It is important to have complex sine generators ( referred to as local oscillators (LO) from this point onwards ) if the frequency offset and phase offset blocks in a channel model produce a complex signal even if the input is a purely real signal. If the LO at the receiver is purely real, the imaginary component of the signal will be ignored.
A Phase Shift Keying (PSK) transmitted signal is of the form exp( j*2*pi*fc*t + p ) where ‘fc’ is the carrier frequency and ‘p’ is the baseband data. In the down-converter block of the receiver, the incoming signal is multiplied by a local oscillator signal which MUST BE of the form exp( -j*2*pi*fc*t + p ). The equations are as follows:
r_x = exp( j*(2*pi*fc*t + p) ); % Received signal
c = exp( - j*(2*pi*fc*t ) ); % LO signal at receiver
p = r_x .* c; % Product signal
Thus,
p = exp( j*p ); % Baseband signal successfully down-converted
This signal passes through the low pass filter and is then fed into the demodulator unit.
If the local oscillator signal was not conjugated, it would be of the form
c1 = exp( j*(2*pi*fc*t ) );
And the product would be
p1 = exp( j*(2*pi*2fc*t + p) ); % Incoming signal further up-converted to carrier frequency 2fc.
If this signal is passed through the low-pass filter, the output would be zero. Hence, it is a rule of thumb to conjugate the local oscillator output at the receiver.
However, in case of the attached model, the result of using a non-conjugated local oscillator signal is NOT zero. This is because:
Carrier frequency fc = 4.8kHz
Sampling frequency fs = 19.2kHz
When the non-conjugated local oscillator is used, the incoming signal is up-converted to 2fc instead of getting down-converted to zero. However, in this case, 2fc is the same as fs/2 (9.6 kHz). That is, the up-converted signal is aliased and the low pass filter passes the signal. Hence, in this very specific case, the output of using a conjugated LO signal is same as that of using a non-conjugated LO signal.
