# GPS CDMA signal acquisition (BPSK modulation)

Sorry, this may be a rookie question: I have a CDMA gps communication in BPSK. Codes have a duration equal to $$1\ \text{ms}$$ and their length is 1023. At the carrier frequency ($$f_c = 1.57524\ \text{GHz}$$), Doppler frequency $$f_d \in [-5, +5]\ \text{kHz}$$.

The receiver is depicted here. The mixer shifts the signal to intermediate frequency $$f_i = 1.3\ \text{MHz}$$. Sampling of the A/D converter is at $$\frac{1}{T_s} = 8\ \text{MHz}$$ and samples are quantized to 1 bit for I and 1 bit for Q. As a result, I have 24000 samples for each channel (so they should "cover" $$3\ \text{ms}$$).

The purpose, of course, is to recognize the transmitter and its Doppler frequency.

In my understanding, it should be sufficient to recollect the transmitted symbols with $$\hat{x_k} = I_k\ \cos(2\pi f_i k T_s) - Q_k\sin(2\pi f_i k T_s)$$ and correlate the sequence obtained with each of the codes (better, their "BPSK" equivalent $$2\ b_k - 1$$) and find the one maximizing this correlation.

I understood that once obtained the data, I can estimate the phase shift between the signal and the local oscillator (thus the doppler) having $$Z_k = I_k \cdot\ Q_k = \frac{1}{8}\ x_k \cdot\ \sin(2 \Delta \theta)$$

Hoping that what I just wrote is correct:

1. are these considerations still correct now that I do have the quantized samples for I and Q and not their actual value?
2. since I/Q samples (and then the $$\hat{x_k}$$) and the codes have a different sampling rate, what's the best option to compute their correlation? I had in mind to run
ifft( fft(code, Nfft) .* conj(fft(xk, Nfft)) )


but having the two signal different sampling, I am pretty sure this will fail.

• In the end I tried to solve it in "reverse". I took the known codes, simulated a sequence of pulses sampled with rate $8\ \text{MHz}$ and took them to the same condition of the received signal (so shifted to an intermediate frequency) and then quantized. Then I computed the circular correlation. I tried letting the intermediate frequency variate $\left[f_i - f_d, f_i + f_d \right]$ and found which of these code-frequency combination returned the best correlation. Still don't know if it is the most correct way, updates will come. – Stefano Jan 24 at 8:26