Blind and Semiblind Channel Estimation - Nonlinear Signal and Image Processing: Theory, Methods and Applications - page 278

where

exp j 2

n

a 0 ,n = α k = j α n a 0 ,n − 1

k =−∞

n exp j 2

n − 2

a 1 ,n = j α α = j α

n a 0 ,n − 2 ,

(8.31)

k

k =−∞

and p 0 ( are the two most significant pulses. More than 99% of

the approximated GMSK signal energy is contained in the p 0 (

t ) and p 1 ( t )

pulse; then,
the signal can be further simplified taking into account only the first term
in the approximation (Equation 8.30)

t )

∞

s ( t ) ≈ a n p 0 ( t − nT ). (8.32)

n =−∞

With this approximation the maximum achievable SNR is 23 dB even in the
noiseless case.

8.5.2 SOS-Based Blind Equalization for GSM

For SOS-based blind equalization, the necessary channel diversity can be ob-
tained by oversampling if only one receive antenna is available. The pulse
shape p 0 , and therefore the linearized GMSK signal, has little excess band-
width beyond the 1 T will

not generate enough diversity. In Reference 50, and later in Reference 52, the
required diversity is obtained using a simple derotation scheme on the re-
ceived signal and then considering the I-Q branches as separate subchannels.
The sequence a n

/ 2 T limit. Oversampling with a rate higher than 1 /

α n a n − 1 is a pseudo-quaternary shift keying (QPSK)
sequence because, at any given time, a n can only take two values rather than
four. It can be written also as a n =

= j

j n a n where a n =± 1isabinary-phase shift-

keying (BPSK) sequence. With this definition, the received baud rate-sampled
signal is given by

∞ ∞

h k j n − k a n − k + w n = j n [ h k j − k ] a n − k + w n . x n = (8.33)

k =−∞ k =−∞

At the receiver part we first derotate the received baud-sampled signal:

∞

j − n x n = [ h k j − k ] a n − k + j − n x n = w n . (8.34)

k =−∞