Characterizing and Comparing Digital Modulations

2.9 Characterizing and Comparing Digital Modulations

Most of the previous material was restricted to PAM. Other modulations and some of their important characteristics are discussed in the sequel.

2.9.1 Linear versus non-linear modulation

A digital modulation scheme is said to be linear if the mapping from the sequence of symbols $m [n]$ to the transmitted signal $s [n]$ (or $s (t)$ ) is linear. Otherwise the modulation is nonlinear. ASK and FSK are examples of linearly and non-linearly modulated signals, respectively. Two modulations of interest are considered in the sequel: PAM and QAM.

For example, observe in Eq. (2.12) that, for an individual PAM symbol, the homogeneity property applies. And when considering the transmission of several PAM symbols using Block (2.16), both upsampling (by $L$ ) and convolution with $p [n]$ are linear operations (homogeneity and additivity apply). As an alternative point of view, the linearity of PAM modulation can be observed considering that $s_{1} (t)$ and $s_{2} (t)$ are the PAM signals corresponding to the symbol sequences $m_{1} [n]$ and $m_{2} [n]$ , respectively. Then, adopting $m [n] = α m_{1} [n] + β m_{2} [n]$ and from Eq. (2.31), the following confirms the linearity:

s (t) = \sum_{n = - \infty}^{\infty} (α m_{1} [n] + β m_{2} [n]) p (t - n T_{sym}) = α s_{1} (t) + β s_{2} (t),

(2.21)

where $α, β \in ℝ$ .

Note that if $m [n]$ should contain only symbols from a given constellation, the values of $α$ and $β$ should be chosen properly, to have $α m_{1} [n] + β m_{2} [n]$ still belonging to the constellation associated to $m [n]$ . But this issue is not important in Eq. (2.21), where our goal is to exemplify the behavior of a linear modulation.

In a similar way, it can be proved that QAM is a linear modulation. As will be seen in Eq. (3.2), a QAM can be written as

s_{QAM} (t) = s_{i} (t) cos (ω_{c} t) + s_{q} (t) sin (ω_{c} t),

where $s_{i} (t)$ and $s_{q} (t)$ are two PAM signals. Applying Eq. (2.21) to both PAMs, one can observe that $s_{QAM} (t)$ is obtained by superposition.

$M$ -ary PSK modulations are also linear simply because they are special cases of QAM. But note that if a phase modulation scheme transmits information via the phase of a single carrier (there is no quadrature scheme with two carriers), this modulation would be nonlinear! An exception to the non-linearity of PSK with a single carrier is a binary PSK with angles 0 and $π$ representing the bits. This case is equivalent to a PAM with amplitudes $\pm 1$ , which is linear.

FSK (binary or $M$ -ary) is a nonlinear modulation because the information signal is within the argument of a sinusoid.

2.9.2 Modulation with memory versus memoryless

When the modulation has memory, the mapping of a symbol to a waveform segment depends on one or more previously transmitted symbols. For example, the PAM modulation given by Eq. (2.13) is memoryless. The memory characteristic is typically introduced for improving the spectrum of the transmitted signal or for improved robustness, and is achieved by encoding the symbol sequences. The next section discusses differential encoding, one of the most used schemes for introducing memory to achieve improved robustness.

2.9.3 Differential encoding

Differential encoding takes as input a bit sequence $b [n]$ of 0’s and 1’s and generates a new bit sequence $d [n]$ . More specifically, the conventional differential encoder at the transmitter performs the operation

d [n] = b [n] \oplus d [n - 1],

(2.22)

where $\oplus$ denotes the exclusive-or in modulo-2 arithmetics and an initial value $d [- 1] \in {0, 1}$ is assumed to be known by both transmitter and receiver. At the receiver, the decoding operation is

\hat{b} [n] = \hat{d} [n] \oplus \hat{d} [n - 1],

(2.23)

where $\hat{d} [n]$ is the received bit sequence and $\hat{d} [- 1] = d [- 1]$ is known. If there are no transmission errors along the channel, then $\hat{d} [n] = d [n]$ and, consequently, $\hat{b} [n] = b [n]$ .

Note that if a transmission error occurs, it does not “propagate” by more than two bits. Eq. (2.23) does not have a feedback and as long as both $\hat{d} [n]$ and $\hat{d} [n - 1]$ are correct, the current bit $\hat{b} [n]$ coincides with $b [n]$ . But differential coding can increase the bit error rate. Consider for example a situation in which errors do not occur in consecutive bits: in this case, the adoption of differential coding would double the error rate.

The initial value $d [- 1]$ is not that important in practice given that $\hat{d} [- 1] \neq d [- 1]$ will only affect the decoding of the first bit $\hat{b} [0]$ .

Listing 2.11 gives an example of differential decoding, which generates the array differentialBits from the input array bits, assuming “1” as initial value. The arrays at the transmitter are:

bits =                      [0     1     0     1     1     0     0]
differentialBits =    ("1") [1     0     0     1     0     0     0]

In case the receiver (erroneously) adopt $\hat{d} [- 1] = 0$ in this example, the error $\hat{b} [0] \neq b [0]$ would occur as can be seen by comparing the original array bits with decodedBits below:

differentialBitsHat = ("0") [1     0     0     1     0     0     0]
decodedBits =               [1     1     0     1     1     0     0]

Listing 2.11: MatlabOctaveCodeSnippets/snip_digi_comm_differential_code.m

1%% Differential encoding at transmitter 
2bits=[0 1 0 1 1 0 0] %bits to be transmitted 
3N=length(bits); %number of bits 
4initialValue = 1; %assume the initial bit is 1 
5differentialBits=zeros(size(bits)); %pre-allocate space 
6differentialBits(1)=xor(initialValue,bits(1)); %initialize 
7for i=2:N %for all other bits 
8   differentialBits(i)=xor(differentialBits(i-1),bits(i)); 
9end 
10m=2*(differentialBits-0.5); %convert {0,1} to {-1,1} 
11%% Differential decoding at receiver 
12mHat=m; %assume no errors: no noise and perfect decoding at receiver 
13differentialBitsHat=(mHat+1)/2; %convert polar {-1,1} to {0,1} 
14%Use ~initialValue to impose an error in initialValue (first bit) 
15decodedBits=xor(differentialBitsHat,[~initialValue ... 
16    differentialBitsHat(1:N-1)]) 
17errors = bits - decodedBits %this is all zeros if no errors occur

Observe in Listing 2.11 that the vector of symbols m is created based on differentialBits, not bits.

To better understand the name differential code, it is useful to note that Eq. (2.23) is equivalent to performing a first order difference (“derivative”) $\hat{x} [n] = ŷ [n] - ŷ [n - 1]$ , but using modulo-2 arithmetic. And to check why decoding works, represent $\hat{x} [n] = ŷ [n] - ŷ [n - 1]$ by $H (z) = 1 - z^{- 1}$ and observe that the receiver performs:

ŷ [n] \to H(z) \to \hat{x} [n] .

It is possible to recover the original $x [n]$ because the transmitter used $y [n] = x [n] + y [n - 1]$ . This corresponds to using the system $1 ∕ H (z) = 1 ∕ (1 - z^{- 1})$ at the transmitter, i. e.

x [n] \to 1 H(z) \to y [n]

such that the receiver implements the inverse of the transmitter system. The commands x=rand(1,4),y=filter(1,[1 -1],x),xh=filter([1 -1],1,y) also help to observe that the receiver is able to recover the original signal. In this case, filter assumes that $y [- 1] = 0$ for both transmitter and receiver.

One advantage of the differential code is that the transmitted waveform $s (t)$ can be inverted ( $- s (t)$ ) without incurring in decoding errors with the exception of, maybe an error at the first symbol given the receiver would not know it should adopt $- \hat{d} [n]$ instead of $\hat{d} [n]$ . This is important when the waveform passes through several stages (distinct equipment) and the polarity may be unknown at some of them or when PSK is used and an error in carrier recovery inverts the bits. To observe why this “polarity insensitive” property is obtained, note that Eq. (2.23) depends on the “difference” between $\hat{d} [n]$ and $\hat{d} [n - 1]$ . For example, in case a sequence [0 1] is flipped to [1 0], the result is the same.

The “polarity insensitive” property of differential encoding can be checked by changing Listing 2.11 to use the command decodedBits=xor( $\sim$ differentialBits,[initialValue $\sim$ differentialBits(1:N-1)]) where the $\sim$ operation inverts the bits. In this example, the first bit will correspond to an error because initialValue was not flipped to mimic a realistic operation.

The nomenclature related to differential encoding can be rather confusing because some modulation schemes perform differentiation using the FIR typically found at the receiver (instead of the IIR in Eq. (2.22)). In other words, the transmitter uses Eq. (2.23) to obtain the so-called differentially-decoded sequence $d' [n] = b [n] \oplus b [n - 1]$ .

2.9.4 Brief comparison of modulations

Linear modulations such as QAM are typically more efficient with respect to required bandwidth than nonlinear modulations such as FSK. On the other hand, FSK is more robust to nonlinearities of the channel and of the transmitter power amplifier. Besides, as discussed in Section 3.8, FSK can be classified as an orthogonal modulation, which are typically power efficient. Therefore, linear modulations are often combined with orthogonal modulations in schemes such as the multicarrier system discussed in Chapter 6.

Some of the techniques used to improve the performance of linear modulations are:

Pulse shaping to improve the spectrum properties.
Differential modulation to improve robustness to phase uncertainty and enable noncoherent demodulation.
Gray coding to improve BER for a given SER.
Offset modulation to have phase transitions avoiding the center of the constellation diagram and remove some design constraints on the power amplifier.

Some examples of digital modulation techniques are listed below, with their variants:

Phase-shift keying (PSK)
- Binary PSK (BPSK), adopts $M = 2$ symbols
- Quadrature PSK (QPSK), adopts $M = 4$ symbols
- $M$ -PSK
- Differential PSK (DPSK)
- Differential QPSK (DQPSK)
- Offset QPSK (OQPSK)
- $π$ /4-QPSK
Frequency-shift keying (FSK)
- Multi-frequency shift keying ( $M$ -FSK)
- Continuous-phase frequency-shift keying (CPFSK)
- Gaussian frequency shift keying (GFSK)
Amplitude-shift keying (ASK)
- On-off keying (OOK)
Quadrature amplitude modulation (QAM)
Continuous phase modulation (CPM)
- Minimum-shift keying (MSK)
- Gaussian minimum-shift keying (GMSK)
Orthogonal frequency-division multiplexing (OFDM) modulation
- Discrete multitone (DMT)

Few comments are provided in the sequel. DPSK is inexpensive, but has an inefficient use of bandwidth. It is very robust and extensively used in satellite communications. QPSK is twice as spectrally efficient than BPSK but the receiver is more complex. OQPSK reduces the envelope variation. $π$ /4-QPSK reduces envelope variations to a lesser extent than OQPSK but can be noncoherently detected. QAM is suitable for channels with good linearity and that can afford relatively high power, such as telephone channels and point-to-point microwave links. In CPM, the carrier phase varies in a continuous manner and creates a constant envelope. It has a compact spectrum with a narrow mainlobe and fast sidelobe roll-off. MSK is a variation of CPM that adopts a minimum spacing that allows two frequencies to be orthogonal. The GMSK is a MSK with an improved spectral efficiency due to the adoption of a Gaussian lowpass filter. DMT is used in DSL while OFDM is used in wireless communications. Table 2.2 illustrates standardized systems that adopted some of the mentioned modulations.

Table 2.2: Modulations used in telecommunication standards.


Standard(s)	Modulation(s)

GSM/DCS1800/PCS1900/DECT (Europe)	GMSK
IS-54/IS-136 (North America)	$π ∕ 4$ -DQPSK
IS-95 (North America)	QPSK / DQPSK
PDC/PHS (Japan)	$π ∕ 4$ -DQPSK
Bluetooth	GFSK, $π ∕ 4$ -DQPSK, 8-DPSK
IEEE 802.11a/802.11g	PSK, QAM, OFDM
Zigbee	BPSK, OQPSK
ADSL2+, VDSL2, G.fast	DMT
3GPP LTE	OFDM

From this very brief discussion, one can note that there are many techniques for achieving improved modulations. The following strategy is adopted in this text: we first concentrated on PAM and we will now focus on QAM. Later we will study OFDM. This strategy aims at introducing many basic concepts used in current telecommunication standards and the reader can complement her/his knowledge with readings about specific modulations of interest.