[Discuss-gnuradio] USRP complex multiplication question

[John Andrews]

Dear All,

I have a question regarding the first complex multiplication process with the IF sinusoid in the USRP. Which of the scheme mentioned below is actually happening in the USRP?

Let's say that two ADCs provide the sampled received signal to the next stage of processing. Let's call the sampled stream of ADC_0 as A and sampled stream of ADC_1 as B.

SCHEME-1

The NCO produces two sinusoids that are in quadrature phase with each other. These are sin(2*pi*f*t) and cos(2*pi*f*t). Let's call the sin(2*pi*f*t)stream as C and

cos(2*pi*f*t) stream as D. The multiplier after the ADC stage does the following.

1. Multiplies the cos(2*pi*fc*t) stream, C,  with the sampled data stream of ADC_0, which is the stream named A and the product is AC.

2. Multiplies the sin(2*pi*fc*t) stream with the sampled data stream of ADC_1, which is the stream named B and the product is BD.

Then sends the resultant product  from these two steps as a complex sample, ( AC + jBD ) 

SCHEME-2

The NCO produces two sinusoids that are in quadrature phase with each other. These are sin(2*pi*f*t) and cos(2*pi*f*t). The multiplier after the ADC stage does the following.

The multiplier after ADC stage then multiplies the ADC outputs with the NCO outputs as

( A + jB ) * ( C + jD) = (AC - BD) + j(AD + BC)

and sends the above output as a complex sample. 

Which one is true?

Thanks