Review of basic ideas from digital communication - Signal Processing and Optimization for Transceiver Systems - page 65

y bp ( t )= x bp ∗ h bp )( t )

=Re ( x lp ∗ h lp )( t ) e jω c t

e jω c t s c ( k ) p ( t − kT )+ jp ( t − kT ) =Re ∗ h lp ( t ) .

k

This can be rewritten as
y bp ( t )=Re
k

kT ) e jω c t ,

s c ( k ) h lp,e ( t − (2 . 105)

where

h lp,e ( t )= p ∗ h lp + jp ∗ h lp ( t )= h r ( t )+ jh i ( t ) . (2 . 106)

Here s c ( k ) ,p ( t ) , and p ( t ) are real, but h lp ( t ) is complex. So the real and imagi-

nary parts p ( t kT ) get mixed up by convolution with h lp ( t ). Nor-

mally the PAM/SSB demodulator just multiplies the received signal by cos ω c t
and performs lowpass filtering. This removes

− kT )and p ( t −

p ( t ) and recovers p ( t ) . But, since

h lp ( t ) mixes up p ( t )and p ( t ) , the receiver now has to be smarter, and must

perform a demodulation with cos ω c t and one with sin ω c t (like a QAM demod-
ulator). This recovers the real and imaginary parts of

s c ( k ) h r ( t kT ) . − kT )+ jh i ( t −

k

So the received digital signal after sampling has the simple form
y d ( n )=
k

s c ( k ) h d ( n − k ) , (2 . 107)

where

h d ( n )= h r ( nT )+ jh i ( nT ) .
One may argue that we can simply keep the real part of y d ( n ) and minimize
complication. But this is like wasting the power of the signal in the imaginary
part. For example, if h d ( n ) happens to be purely imaginary, this is a disaster
because the real part of y d ( n ) has no information about the signal. So we take
the entire complex signal y d ( n ), perform “time-domain equalization” (TDE),
and take the real part to get
y d ( n )=
k

s c ( k ) h d ( n

− k )+ q ( n ) , (2 . 108)

where h d ( n )isrealand q ( n ) is real-valued noise. Thus at this point the digital
signal processor simply sees a real PAM signal s c ( k ) and a real channel h d ( n ).