The Nyquist filter

Data transmission

The general problem is to transmit a succession of symbols (the digital zeros and ones), by the mean of any support: coaxial cable, twisted copper line, electromagnetic waves, optical fibre....  We previously saw the role of the coding which was to adapt the signal to the selected support and to allow the rhythm recovery.

In the particular case which interests us, transmission of a 2,048 Mbit/s signal, biphase coded, on a radioelectric support, the signal to be transmitted has the following shape :

If the bandwidth of the support were infinite, or very large, the signal could  be transmitted as is, without many drawback. Only the broadband noise could annoy us. However, when one transmits a radio signal, one always seeks to control or reduce the bandwidth of the transmission. It is said whereas the channel is band limited. It will act as a filter which will deform the transmitted signal.

Which are the impairments which the channel will bring to the signal?

First of all, the signal will be received with a certain delay. Then, the rise and falling time of the signal will be greater. Lastly, and it is more serious, over-oscillations will interfere with the useful signal. The importance of these phenomena depends closely on the characteristics of the channel: bandwidth and response curve of the equivalent filter.

On the left, the transmitted signal, on the right the received signal:

As the transmitted symbols are quickly followed by an other, the over-oscillations corresponding to a received symbol will disturb the reception of the following symbols. It is said that we observe intersymbol interferences .

Sampling

The received signal has an analogue nature. The problem which one has to manage at the receiving end is to transform the received signal into a signal as conform as possible with the digital transmitted signal, i.e. with the less possible errors.

For that we will decide at regular intervals, if the transmitted symbol is a " 0 " or a " 1 ", by comparing the value of the received signal with a fixed threshold. For example, if at the selected moment the received signal is higher than a certain value we will decide than we receive a " 1 ", in the opposite case a " 0 ". Obviously, the operation must be carried out at the transmission rate, very precisely.

sampling instant:

At the moment of sampling, the decision will be made according to the instantaneous value of the received signal, which is itself the composition of the useful signal, the noise and the intersymbol interference. We should obviously minimize the latter.

How to avoid the effect of the interference intersymbol?

It is very simple! We just need to choose the filtering brought by the channel so that the parasitic over-oscillations is equal to zero at the precise time where the sampling will be carried out.

The Nyquist filters have this property.

To know some more: theoretical aspects, by Jean-Claude Imbeaux, F6AXK
        Thank you for this contribution, very usefull for the achievement of our filter.

 

The achievement of our Nyquist filter

The synthesis of the filter

It will be supposed that the channel filtering is given essentially by the filters we will insert. The CCIR 405-1 pre-emphasis/de-emphasis operations, which are carried out to preserve a total compatibility with ATV analogue signals, cause " a colouring " of the noise and make difficult the theorical study of optimization and the share of the Nyquist filtering between transmitting and receiving. In a first experimental approach we will share equally the filtering between transmitting and receiving.

Assuming F equal to 2.048 MHz and a roll-off of 1 in the formulas introduced above, one obtains the desired response of total the filtering:

As for a previous study on the TV preemphasis, simulations can be carried out very simply by means of a Spice software. There are many free or commercial versions.
For an amateur use, one will use here the free software "Spice Opus" downloaded on
http://fides.fe.uni-lj.si/spice/, and then a small explanatory 5 pages document on http://fides.fe.unilj.si/spice/getstarted/getstarted.html.
The site is very sympathetic and also provide many examples of use. At the end of one hour of try, we are able to launch out simulations.

The filter to be realized should have a low-pass type behavior.   The characteristic impedance will be selected equal to 300 ohms so that the filter can be fed directly from digital circuits... By the mean of two cascaded cells in pi configuration and made " soft ", i.e. damped, we could get exactly the required shape, until a frequency of 3.5 MHz. Unfortunately, the group delay distortion was excessive (nearly 100 ns), which led to significant impairments of the signal.

Indeed, an ideal filter creates a pure and uniform delay. In a real filter, the delay often depends on the frequency. If the highest frequencies are not propagated at the same speed as the low frequencies, the transmitted signal can be passably deformed. It is generally estimated that the variation of the delay in the frequency band to be transmitted should be lower than 20% duration of the symbol. In our particular case where the modulation rate of the biphase signal is 4 MBauds, the duration of the elementary symbol is of 250 ns, and the group delay distortion must be lower than 50 ns.

After some atempts in simulation one obtains a result not too different from that expected, with a simple pi cell, compensated by the dipole R1 L2.

On the left, the source, with its internal resistance of 300 ohms. On the right side, the receiver whose internal resistance must be also equal to 300 ohms.

The Spice "netlist" can be written:

Nyquist filter, group delay
* to nyquist.cir file
control
ac flax 101 45KHz 4.5Meghz
let phase = vp(4) + 2*pi*(vp(4) lt 0) - 2*pi
stud phase
GD = -1e9*2*(phase[1,100 ] - phase[0,99])/(45000*2*pi)
stud GD xlabel f[Hz ] ylabel group_delay[ns ]
unlet phase
stud 4*(vm(4)^2) xlabel f[Hz ] ylabel response ylimit 0 1.1
endc
* voltage resources
v1 1 0 cd. 0V ac 1 sin 0 1V 10megHz
* resistors
laughed 1 2 300
ro 4 0 300
R1 2 5 1500
* capacitors
C1 2 0 150pF
C2 4 0 150pF
* inductances
L1 2 4 27uH
L2 5 4 47uH
end

One backs up the netlist under the file name "nyquist.cir". While launching Spice Opus with the command "source nyquist.cir" one gets, for two cascaded filters (Nyquist transmit + Nyquist receive) the following response curve:

 

 

And the group delay curve :

The average group delay we get with two filters (transmit + receive) stays between 173 ns and 178 ns. The group delay is constant at about 4 ns in the frequency band from 0.5 MHz to 3.5 MHz, which is excellent.

The deformations of the signal (cf eye pattern) are probably due to the variations of the amplitude response compared to the theoretical curve. For our application, we wish a simple filter structure. We will thus stay there for the moment, while keeping for later a better total optimization of the Nyquist filtering, in particular by the setting of two compensated cascaded cells, similar to that described above.

To summarize this page :

When one wishes to transmit high data rates, it is necessary to control the amplitude-frequency response and the phase-frequency response of the data communication channel. It's a good idea when one tries to transmit television signals either: -). In digital transmission, the Nyquist filter is used to cancel the effects of intersymbol interferences.

For a 2 Mbit/s digital transmission, biphase coded, the final practical schematic diagram of the filter to be inserted, once in the video path to the transmitter and one other time in the received video path, is the following:

The input and output impedances should be equal to 300 ohms.

B5+ et 73 de Jean-François Fourcadier, F4DAY

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