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They have What they are missing: stuff aiming up. From the perspective overhead.

Pulse Shapes — Constellation Modulator

sps = 16

parameters to the root raised cosine filter sample rate symbol rate alpha num taps: 11 * sps (is popular)

constellation.airity

Frequency locked loops (FLL)

automate the slider that adjusts the offset

The received spectrum isn't centered around zero. I want the spectrum that is centered around zero.

Do a band-edge filter on the spectrum offers balance metaphor turns on and then falls offers

You can create an entire filter in the time domain. Take incoming data stream, pass through filter, Smooth error signal, and then nudge the

AGC = automatic gain control rate 100u reference 1 gain 1 max gain 65.536k

FLL Band-edge needs samples per symbol the filter rolloff factor (alpha) prototype filter size sps * 2 + 1 loop bandwidth: 2*math.pi / sps / 100

Need a shift that is

How do you actually form a filter in the time domain to get that response. There are some nice mathematical forms of the humps.

Lesson 18 Symbol Timing Synchronization (clock recovery)

error signal is the data times the slope or sign(x) x-dot

For this to work, you need to have transitions. y[n] y'[n] maximum likelihood sps expected TED gain 1.0 loop bandwidth 0.045 damping factor 1.0 maximum deviation 1.5 output samples/symbol 1 interpolating resampler: polyphase filterbank, matched filter nfilts (number of filters) rcc_taps

RRC Filter Taps gain: nfilts sample rate nfiltssamp_rate symbol rate: samp_rate/sps Excess BW: alpha num taps: 11sps*nfilts [spans 11 symbols]

Carrier Phase Synchronization with a Costas Loop

By end of today, we have all the tools: timing synchronizations carrier phase synchronization

No matter how good your clocks, you have to deal with frequency offsets and phase offsets

Simplified diagram from last time

Symbol Sync block using a polyphase filterbank, MF Now we look at the Costas loop clocks matched to something like a part per billion

How does the Costas loop do its thing producing (at least for BPSK) to a purely real?

  1. If we knew theta, we could "derotate" by ze^{-i\theta}
  2. If we can estimate θ, we can drive it to zero with a control loop

data: z = x + iy = e^{i\phi} a = a_I + i a_Q = e^{i \phi_a}

$$z a^* = e^{i(\phi - \phi_a)} $$ $$\theta = \arg(z)$$

For small values of θ, we can use the small-angle approximation. We're going to use $$\theta \approx \Im(z a^*)$$.

Costas loop needs to know the number of constellation points

constellation.arity() gives the order

Resolving phase ambiguity and differential encoding

Two techniques to handle.

  1. send a unique word with a lot of transitions; Costas loop will lock. Then ask did I get the right sign (BPSK) or the right one of 4 phases (QPSK).
  2. Differential encoding

for BPSK, start in some state. To send 0, stay in the state. To send a 1, switch states. **