Demodulation Error

Hi everyone,

I'm trying to demodulate the following in order to obtain "signal" -

cos(cos(2*PI*11025*i/88200.0) + signal)

I'm using a signal of - 

cos(2*PI*1102.5*i/88200.0) 

so I'm actually demodulating - 

intensity[i] = cos( cos(2*PI*11025*i/88200.0) +
cos(2*PI*1102.5*i/88200.0))

To demodulate this signal, I consider two bands. The first band is
around 11025Hz, and the second is around 2*11025Hz.

The first stage of the demodulation scheme involves individually
shifting these two bands into the base band, as follows -

for( i=0;i<SIZE;i++)
{
	mix1[i] = cos(w(11025)*i/double(SR))*intensity[i];
	mix2[i] = cos(2*w(11025)*i/double(SR))*intensity[i];
}
	
// low pass the base band, to remove all other bands
filterInit(11025/2.0,filter1,b);   
for(i=0;i<SIZE;i++)
  mix1[i] = filterLP(mix1[i], filter1, b);

filterInit(11025/2.0,filter2,b);
for(i=0;i<SIZE;i++)
  mix2[i] = filterLP(mix2[i], filter2, b);

Mix1 and mix2 above are equal to -

J1(1)sin(signal) = J1(1)sin(cos(2*PI*11025*i/88200.0)) 

J2(1)cos(signal) = J2(1)cos(cos(2*PI*11025*i/88200.0)) 

Where J is a bessel function.

I then diff and cross multiply, add the result, and intergrate that to
get "signal" -

diff1[0] = 0;
for(i=1;i<SIZE;i++)
  diff1[i] = (mix2[i] - mix2[i-1]) * mix1[i];

diff2[0] = 0;
for(i=1;i<SIZE;i++)
  diff2[i] = (mix1[i] - mix1[i-1]) * mix2[i];

for(i=0;i<SIZE;i++)
  sum[i] = diff1[i] - diff2[i];

output[0] = 0;
for(i=1;i<SIZE;i++)
  output[i] = sum[i] + /*0.995*/	sum[i-1];

the maths of which is -

the integral of 

J1(C)*J2(C)*diff(signal)*(cos(signal)^2) +
J1(C)*J2(C)*diff(signal)*(sin(signal)^2)

//cos^2 + sin^2 = 1

This algorithm worked fine when I simulated it, but now I've realized
that it doesn't work when my original signal has a phase shift - such
as -

intensity[i] = cos( cos(2*PI*11025*i/88200.0) +
cos(2*PI*11025*i/88200.0))
intensity + 5 = intensity. // shift by 5 samples
 
  
The problem is that the modulation and the local oscillators, both at
11025Hz, are not in phase.

Another way of causing this problem is if I change my code to -

for( i=0;i<SIZE;i++)
{
	mix1[i] = cos(w(11025)*(i+7)/double(SR))*intensity[i];
	mix2[i] = cos(2*w(11025)*(i+7)/double(SR))*intensity[i];
}


What might I do to avoid this problem? This demodulation scheme is
refered to as homodyne demodulation using phase generated carrier
(avoids signal fading)

Regards.

I imagine the carrier signal is produced by the demodulator to be
in-phase with the local oscillators.

bg_ie@yahoo.com (Barry) wrote in message news:<731cea69.0308161329.6de57893@posting.google.com>...
> Hi everyone,
> 
> I'm trying to demodulate the following in order to obtain "signal" -
> 
> cos(cos(2*PI*11025*i/88200.0) + signal)
> 
> I'm using a signal of - 
> 
> cos(2*PI*1102.5*i/88200.0) 
> 
> so I'm actually demodulating - 
> 
> intensity[i] = cos( cos(2*PI*11025*i/88200.0) +
> cos(2*PI*1102.5*i/88200.0))
> 
> To demodulate this signal, I consider two bands. The first band is
> around 11025Hz, and the second is around 2*11025Hz.
> 
> The first stage of the demodulation scheme involves individually
> shifting these two bands into the base band, as follows -
> 
> for( i=0;i<SIZE;i++)
> {
> 	mix1[i] = cos(w(11025)*i/double(SR))*intensity[i];
> 	mix2[i] = cos(2*w(11025)*i/double(SR))*intensity[i];
> }
> 	
> // low pass the base band, to remove all other bands
> filterInit(11025/2.0,filter1,b);   
> for(i=0;i<SIZE;i++)
>   mix1[i] = filterLP(mix1[i], filter1, b);
> 
> filterInit(11025/2.0,filter2,b);
> for(i=0;i<SIZE;i++)
>   mix2[i] = filterLP(mix2[i], filter2, b);
> 
> Mix1 and mix2 above are equal to -
> 
> J1(1)sin(signal) = J1(1)sin(cos(2*PI*11025*i/88200.0)) 
> 
> J2(1)cos(signal) = J2(1)cos(cos(2*PI*11025*i/88200.0)) 
> 
> Where J is a bessel function.
> 
> I then diff and cross multiply, add the result, and intergrate that to
> get "signal" -
> 
> diff1[0] = 0;
> for(i=1;i<SIZE;i++)
>   diff1[i] = (mix2[i] - mix2[i-1]) * mix1[i];
> 
> diff2[0] = 0;
> for(i=1;i<SIZE;i++)
>   diff2[i] = (mix1[i] - mix1[i-1]) * mix2[i];
> 
> for(i=0;i<SIZE;i++)
>   sum[i] = diff1[i] - diff2[i];
> 
> output[0] = 0;
> for(i=1;i<SIZE;i++)
>   output[i] = sum[i] + /*0.995*/	sum[i-1];
> 
> the maths of which is -
> 
> the integral of 
> 
> J1(C)*J2(C)*diff(signal)*(cos(signal)^2) +
> J1(C)*J2(C)*diff(signal)*(sin(signal)^2)
> 
> //cos^2 + sin^2 = 1
> 
> This algorithm worked fine when I simulated it, but now I've realized
> that it doesn't work when my original signal has a phase shift - such
> as -
> 
> intensity[i] = cos( cos(2*PI*11025*i/88200.0) +
> cos(2*PI*11025*i/88200.0))
> intensity + 5 = intensity. // shift by 5 samples
>  
>   
> The problem is that the modulation and the local oscillators, both at
> 11025Hz, are not in phase.
> 
> Another way of causing this problem is if I change my code to -
> 
> for( i=0;i<SIZE;i++)
> {
> 	mix1[i] = cos(w(11025)*(i+7)/double(SR))*intensity[i];
> 	mix2[i] = cos(2*w(11025)*(i+7)/double(SR))*intensity[i];
> }
> 
> 
> What might I do to avoid this problem? This demodulation scheme is
> refered to as homodyne demodulation using phase generated carrier
> (avoids signal fading)
> 
> Regards.