- Binary Phase Shift Keying (BPSK): 1 bit per symbol
- Quaternary Phase Shift Keying (QPSK): 2 bits per symbol
- 8-Phase Shift Keying (8PSK): 3 bits per symbol

- 4-position Quadrature Amplitude Modulation (4-QAM): 2 bits per symbol
- 16-position Quadrature Amplitude Modulation (16-QAM): 4 bits per symbol
- 64-position Quadrature Amplitude Modulation (64-QAM): 6 bits per symbol

- Purpose: limit the bandwidth of the transmission
- Happens after bit-to symbol mapping above
- Have to avoid introducing inter-symbol-interference (ISI)
- Popular implementation: raised cosine filter (RC)
- One-half implemented on transmitter end, other half implemented on receive end
- Each half is a root-raised cosine filter (RRC) filter
- When combined, the RRC filters provide zero-ISI

- Modulates and Demodulates a signal
- Additionally moves between the symbol rate f_symbol and the DAC or ADC sampling rate f_system
- Digital Upconverter (DUC): transmitter, f_symbol -> f_system through filtering
- Digital Downconverter (DDC): receiver, f_system -> f_symbol
- Multirate R_multirate = f_system/f_symbol

- On receive side, need to account for phase and frequency offset
errors
- Phase error calculation:
- Decision Directed: transmitted data symbols unknown, phase error
is generated based on symbol decision (closest symbol to received
sample)
- Generally use this method
- Possible that there is still phase rotation (e.g., a multiple of 90 degrees)

- Data aided: receiver knows transmitted symbols, derives phase error based on that

- Decision Directed: transmitted data symbols unknown, phase error
is generated based on symbol decision (closest symbol to received
sample)

- Goal to sample at an interval at the proper time to get the max signal amplitude
- Done using a timeing error detector (TED)
- Two methods: oversampling and interpolation
- oversampling: sample at rate much higher than symbol rate, then select the samples closest to the maximum effect points
- Interpolation: oversample at small amount (e.g. 2x symbol rate) and interpolate between samples

- TED produces early and late signals, then subtracts between them
to get a 'punctual' (on time) sample time
- Combined timing and carrier synchronization looks like:
- Should make I and Q amplitudes converge:

- Calculates an initial frequency offset, for a large value (happens before fine frequency sync)
- Uses FFT on input, raised to 4th power, then selecting FFT bin with highest magnitude as the desired signal

- Once you have a proper PSK (e.g. BPSK or QPSK) signal plot, there is a chance the phase is incorrect, e.g. rotated to the wrong constellation points. This means your resulting bits will be incorrect since they are derived from the wrong constellation point
- Two ways to handle this: differential encoding or unique word transmission

- Encodes input bits so that they can be recovered after incorrect receive due to PSK phase offset
- For BPSK, you use the current bit and previous encoded bit (Default 0) and XNOR them:
- To decode, you do the same thing except your previous bit to use
is BEFORE decoding
- If the phase is incorrect on the received symbols, then the first decoded bit will be incorrect, but the rest will be correct
- QPSK encoding and decoding is similar, except you use the last 2 bits
instead of just the last bits, and the output of encoding/decoding
is two bits instead of 1 bit

- You transmit a fixed/known bit sequence before your actual payload. On the receive end, we look for that sequence, and if it's not found, check the resulting sequence if we rotate the phase/bit mapping ourselves. Once we find the phase mapping that results in the correct sequence, we apply said phase mappnig to the rest of the input as well
- For BSPK this looks like:
- For QPSK this looks like: