Plot the autocorrelation function and show that other than

Question

Assignment 1. Generator and Degradations

Part 1. Signal Generator

• Design Gold code of size 1023 (GPS C/A code) using HW2 matlab codes. Two M-codes are used which are generated by two generating polynomials: G1: g(D) = 1 + D³ + D¹⁰ and G2: g(D) = 1 + D² + D³ + D⁶ + D⁸ + D⁹ +D¹⁰.

• Plot the autocorrelation function and show that other than peak there are only three values in the autocorrelation. What are the values of the autocorrelation function?

• Increase the number of samples per chip to 4 samples (e.g. 1-> 1111, 0-> 0000) and plot autocorrelation function again.

Generate 5 periods of oversampled signal

o Add Doppler sinusoid by multiplying code signal to a sinusoid. Assume one code period is 1ms. Multiply your signal to a complex sinusoid exp(2Π jfnt_s) where t_s is the sampling period, n is the sample index.

o Introduce a code-phase shift (0-4092): starting from a sample index (n_code_phase) in the range (0-4092) extract 4 periods of the signal from previously generated 5 periods. At this stage you will have a signal with certain code-phase shift and Doppler frequency modulation, oversampled (4 samples per chip). Add AWGN noise similar to HW2. Keep this signal generator for simulations in all design projects.

Part 2. Study degradation effects (Set AWGN noise to "0" (no noise)

o Crosscorrelate one oversampled code period with Doppler sinusoidal modulation with Doppler free oversampled code. Starting with frequency "0" increase the frequency and demonstrate autocorrelation peak degradation. At what frequency the peak is reduced two times? As values are complex - plot absolute values for crosscorrelation to see the peak?

o Study degradation effects because of code Doppler. Use one code period of the generated signal with Doppler frequency "0" (no sinusoidal Doppler). Generate another signal which has a code Doppler in it. Cross-correlate them and see degradations. At what Doppler frequencies code Doppler effects become noticeable?

o The code Doppler is obtained by stretching/expanding the signal in proportion to relative speed (see lecture notes on the phenomena). When such signal is sampled then there will be no exact 4 samples per chip. The following diagram shows the way to obtain a sampled signal with code Doppler. We have to sample a stretched continuous code to account for the code Doppler.

• One should have certain amount of samples per code period. E.g., with 4 samples per chip we will have 4092 samples per code period. The idea is for each sample to calculate an appropriate code value. In Doppler free case we will always have correct alignment of samples and code chips, while, in general, samples are not aligned with chips in the presence of code Doppler. Let us assume that the chip duration is changed due to Doppler and is now d chip T _ . Then assume that the number of samples per original chip was k . Then sampling interval is ΔT = T_chip/k. Then the nth sample corresponds to [nΔT_s/T_{chip_d}]chip.

Assignment 2. Acquisition

• Signal generator from previous design project is used in this project. We acquire a signal with shifted code phase and Doppler sinusoidal modulation as generated in Design Project I. Set a code phase to 556, Doppler frequency to 1kHz. Add AWGN noise as your generator can do it.

• Have two "for" loops: (1) Loop 1: multiply received signal to complex sinusoids exp(-2Πjfnt_s)- to wipe-off Doppler modulation, select sinusoids [-2kHz -1.5kHz, -1kHz, 0kHz, 1kHz, 1.5 kHz, 2kHz] as candidate sinusoids; (2) Loop 2. For each candidate frequency apply code correlator as explained below. You will get 3D picture, e.g. all code phases on x-axis, all frequencies on y-axis, and correlation values on z-axis. Set noise to "0", get a perfect peak, increase noise to have noise floor increased. Plot the picture and report.

• Use two correlator design approaches (1) parallel set of ordinary correlators. (2) matched filter. For matched filter conv.m routine can be used.

• Assume certain number of coherent and noncoherent stages (take 2 and 2). Calculate correlation value and combine coherently and noncoherently several periods of autocorrelation as is shown on the diagrams on next page. 

Assignment 3. Tracking

Transmitter

• Use DSSS signal model: oversampled code (4 samples per chip) and Doppler sinusoidal modulation (no code Doppler).

Signal duration is one code period.

Receiver
• Carrier Tracking. Set the code delay equal to 0 (no delay) which means a perfect alignment of code and replica. Implement an ordinary correlator (integrate&dump unit) for one code duration wiping-off the code. On the output of the correlator implement carrier loop discriminators and get characteristic curves:

o Common PLL (atan2 based) discriminator (group 1 from the student list)

o Costas PLL (atan based) discriminator (group 2 from the student list)

o FLL (atan2 based) discriminator. For FLL characteristic one should take two consecutive correlator outputs. So take two code periods. (group 3 from the student list)

• Code tracking. Set the Doppler shift effects in frequency to 0 or disable these feature. Implement two correlators (early and late) by correlating the transmiter signal with shifted replica codes. Use the correlator outputs to obtain (early minus late envelop normalized by early plus late envelope) discriminator characteristic curves. Obtain the same figure with one of envelope approximation methods (JPL or Robertson).

Combine both figures using Matlab "hold on" command and include the figure in the report