Lab 5 - Digital Modulation: Symbol Synchronization

Lab 5 - Digital Modulation:
Symbol Synchronization
ECE531 – Software Defined Radio

1 Overview and Objectives 2
2 Pulse Shaping and Matched Filtering 2
2.1 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 Timing Error 4
3.1 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4 Symbol Timing Compensation 6
4.1 Gardner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.2 Müller Mueller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.3 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5 Adding Pieces Together 12
5.1 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
6 Automatic Timing Compensation With Pluto 13
6.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
6.2 Testing on Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
7 Lab Report Preparation & Submission Instructions 14
1
1 Overview and Objectives
This laboratory will introduce the concept of timing offset between transmitting and receiving
nodes. Specifically, a simplified error model will be discussed along with three standard recovery
methods which have different performance objectives. Moreover, MATLAB® will also be used
as development tool for digital communication systems, especially the construction of a prototype
software-defined radio (SDR) based simulation.
This lab assignment follows the background theory discussed in textbook Chapter 6. Matlab
source code and implementation examples helpful for this lab are found in this chapter.
2 Pulse Shaping and Matched Filtering
0
Frequency (Hz) ×105
-15
-10
-5
0
5
10
15
20
25
30
PSD (dB)
Filtered
Original
Figure 1: Frequency spectrum of PSK signal before and after pulse shaping.
The topics of matched filters and transmit filters are introduced here to discuss how they are used
in practices and demonstrate effects they help alleviate. In digital communications theory when
matched filtering is discussed it is typically called pulse shaping at the transmitter and matched
filtering at the receiver for reference. The goal of these techniques is three fold: first to make the
signal suitable to be transmitted through the communication channel mainly by limiting its effective
bandwidth, increase the SNR of the received waveform, and to reduce intersymbol interference (ISI)
from multi-path channels and nonlinearities.
By filtering a symbols sharp phase and frequency transitions are reduced, resulting in a more
compact and spectrally efficient signal. Figure 1 provides a simple example of a DBPSK signal’s
frequency representation before and after filtering with a transmit filter. The filter used to generate
this figure was a square-root raised cosine (SRRC) filter, which is a common filtered use in
communication system. Details of the raised cosine filter are provided in Chapter 2 (§2.7.5), but the
2
more common SRRC has the impulse response:
h(t) =



1
√
Ts

1 − β + 4
β
π

, t = 0
β
√
2Ts
 1 +
2
π

sin 
π
4β

+

1 −
2
π

cos 
π
4β
  , t = ±
Ts
4β
1
√
Ts
sin 
π
t
Ts
(1 − β)

+ 4β
t
Ts
cos 
π
t
Ts
(1 + β)

π
t
Ts
"
1 −

4β
t
Ts
2
# , otherwise
(1)
where Ts
is the symbol period and β ∈

0, 1

is the roll-off factor.
The SRRC is used since it is a Nyquist type filter, which produces zero ISI when sampled
correctly [1]. We can demonstrate the effect of ISI by introducing a simple nonlinearity into the
channel and consult the resulting eye diagrams which were introduced in Section 2.4.1. Nonlinearities
cause amplitude and phase distortions, which can happen when we clip or operate at the limits of
our transmit amplifiers. For more details on the used model consult [2], but other models exists
such as [3]. In Figure 2 of observe the effects of ISI as the eye becomes compressed and correct
sampling becomes difficult to determine.
-0.5 0 0.5
Time
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Amplitude
Original Symbols
-0.5 0 0.5
Time
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
After ISI Channel
-0.5 0 0.5
Time
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
After RX Filtering Figure 2: Eye diagrams of QPSK signal effected by nonlinearity causing ISI, which is reduced by SRRC matched
filtering.
The final aspect of the matched filters we want to discusses or provide insight into is SNR
maximization. This argument logically comes out of the concept of correlation. Since the pulsedshaped/filtered signal is correlated with the pulse shaped filter and not the noise, matched filtering
will have the effect of SNR maximizing the signal. Creating peaks at central positions of receive
pulses. We demonstrate this effect in Figure 3, where we present data transmitted with and without
pulse shaping under AWGN. In the middle plot we observe a signal closely related to the originally
transmitted sequence, even under high noise. However, without pulse shaping even visually the
evaluation of the transmitted pulse becomes difficult. We even observe demodulation errors in this
third plot without anytime timing offset introduced.
3