Some background to Root Cosinus Shaped impulses
This is a htmlfile 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 d_{d}(t) is convoluted by a the impulse response h_{t}(t) of a transceiving filter (in PSK31 h_{t}(t) has the form of a raised cosinus)
s_{t}(t) = d_{d}(t)*h_{t}(t)


(0.1) 
at the receiver side again s_{t}(t) is convoluted with the impulse response h_{r}(t) of the receiving filter.




(0.2) 


d
_{d}(t)*h_{t}(t)*h_{r}(t)


(0.3) 



In AWGN best filter is a matched filter. For intersymbol free reception s_{e}(t) at the receiver should follow the first nyquist criterion (which means zero signal of preceedat integer bitsample times). filters with a cosinusrolloff spectral slope like





(0.4) 



1
2 

é
ê
ë 
1+cos 
æ
ç
è 

p
2r 
( 
w
w_{n} 
(1r) 
ö
÷
ø 

ù
ú
û 


for 1r £ 
w
w_{n} 
£ 1+r 

(0.5) 




(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 f_{g} = w_{n} and r = 1 is a cosinus shaped filter slope from 0¼2w_{n}.
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
H_{t}(jw) ·H_{r}(jw)
= H_{c}(jw) 

(0.7) 
H_{t}(jw), H_{r}(jw) are the fouriertransforms of h_{t}(t),h_{r}(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
h_{0}(t) = 
4r 
t
T 
cos 
é
ê
ë 
p(1+r) 
t
T 

ù
ú
û 
+sin 
é
ê
ë 
p(1r) 
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 FIRFilter. 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 rrcfilter 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 timeshaped PSK) and the eye opening, where eyeopening means (in difference to your opinion) the HORIZONTAL eyeopening in the eye diagram. HORIZONTAL eyeopening shows the sensitivity to failures in the bitsamplingtime, which are fatal at ideal LOWPASS (r = 0) and are ideal for r = 1. The vertical eyeopening is always optimal, because of the ISIfree 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 smeter award.
© 19992001 and all wrong english by Michael, DL6iAK
File translated from T_{E}X by T_{T}H, version 2.00.
On 27 Jan 1999, 11:38.
