Symbol Synchronization

Please reference the lecture slides for workshop 3 as needed.

• Workshop 03 - Symbol Sychronization

3.1 - Constellation Modulator

In this activity you will implement a basic digital transmitter that streams a continuous sequence of random QPSK symbols. We will use this basic transmitter to investigate some of the key subsystems in a practical digital receiver. Consider the following block diagram of a pulse modulation communication systems.

Key elements in our transmit chain include the following.

• Symbol mapper – maps a stream of input bits to a stream of complex-valued symbols taken from some constellation, e.g., BPSK, QPSK, or 16QAM.
• Upsampler – inserts zeros between our symbols to increase the sampling rate. We must oversample our signal when using a pulse shape with excess bandwidth. Why?
• Pulse-shaping filter – filters our upsampled symbol sequence to ensure the signal is bandlimited. We often use filter coefficients, g[k], defined by a Root Raised Cosine (RRC) pulse with a configurable excess bandwidth parameter.
• Digital-to-analog converter (D/A) – converts baseband digital waveform (discrete-valued, discrete-time) to a baseband analog waveform (continuous-valued, continuous-time).
• Upconversion – applies quadrature modulation to shift our complex signal directly from baseband (centred about 0 Hz) to passband (centred about our carrier frequency f_c).

Implement the GRC flowgraph as depicted above. It should be clear what variables to used in configuring the various blocks but make sure you define the sample rate variable samp_rate as a function of the symbol rate and the number of samples per symbol, i.e.,samp_rate = symb_rate * sps. Note the scaling factor applied at the output of Constellation Modulator. This is to normalize the QPSK constellation to unit energy as points in the constellation are at (\pm \sqrt{2} \pm j\sqrt{2}).

FLUX Questions:

1. Run your flowgraph and try a few different values for the samples-per-symbol (sps) and the excess bandwith (excess_bw) parameters. How does the output spectrum change?
   Note: The Constellation Modulator parameters CANNOT be updated while running the flowgraph.
2. Increase the symbol rate to 500 kSymb/s and then rerun your flowgraph. Does the change in output spectrum align with your expectations?

3.2 Manual Symbol Synchronization

You will now focus only on recovering the symbol timing of the transmitted waveform. We will add a symbol timing recovery subsystem that takes as input the output of the received matched filter and outputs the sample closest to the optimal sampling time.

To begin, you will manually select the best sampling offset, \hat{k}, based on visual observation.

The Root Raised Cosine Filter implements the matched filtering with a digital RRC pulse. The Keep 1 in N block decimates our waveform by a factor of sps down to the symbol rate. By adjusting the delay variable we can change the sample offset that is passed by the decimator, allowing us to adjust the symbol sampling point to be as close as possible to the optimal timing.

Update your receiver flowgraph based on the manual symbol timing recovery system as depicted in the figure above. Parameters for the RRC filter are listed below for your convenience.
Note: The delay variable must be an integer and NOT a float for the Delay block to function properly.

FLUX Questions:
3. Run your flowgraph and observe the constellation on the QT GUI Constellation Sink. How does the constellation change as you adjust the delay?
4. Increase the oversampling rate by trying values of 4 and 8 for sps. How does the oversampling rate affect the constellation at different delays?

Before automating the symbol timing recovery process, we would first like to determine a metric to quantify the deviation of the received constellation from the ideal points. A common metric employed by communication engineers to assess degradation in a symbol constellation is the error vector magnitude or EVM. It is a useful measurement of performance both at the transmitter and the receiver. At the transmitter it captures phase and amplitude distortions introduced by the analog front-end, while at the receiver it captures these effects plus interference, noise, and residual errors after impairment compensation.

As show in the figure above, the error vector of a given received symbol is the distance from that symbol (red x) to its ideal location in the complex plane (blue x). The EVM of a group of M observed symbols will be defined as the RMS average magnitude of their error vectors normalized by the maximum ideal constellation magnitude. It is typically expressed as a percentage.

EVM = \frac{ \sqrt{\frac{1}{M} \sum_{i=0}^{M-1} |y[i] - a[i]|^2 } }{\max_j |c_j|} \text{ where } a[i]\in\{c_0, c_1, \dots\}

Here, c_j are the ideal points in our constellation, a[i] is the ideal location of the symbol transmitted at time i and y[i] is the actually transmitted or received symbol at time i. You will notice there are parallels between the definition of EVM and SNR. An increase in EVM will result in a decreaes in SNR and an increase in the observed BER. GNU Radio has a block for computing the EVM that we can include in our flowgraphs.

Add the EVM Measurement block to your flowgraph to measure the EVM of your received constellation, connecting the output to a QT GUI Number Sink. Note that EVM is expressed as a percentage and thus falls between 0 and 100.
Hint: You can add exponential averaging to QT GUI Number Sink by entering an average parameter value between 0.0 and 1.0.

FLUX Questions:
5. How does the EVM vary as you change the delay value of your manual timing recovery system?
6. How does the EVM change if you first increase the oversampling factor sps and then vary the delay parameter?

3.3 Timing Recovery Loop

You will now replace your manual symbol timing correction with GNU Radio's Symbol Sync block. This block is a feedback-based timing correction block composed of a timing error detector (TED), proportional-integral (PI) loop filter, and sample interpolator. There are multiple options for both the TED and interpolating filter. For this workshop you will use the Early Late TED (ELTED) and 8-tap MMSE interpolator.

Construct the GNU Radio flowgraph as depicted above. Note that we have included a Channel Model block to create a sample clock offset between the transmit and receive chains. If the sampling rate of the input to Channel Model is F_{in}, then sample rate of the output will be at F_\text{out} = \epsilon \cdot F_\text{in}.

FLUX Questions:
7. What EVM do you measure after symbol synchronization?
8. Try different values of epsilon in the Channel Model block. At what epsilon does your symbol synchronizer no longer work?

3.4 RF Loopback

Now that we have investigated symbol timing recovery in simulation, let's move on to see if our system works with our real-world hardware! Create an RF loopback by physically connecting the output of the TX1 port to the RX1 port with a 30 dB attenuator inline. A friendly reminder that...

Important Note: To avoid violating governmental regulations and damaging the hardware, please adhere to the following guidelines.

1. Never transmit or receive on any band on which you are not licensed to operate.
2. Always terminate TX and RX ports with a 50 Ohm load - either an antenna or SMA terminator.
3. Never connect a TX port directly to an RX port without proper attenuation of at least 30 dB.

We will use the following flowgraph to create our RF loopback experiment.

Implement the RF loopback flowgraph as depicted in the figure above. There are a few important things to note when creating this flowgraph. First, the bladeRF expects sample values in the range (-1.0, 1.0) and thus we have provide additional scaling in Multiply Const block after Constellation Modulator. Second, in future workshops we will address the issue of channel estimation and carrier offset correction. Because we do not have these subsystems present, the received constellation has a fixed amplitude scaling and random phase shift relative to the ideal constellation points. You will manually correct for these by observing the output constellation. Update the Multiply Const block such that Constant is amplitude * np.exp(1j*phase). This expression uses the Numpy Python package which must be included in an Import block as follows: import numpy as np.

FLUX Question:
1.Run your flowgraph and observe the received signal constellation. Adjust the amplitude and phase scaling variables to align the received symbol constellation with the transmitted constellation. This should minimize the EVM. What EVM do you measure after symbol synchronization in your RF loopback?