Digital Receivers: Symbol-Timing Recovery for QPSK

Introduction

This receiver exercise introduces the primary components of a QPSK receiver with specific focus on symbol-timing recovery. In a receiver, the received signal is first coherently demodulated and low-pass filtered (see Digital Receivers: Carrier Recovery for QPSK) to recover the message signals (in-phase and quadrature channels). The next step for the receiver is to sample the message signals at the symbol rate and decide which symbols were sent. Although the symbol rate is typically known to the receiver, the receiver does not know when to sample the signal for the best noise performance. The objective of the symbol-timing recovery loop is to find the best time to sample the received signal.

Figure 1 illustrates the digital receiver system. The transmitted signal coherently demodulated with both a sine and cosine, then low-pass filtered to remove the double-frequency terms, yielding the recovered in-phase and quadrature signals, s_I[n] and s_Q[n]. These operations are explained in Digital Receivers: Carrier Recovery for QPSK. The remaining operations are explained in this module. Both branches are fed through a matched filter and re-sampled at the symbol rate. The matched filter is simply an FIR filter with an impulse response matched to the transmitted pulse. It aids in timing recovery and helps suppress the effects of noise.

**Figure 1:** Digital receiver system

If we consider the square wave shown in Figure 2 as a potential recovered in-phase (or quadrature) signal (i.e., we sent the data [+1,-1,+1,-1,…]) then sampling at any point other than the symbol transitions will result in the correct data.

**Figure 2:** Clean BPSK waveform.

**Figure 3:** Noisy BPSK waveform.

However, in the presence of noise, the received waveform may look like that shown in Figure 3. In this case, sampling at any point other than the symbol transitions does not guarantee a correct data decision. By averaging over the symbol duration we can obtain a better estimate of the true data bit being sent (+1 or -1). The best averaging filter is the matched filter, which has the impulse response u[n]−u[n−T_symb], where u[n] is the unit step function, for the simple rectangular pulse shape used in Digital Transmitter: Introduction to Quadrature Phase-Shift Keying. 1 Figure 4 and Figure 5 show the result of passing both the clean and noisy signal through the matched filter.

**Figure 4:** Averaging filter output for clean input.

**Figure 5:** Averaging filter output for noisy input.

Note that in both cases the output of the matched filter has peaks where the matched filter exactly lines up with the symbol, and a positive peak indicates a +1 was sent; likewise, a negative peak indicates a -1 was sent. Although there is still some noise in second figure, the peaks are relatively easy to distinguish and yield considerably more accurate estimation of the data (+1 or -1) than we could get by sampling the original noisy signal in Figure 3.

The remainder of this handout describes a symbol-timing recovery loop for a BPSK signal (equivalent to a QPSK signal where only the in-phase signal is used). As with the above examples, a symbol period of T_s=16 samples is assumed.

Early/late sampling

One simple method for recovering symbol timing is performed using a delay-locked loop (DLL). Figure 6 is a block diagram of the necessary components.

**Figure 6:** DLL block diagram.

Consider the sawtooth waveform shown in Figure 4, the output of the matched filter with a square wave as input. The goal of the DLL is to sample this waveform at the peaks in order to obtain the best performance in the presence of noise. If it is not sampling at the peaks, we say it is sampling too early or too late.

The DLL will find peaks without assistance from the user. When it begins running, it arbitrarily selects a sample, called the on-time sample, from the matched filter output. The sample from the time-index one greater than that of the on-time sample is the late sample, and the sample from the time-index one less than that of the on-time sample is the early sample.Figure 7 shows an example of the on-time, late, and early samples. Note in this case that the on-time sample happens to be at a peak in the waveform. Figure 8 and Figure 9 show examples in which the on-time sample comes before a peak and after the peak.

The on-time sample is the output of the DLL and will be used to decide the data bit sent. To achieve the best performance in the presence of noise, the DLL must adjust the timing of on-time samples to coincide with peaks in the waveform. It does this by changing the number of time-indices between on-time samples. There are three cases:

In Figure 7, the on-time sample is already at the peak, and the receiver knows that peaks are spaced by T_symb samples. If it then takes the next on-time sample T_symb samples after this on-time sample, it will be at another peak.

In Figure 8, the on-time sample is too early. Taking an on-time sample T_symb samples later will be too early for the next peak. To move closer to the next peak, the next on-time sample is taken T_symb+1 samples after the current on-time sample.

In Figure 9, the on-time sample is too late. Taking an on-time sample T_symb samples later will be too late for the next peak. To move closer to the next peak, the next on-time sample is taken T_symb−1 samples after the current on-time sample.

The offset decision block uses the on-time, early, and late samples to determine whether sampling is at a peak, too early, or too late. It then sets the time at which the next on-time sample is taken.

**Figure 7:** Sampling at a peak.

**Figure 8:** Sampling too early.

**Figure 9:** Sampling too late.

The input to the offset decision block is on-time(late−early), called the decision statistic. Convince yourself that when the decision statistic is positive, the on-time sample is too early, when it is zero, the on-time sample is at a peak, and when it is negative, the on-time sample is too late. It may help to refer to Figure 7, Figure 8, and Figure 9. Can you see why it is necessary to multiply by the on-time sample?

The offset decision block could adjust the time at which the next on-time sample is taken based only on the decision statistic. However, in the presence of noise, the decision statistic becomes a less reliable indicator. For that reason, the DLL adds many successive decision statistics and corrects timing only if the sum exceeds a threshold; otherwise, the next on-time sample is taken T_symb samples after the current on-time sample. The assumption is that errors in the decision statistic caused by noise, some positive and some negative, will tend to cancel each other out in the sum, and the sum will not exceed the threshold because of noise alone. On the other hand, if the on-time sample is consistently too early or too late, the magnitude of the added decision statistics will continue to grow and exceed the threshold. When that happens, the offset decision block will correct the timing and reset the sum to zero.

Sampling counter

The symbol sampler maintains a counter that decrements every time a new sample arrives at the output of the matched filter. When the counter reaches three, the matched-filter output is saved as the late sample, when the counter reaches two, the matched-filter output is saved as the on-time sample, and when the counter reaches one, the matched-filter output is saved as the early sample. After saving the early sample, the counter is reset to either T_symb−1, T_symb, or T_symb+1, according to the offset decision block.

MATLAB Simulation

Because the DLL requires a feedback loop, you will have to simulate it on a sample-by-sample basis in MATLAB.

Using a square wave of period 32 samples as input, simulate the DLL system shown in Figure 6. Your input should be several hundred periods long. What does it model? Set the decision-statistic sum-threshold to 1.0; later, you can experiment with different values. How do you expect different thresholds to affect the DLL?

Figure 10 and Figure 11 show the matched filter output and the on-time sampling times (indicated by the impulses) for the beginning of the input, before the DLL has locked on, as well as after 1000 samples (about 63 symbols' worth), when symbol-timing lock has been achieved. For each case, note the distance between the on-time sampling times and the peaks of the matched filter output.

**Figure 10:** Symbol sampling before DLL lock.

**Figure 11:** Symbol sampling after DLL lock.

DSP Implementation

Once your MATLAB simulation works, DSP implementation is relatively straightforward. To test your implementation, you can use the function generator to simulate a BPSK waveform by setting it to a square wave of the correct frequency for your symbol period. You should send the on-time sample and the matched-filter output to the D/A to verify that your system is working.

Extensions

As your final project will require some modification to the discussed BPSK signaling, you will want to refer to the listed references, (see Proakis and Blahut, and consider some of the following questions regarding such modifications:

How much noise is necessary to disrupt the DLL?

What happens when the symbol sequence is random (not simply [+1,-1,+1,-1,…])?

What would the matched filter look like for different symbol shapes?

What other methods of symbol-timing recovery are available for your application?

How does adding decision statistics help suppress the effects of noise?

FOOTNOTES

For digital communications schemes involving different pulse shapes, the form of the matched filter will be different. Refer to the listed references for more information on symbol timing and matched filters for different symbol waveforms.