV YD TU 73 K
YD XDV GL 73 SK

At very slow speeds (dot lengths of 30 seconds and more) it is adviced to keep the exchanged information even shorter, in order to keep the duration of a QSO within reasonable limits :

CQ G3AQC K
G3AQC VA3LK K
VA3LK AQC OO K
AQC LK OO K
LK AQC SK

7. Literature

The Scientist and Engineer's Guide to Digital Signal Processing
By S.W. Smith (California Technical Publishing, ISBN 0-9660176-3-3, 1997)
This book with 640 pages and over 500 illustrations is an excellent introduction to DSP. It can be downloaded for free (PDF files) or can be purchased as a hardcover book for 64USD +P&P (may 2000).
Look here for details and download.
(information provided by Andy Talbot, G4JNT)

8. Acknowledgements

My thanks go to :

Alan Melia (G3NYK) for checking spelling and grammar
Geri Kinzel (DK8KW) and John Andrews (W1TAG) for the measurements to compare QRSS with normal (aural) CW
Alberto di Bene (I2PHD) for his comments on averaging techniques
Andy Talbot (G4JNT) for the literature information

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