DL6iAK

### Some background to Root Cosinus Shaped impulses

This is a html-file compiled from TEX, because its more easy for me to write the formulas in TEX. If you don't like formulas, you can emergency exit here.

The reason for root cosinus shaping:

The data impuls dd(t) is convoluted by a the impulse response ht(t) of a transceiving filter (in PSK31 ht(t) has the form of a raised cosinus)

 st(t) = dd(t)*ht(t)
(0.1)

at the receiver side again st(t) is convoluted with the impulse response hr(t) of the receiving filter.

 se(t)
 =
 st(t)*hr(t)
(0.2)
 =
 d d(t)*ht(t)*hr(t)
(0.3)

In AWGN best filter is a matched filter. For intersymbol free reception se(t) at the receiver should follow the first nyquist criterion (which means zero signal of preceedat integer bitsample times). filters with a cosinus-roll-off spectral slope like

 Hc(jw)
 =
 1
 for |w|wn £ 1-r
(0.4)
 Hc(jw)
 =
 1 2 é ê ë 1+cos æ ç è p 2r ( w wn -(1-r) ö ÷ ø ù ú û
 for  1-r £ |w| wn £ 1+r
(0.5)
 Hc(jw)
 =
 0
 for |w| wn ³ 1+r
(0.6)

will have an impuls response following the nyquist criterion I. r is in the range 0¼1, which means r = 0 is an ideal lowpass at fg = wn and r = 1 is a cosinus shaped filter slope from 0¼2wn.
To get an impuls at the receivers output to the decision logic, which follows the nyquist criterion I and uses a matched filter as receiving filter means

 Ht(jw) ·Hr(jw) = Hc(jw)
(0.7)

Ht(jw), Hr(jw) are the fouriertransforms of ht(t),hr(t). To follow this equation, you have to use a transmitting and receiving filter which are root cosinus shaped.

the impulse response of such a filter will be calculated by

h0(t) =
 4r t T cos é ê ë p(1+r) t T ù ú û +sin é ê ë p(1-r) t T ù ú û

 é ê ë 1-(4r t T )2 ù ú û pt
(0.8)

where T means the symbol time. Such a filter has a infinite impulse response, so it must be windowed to implement it with a FIR-Filter. The impulse response looks much more complicated, than the raised cosinus form of standard PSK. It looks like a damped sin(x)/x function of infinite length.

My rrc-filter is of length 24T and parameter r = 0.6. Its windowed with a hanning window. I didn't experiment with the value of r, but r = 0.6 seems to be a good compromise in spectral bandwidth (its smaller than your time-shaped PSK) and the eye opening, where eye-opening means (in difference to your opinion) the HORIZONTAL eye-opening in the eye diagram. HORIZONTAL eye-opening shows the sensitivity to failures in the bit-sampling-time, which are fatal at ideal LOWPASS (r = 0) and are ideal for r = 1. The vertical eye-opening is always optimal, because of the ISI-free matched filter pair and the nyquist I impuls after the receiving filter.

There is some overshoot in amplitude when using this type of filters, so the headroom in the PA must be higher, than in the time shaped version. Since PSK is a excellent low power mode, it think this not a disadvantage, only for those running big guns, which allways want to win the end of the s-meter award.

© 1999-2001 and all wrong english by Michael, DL6iAK

File translated from TEX by TTH, version 2.00. On 27 Jan 1999, 11:38.