Theoretical aspects of the filtering for digital transmission

application to the FM transmission two-phase

by Jean-Claude Imbeaux,   F6AXK

(bad) translation by J.F. Fourcadier F4DAY

 

Nyquist Filtering

The Nyquist filtering is intended to eliminate the interference between symbols, which allows a simple decision, symbol by symbol. For a sampling at the moments kT, the Nyquist criterion in the temporal field is nyq(kT) = 0, except for k=0.

This results in a symmetry of frequency response Nyq(f) around the point of frequency f=1/2T.

For biphase transmission, one seeks to cancel the impulse response all the kT/2, that is to say nyq(kT/2)=0. The symmetry of Nyq(f) is then around the frequency point 1/T.

At the transmission end one starts from a rectangular pulse of a T/2 duration, which must be taken into account in the total response of the transmission chain.

The combination of transmission filtering and reception filtering must be equivalent to a Nyquist compensated filtering, that is to say  :

NyqComp(f) = Nyq(f) / (sin( p * fT/2)/( p * fT/2))

 

 

filtering share between emission and reception

For a linear transmission chain, with white noise, the filtering which makes it possible to minimize energy to be transmitted on the channel, for a fixed signal/noise ratio (or an error rate) after reception filtering, is such as  :

He(f) = Hc(f) ^1/2 / G (f)¼ and Hr(f) = Hc(f) ^1/2 * G (f)¼

with Hc(f) = He(f) * Hr(f)  ; combination of filterings emission and reception

G (f) = spectral power density of the signal before filtering emission

For a traditional digital transmission, with independent rectangular pulses of duration T, and Nyquist criterion, this led to  :

He(f) = Nyq(f) ^1/2 / (sin( p * fT)/( p * fT)) and Hr(f) = Nyq(f) ^1/2

The Nyquist filtering is shared between emission and reception. Moreover emission filtering is compensated to take account of the initial rectangular impulse.

For a biphase transmission, it is not possible to satisfy the optimal sharing of filtering, because the spectrum G (f) is null for f=0. The advantage of having a null DC component thus results in a degradation of the performances compared to the traditional case.

It is possible to compare three particular cases theoretically  :

a) total filtering (Nyquist Compensated) shared in an identical way between emission and reception , is  :

He(f) = Hr(f) = NyqComp(f) ^1/2 = (Nyq(f)/(sin( p * fT/2)/( p * fT/2))) ^1/2

 

b) filtering of shared Nyquist, but compensation entirely with the emission , is  :

He(f) = Nyq(f) ^1/2 / (sin( p * fT/2)/( p * fT/2)) and Hr(f) = Nyq(f) ^1/2

 

c) filtering of shared Nyquist, but compensation entirely with the reception , is  :

He(f) = Nyq(f) ^1/2 and Hr(f) = Nyq(f) ^1/2 / (sin( p * fT/2)/( p * fT/2))

Calculation shows that the case a) is the best of the three, but the difference is very weak (0,3 dB better than the case b) which is worst).

 

 

Two-phase transmission / FM

The FM transmission with preemphasis and deemphasis of the biphase signal still complicates the situation, for two reasons  :

- the transmission chain is not linear (FM modulation)

- the noise is not white after deemphasis

The first point makes that it becomes impossible to define by theoretical calculation, the optimal sharing of filtering between emission and reception (the second point, on the nonwhite noise, can be taken into account theoretically).

Only knowledge is that of total filtering NyqComp(f)  ; cf §1.

Then, to go further, it would be necessary to use simulation software, by including the FM modulation-demodulation.

---

 

B5+ et 73 de Jean-François Fourcadier, F4DAY

back to the welcome page

© 2000-2004  J.F. Fourcadier F4DAY