Basic Wireless Communication System Review

2. Review • The very basic wireless communication system

3. Review • Filters are used to translate the bits into baseband waveforms. We use RRC filers. • This process is called ``pulse shaping.’’ • Every symbol (in BPSK, each symbol is either -1 or 1) will generate a waveform, which depends on the *impulse response* of the low pass filter (in BPSK, it is either just the impulse response or the impulse response times -1). • Remember the shape of the impulse response! • The transmitter sends a symbol to the LPF every T seconds. T is called the symbol time. In GNU SDR, T is 2us.

4. Review • The signals we can actually send is the following, where I(t) and Q(t) are the baseband waveforms. In BPSK, the cosine is multiplied with a waveform, the sine is not.

5. Review • The baseband waveform is the addition of the impulse response waveforms generated by the data symbols. • The impulse response is usually written as h(t). • Let the data symbols sent to the low pass filter be x[n] at time n. x[n] will add a voltage of x[n]h(t-nT) to the baseband waveform at time t.

6. Review • Multi-path. Signals travel infinite number of paths to reach the receiver. The received signal is the addition of signals from all paths: • where A is the attenuation, \tau is the delay. • After received the signal, we will first multiply it with sine and cosine wave, and pass it through the LPF, as explained before.

7. Review • Complex channels – because we can send both a sine and a cosine wave, which can be conveniently represented as a complex wave. • The sender and receiver will try to use the same frequency, but they cannot. Consider the cosine branch and consider a single path. • So, after the low pass filter, it becomes • Similarly, the sine branch will produce • Therefore, the cosine and sine branch can be regarded as a complex number

8. Review • To establish the communication, we will first have to reproduce the baseband waveform. Which means that we have to get rid of , a process called carrier phase tracking. • We take samples from the received baseband waveform to get • Assume the samples are taken at perfect time, i.e., when the impulse response is at the peak, so the sample should be either 1 or -1 in the BPSK case. The ideal sample is called “constellation point.” Extract the phase error and adjust the phase.

9. Review • Timing synchronization – how to take samples at the correct time. • The basic Mueller and Muller algorithm: u=a_{k-1} x_k – a_{k} x_{k-1}. • u is the timing error. • In this example, if off by a delta, x_{k-1} will be deducted by the impulse response at T+delta, which is negative, while x_{k} will be added by the impulse response at -T+delta, which is positive. k-1 k K+1

10. Review • While • The first line is the one that does the work. In BPSK, it becomes

11. Review • The current GNU SDR implementation uses the optimized mm algorithm. Assume a sample error of \delta. The sample at time n is • The timing error is calculated as • The reason is that

12. Review • Dealing with multipath.

13. Review • Important thing to remember is that even with perfect carrier phase tracking and symbol timing, when taking a sample, it will contain some thing from the neighboring symbols. • At time 0, the path with delay will contribute to this sample, but for k!=0, also • And there are infinite number of paths. • In other words, if you take a sample, will be something like , where g[i] is the summation of all paths.

14. Review • The red curve represents the other path. k-1 k K+1

15. Review • We can use equalization. The basic idea is to subtract the contributions of non-relevant samples from the current sample. • A simple yet okay algorithm is LMS. Start with an arbitrary for all i, usually =1 and =0 for all other i. • Let . Then , where • Basically, if e< 0 while s_i>0, it means that c_i is not large enough… • The idea is to minimize the mean square of the error.

16. Review • A signal can also be represented in the frequency domain.

17. Review • LTI system. • A linear system is a system such that • Suppose . A linear system is “time invariant” if .

18. Review • DFT: • IDFT: • Convolution:

19. CDMA • Code Division Multiplexing

20. CDMA • Used in 3G networks. • Direct Sequence Spread Spectrum: spread a data bit into multiple chips. • Each sender has a unique chip sequence, that is *orthogonal* with other chip sequences.

21. Simple Examples of CMDA • A: (-1 -1 -1 +1 +1 -1 +1 +1 ) • B: (-1 -1 +1 -1 +1 +1 +1 -1 ) • C: (-1 +1 -1 +1 +1 +1 -1 -1 ) • D: (-1 +1 -1 -1 -1 -1 +1 -1 )