1
EN1053 Introduction to Telecommunications
Notes on
Elementary Concepts in Telecommunications
By
Prof. Dileeka Dias
Department of Electronic & Telecommunication Engineering
4. MODULATION & MULTIPLEXING
Part I - Modulation
University of Moratuwa
3 Chapter 5
Modulation
5.1 What is Modulation and why is it needed ?
We have encountered the word modulation in connection with FM (Frequency Modulation) and AM (Amplitude Modulation) radio. Modulation and its inverse demodulation, make it possible to tune in to one radio or TV program when many are on the air. Modulation is an essential element in communications. It enables many independent messages to be sent over a communication channel. The modulation and demodulation process is shown in Figure 2.4.1.
Modulator
Carrier
Information-bearing signal Channel
Demodulator Recovered signal
Figure 5.1 Modulation and Demodulation
As we have seen, any signal can be represented as the sum of sinusoids of different frequencies. The bandwidth of the signal is the range of frequencies of these constituent sine waves. A signal can be shifted up or down in frequency by shifting the frequencies of its components.
Consider a signal with sinusoidal component frequencies extending from zero to some upper limit W. Such a signal is commonly called a baseband signal. It is the basic signal that we start off with, in transmitting information. The baseband signal represents the signal in its original frequency range.
A signal can be shifted from its baseband range of frequencies to a higher frequency range without altering its information content. A baseband signal from 0 to 4 kHz for example, might be shifted to the 60 to 64 kHz range. Shifting signals from their baseband range of frequencies to another, higher range of frequencies is accomplished through modulation. The shifting may be done linearly or nonlinearly.
Communication systems most often do not carry signals in baseband form. There are several reasons for this.
The need to carry multiple signals over a channel
Suppose we have a single channel between two distant cities, and we wish to transmit more than one signal over this. If we transmitted one signal at baseband frequencies, we would use up the frequency range and will not be able to transmit another baseband signal. In such a case, with baseband transmission, each signal must be sent over a different pair of wires.
Antenna size
The wavelength of a radio signal is the velocity of light divided by frequency. Baseband signals have low frequencies and therefore long wavelengths. For effective transmission and reception, antennas need to have lengths of a quarter of a wavelength or more. At the highest telephone
4 baseband frequency, 4 kHz, a wavelength would be 75,000 m. At this frequency, it will not be possible to have antennas.
Through modulation, we can move baseband signals to non-overlapping frequency bands for transmission over a common channel, and also to frequency ranges where it is possible to have antennas of reasonable size.
The two basic categories into which modulation techniques fall are continuous wave modulation schemes, and pulse modulation schemes. In continuous wave modulation schemes, the information-bearing signal (the modulating signal) varies a sinusoidal signal of a much higher frequency, called a carrier signal. The variation may be in amplitude, frequency or phase. In traditional continuous wave modulation schemes, both the modulating signal as well as the carrier are analog signals. These will be described in Section 5.2. In more modern digital modulation techniques, a digital signal modulates an analog carrier. These are described in Section 5.3.
In pulse modulation schemes, the carrier is a periodic pulse train whose pulse widths, positions or amplitudes are varied by the information-bearing signal which is analog. These are described in Section 5.4.
Pulse Code Modulation (PCM) is described in Annex A. Even though it is not correct to say that this is a modulation scheme in the same sense as the other techniques, this important technique is included in this section for the sake of completeness.
5.2 Fundamentals of Continuous Wave Modulation Schemes
Continuous wave modulation schemes carry the information in the message signal by varying the amplitude, phase or frequency of a carrier signal. In the process, the message signal energy is distributed around the carrier frequency.
A sinusoidal carrier can be represented as
A f t
t
c( ) ccos2 c (5.1)
where A is the amplitude fc is the frequency and is the phase. When serving as a carrier, it may
be modulated by varying the amplitude, frequency or phase in direct proportion to the message signal. These types of modulation are called amplitude modulation (AM), frequency modulation (FM) and phase modulation (PM) respectively. Collectively, FM and PM are called angle modulation. Figure 5.2 depicts AM and FM.
AM is a linear modulation scheme, which means that the modulating signal spectrum is linearly translated during the process of modulation. PM and FM are examples of nonlinear modulation techniques. In these techniques, there is no linear relationship between the signal spectra before and after modulation.
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(a) AM (b) FM
Figure 5.2 Amplitude Modulation (AM) and Frequency Modulation (FM)
In this section, we will study the different fundamental modulation techniques using a modulating signal x(t). For mathematical convenience, we will normalize all modulating signals to have a magnitude not exceeding unity:
x(t) 1 (5.2a)
This normalization puts an upper limit on the average modulating signal power ,
Sx x2(t) 1 (5.2b)
when we assume x(t) to be a deterministic power signal. Both energy signals and power signal models can be used for x(t).
where
T T dt t x T t x 2 2 2 ) ( 1 ) ( (5.2c)6
5.2.1 Amplitude Modulation
There are several variations of amplitude modulation, each having its own advantages and disadvantages. The common feature is that the carrier amplitude is varied in proportion to the modulating signal. The information is carried in the envelope, or the time-varying amplitude of the modulated signal.
Conventional AM (Double Sideband Large Carrier AM)
If Ac denotes the unmodulated carrier amplitude, modulation by x(t) produces the modulated
envelope
1 ( )
)
(t A x t
A c (5.3)
where is a positive constant called the modulation index. The complete AM signal is given by:
t t x A t A t t x A t x c c c c c c c cos ) ( cos cos ) ( 1 ) ( (5.4)Figure 5.3 shows the modulator structure implementing the above AM signal. Figure 5.4 shows a typical message and the resulting AM signal for two values of . The envelope reproduces the shape of x(t) if fc W and 1 where W is the bandwidth of x(t).
When these conditions are satisfied, the signal may be extracted from the carrier using a simple envelope detector.
Figure 5.3. Amplitude Modulator
X X x(t) Ac cos ct Modulating Signal Carrier + X Modulated Signal Ac [1+x(t)]cos ct
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Figure 5.4 Amplitude Modulation (AM) waveforms
The constraint 1 ensures that the carrier envelope does not go negative. With 100% modulation ( 1) the envelope varies between 0 and Amax. Overmodulation (1) causes phase
reversals and envelope distortion as illustrated in Figure 5.4 The Fourier Transform of (5.4) gives,
( ) ( )
2 ) ( ) ( 2 1 ) ( c c c c c c c f A f f f f A X f f X f f X (5.5)The AM spectrum given by this equation is sketched in Figure 5.5. The AM spectrum consists of impulses at the carrier frequency and symmetrical sidebands centered at fc. The presence of an
upper and a lower sideband accounts for the name double sideband amplitude modulation (DSB). The transmission bandwidth using this technique is
8 W
B
AM
T 2 (5.6)
This implies that AM requires twice as much bandwidth to transmit x(t) at than at baseband without any modulation. Transmission bandwidth is an important consideration in comparison of
modulation techniques.
Figure 5.5 The AM Spectrum
Another important consideration is the average transmitted power given by,
2 2 cos 1 ) ( ) ( 2 1 cos ) ( 1 ) ( 2 2 2 2 2 2 2 t t x t x A t t x A t x S c c c c c TAM using (5.4).
x t x t
t A t x t x A ST c c c AM 2 1 2 ( ) ( ) cos2 1 ) ( ) ( 2 1 2 1 2 2 2 2 2 2 (5.7)9 whose second term averages to 0 under the condition fc >>W. Thus, if x(t) 0and
x S t x2( ) , then ) 1 ( 2 1 2 2 x c T A S S AM (5.8) We can express the above equation as:
sb c T P P S AM 2 (5.9a)
where the unmodulated carrier power Pc and the sideband power Psb are given respectively by,
2 2 1 c c A P and Psb Ac2 2Sx 2SxPc 2 1 4 1 (5.9b)
The modulation constraint x(t) 1 requires that 2Sx 1, so Psb 12Pc and T sb T c S P S P 2 1 2 Psb 14ST (5.10)
Consequently, at least 50% of the total transmitted power resides in a carrier term that is independent of x(t), and thus conveys no information.
However, this modulation scheme is used in AM broadcasting due to the simplicity of the demodulator. Carrier frequencies in the range 540 to 1600 kHz are assigned for AM broadcasting, with a carrier spacing of 10 kHz. The bandwidth of 10 kHz for each channel allows the transmission of signals having a bandwidth of approximately 5 kHz.
5.2.2 Double Sideband Suppressed Carrier (DSB-SC) AM
The wasted carrier power in conventional AM can be eliminated by setting = 1 and suppressing the unmodulated carrier frequency component. The resulting modulated wave becomes: t t x A t xc( ) c ( )cosc (5.11)
which is called double-sideband suppressed-carrier modulation. The frequency domain representation of the above signal is simply,
c
c
c f X f f X f f X 2 1 2 1 ) ( (5.12)The DSB-SC spectrum looks similar to that of conventional AM shown in Figure 5.5 without the unmodulated carrier impulses. The transmission bandwidth remains unchanged at
W
BTDBSC 2 (5.13)
This modulation process and the spectrum are shown in Figure 5.6.
Although AM and DSB-SC are quite similar in the frequency domain, they are different in the time domain as illustrated in Figure 5.7. The DSB-SC envelope and phase are:
10 ) ( ) (t A x t A c and 0 180 0 ) (t 0 ) ( 0 ) ( t x t x (5.14)
The envelope takes the shape of |x(t)|, rather than x(t), and the modulated wave undergoes a phase reversal whenever x(t) crosses zero. Recovery of x(t) cannot be accomplished by an envelope detector, and calls for a more sophisticated demodulation process.
However, since there is no unmodulated carrier component in the modulated signal, all the transmitted power is in the information-bearing sidebands. Thus:
x c sb T P A S S SC DSB 2 2 1 2 (5.15)
DSB-SC makes better use of the available transmit power from a given transmitter.
The foregoing considerations suggest a trade-off between power efficiency and demodulation methods. DSB-SC conserves power, but requires complicated demodulation circuitry, whereas AM requires increased power, yet permits simple envelope detection.
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Figure 5.7 DSB-SC waveforms 5.2.3 Single Sideband AM (SSB)
Both AM and DSB are techniques which double the signal bandwidth on modulation. Since the information of the message is preserved in duplicate in both the upper and lower sidebands, only one of these sidebands is necessary to represent exactly the message at the receiver. Modulation methods that transmit only one of the sidebands is called Single Sideband (SSB).
The simplest method to generate SSB signals is to generate a DSB signal and filter out one of the sidebands as shown in Figure 5.8.
The transmit bandwidth of SSB is given by,
(5.16)
The transmit power of SSB is given by,
x c sb T P A S S SSB 2 4 1 (5.17)
Thus, SSB conserves transmission bandwidth as well as making efficient use of transmit power. However, the problem with SSB is the difficulty of filtering out one sideband. This is particularly critical for signals having significant low frequency content. The sideband filter tends to remove and/or distort the low frequency components of the sideband that is to remain.
W B
SSB
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Figure 5.8 The SSB modulator and spectrum 5.2.4 Vestigial Sideband AM (VSB)
Consider the transmission of a very large bandwidth signal having significant low frequency content (e.g., video, facsimile, data). Bandwidth conservation argues for the use of SSB. But practical SSB systems have poor low frequency response as discussed above. DSB works well for low message frequencies, but the transmission bandwidth is twice that of SSB. VSB is a compromise between the two.
VSB is derived by filtering DSB or AM in such a way that one sideband is passed almost completely, while a part (a vestige) of the other sideband is included. In block diagram form, the generation of VSB is similar to that of SSB shown in Figure 5.8, except for the sideband filter characteristics.
The key to VSB is the sideband filter. Figure 5.9 compares the sideband filter characteristics in SSB and VSB. While the exact shape is not crucial in VSB, it must have a relative response of ½ at the carrier frequency and odd symmetry about this point. If the transition bandwidth of the sideband filter in VSB is , the transmission bandwidth is given by,
(5.18)
For small , VSB approximates SSSB, and for large , it approximates DSB-SC. The transmit power is given by,
x c T x c S S A S A VSB 2 2 2 1 4 1 (5.19) The VSB signal spectrum is shown in Figure 5.10.
W W
B
VSB
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Figure 5.9 Sideband filter characteristics for SSB and VSB
Figure 5.10 The VSB spectrum
If an AM (conventional) signal is applied to a VSB filter, the resulting modulation scheme is called VSB plus carrier. The unsuppressed carrier allows envelope detection while retaining the bandwidth conservation characteristics. This type of modulation is used in TV transmission.
14 Figure 5.11 shows a comparison of the different AM techniques using a single tone of frequency fm as the modulating signal. This is called tone modulation or sinusoidal modulation, and is often
used as a tool for the study of modulation techniques.
Figure 5.11 Spectra of the different AM techniques for tone modulation 5.3 Angle Modulation
As in the case of AM, this too is a type modulation scheme where both the modulating signal and the carrier are analog signals.
Phase modulation (PM) and frequency modulation (FM) are special cases of angle modulation. The angle modulated signal may be written as:
(5.20)
( )
cos ) (t A t t xc c c Frequency fc -fc 0 (a) AM Amplitude Frequency -fc 0 (b)DSB-SC Amplitude Frequency fc -fc 0 (c) SSB Amplitude Frequency fc -fc 0 Amplitude (d) VSB fc+fm fc-fm fc fc+fm fc-fm fc+fm fc+fm fc-fm15 For PM, the phase is directly proportional to the modulating signal. That is,
(5.21)
where the proportionality constant p is called the phase sensitivity or the phase deviation
constant of the modulator.
For FM, the modulated signal phase is proportional to the integral of x(t) :
(5.22) where f is the frequency deviation constant of the modulator in radians/volt-sec. Figure 5.12
shows examples of PM and FM signals.
Figure 5.12 Examples of PM and FM
For the case of FM, using equation (5.20), we obtain the instantaneous frequency as follows:
(5.23) ) ( ) (t px t
t f x v dv t) ( ) (
) ( 2 1 ) ( 2 1 ) ( t x f t t dt d t f f c c i 16 which is why this modulation scheme is called frequency modulation. The instantaneous frequency varies about the carrier frequency in a manner that is directly proportional to the modulating signal.
The peak frequency deviation of an FM signal is a quantity of special interest in communication systems, and is given by,
() 2 1 max t dt d F or max (t) dt d (5.24)The frequency modulation index is given by,
(5.25)
where W is the modulating signal bandwidth in Hz.
5.3.1 FM signal analysis with sinusoidal modulation
Let x(t)acosmt (5.26)
For FM from (5.23), the instantaneous frequency is given by,
t a t x m f c f c i cos ) ( (5.27) The peak frequency deviation
f a
(5.28)
We can rewrite (5.24) as, t m c
i
cos (5.29)
The angle of the FM signal is,
t t t t t m f c m m c sin sin ) ( (5.30) The resulting FM signal is,
t
A t
t
t A t t A t x m f c c m f c c m f c c c sin sin sin sin cos cos ) sin cos( ) ( (5.31)5.3.2 Narrowband and Wideband FM
For small values of f we can write
t t t m f m f m f sin ) sin sin( 1 ) sin cos( (5.32) W W F f 2
17 The condition where f is small enough for these approximations is the condition for
narrowband FM (NBFM). Usually, a value of f <0.2 is taken to be sufficient to satisfy this
condition. If this condition is not satisfied, then the modulation is called wideband FM. Using (5.31) and (5.32) we can write the expression for the narrowband FM signal as:
t t A t A t
xc( ) ccosc f csinm sinc (5.33)
At this point it is instructive to compare (2.4.33) with an AM signal with tone modulation given by. t t A t A t
xc( ) ccosc ccosm cosc (5.34)
Though the two equations are similar, AM and NBFM are very different modulation schemes. As equation (5.34) shows, the modulation is added in phase with the carrier in AM. In contrast, as (2.4.33) shows, in NBFM, the modulation is added in quadrature with the carrier.
Figure 5.13 shows the generation of NBFM and NBPM using equations (5.21), (5.22) and (5.33).
X x(t) A c cos c t Modulating Signal Carrier + NBPM Signal 900 X x(t) A c cos c t Modulating Signal Carrier + NBFM Signal 900 Integrator (a) NBPM (b) NBFM
Figure 5.13 Generation of Narrowband PM and Narrowband FM 5.3.3 Spectra of FM signals
From (5.30) we can express the FM signal in complex notation as:
j t j t
c t j c c m c e e A e A t x sin ) ( Re Re ) ( (5.35)18 The second exponential in(5.35) is a periodic function of time with a fundamental frequency of
m. It can be expanded in a Fourier Series
n t jn n t j m m e F e sin (5.36) where
2 / 2 / sin 1 T T t jn t j n e e dt T F m m (5.37)Making a change of variable mt(2/T)t, we get
e d Fn j( sin n ) 2 1 (5.38) The above function is known as the Bessel function of the first kind, of order n and argument , and is denoted by Jn(). Note that n is an integer and is a positive continuous variable. Someof these functions are plotted in Figure 5.14.
Figure 5.14 Bessel function of the first kind Jn()
The properties of Jn() can be summarized as follows:
1. Jn() are real valued
2. Jn()=J-n(), for n even 3. Jn()=-J-n(), for n odd 4.
n n J 2() 119
n t jn n t j m m e J e sin () (5.39)Then, the FM signal in (5.32) can be written as:
n
t J A e J e A t x c n n c n t jn n t j c c m c 0 cos ) ( ) ( Re ) (
(5.40)From these results, it can be seen that an FM waveform with sinusoidal modulation, in contrast to AM, has an infinite number of sidebands. However, the magnitudes of the higher order spectral components (sidebands) become negligible, and for all practical purposes, the power is contained within a finite bandwidth.
Spectral plots for several different values of are shown in Figure 5.15. Note that can be varied by varying or m as demonstrated in Figure 5.15.
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Figure 5.15 Spectra form FM waveforms with sinusoidal modulation (a) for constant m
(b) for constant 5.3.4 Bandwidth of FM signals
21 How many sidebands are important to the FM transmission of a signal? This will depend on the intended application and the fidelity requirements. A rule commonly adopted is that a sideband is significant if its magnitude is equal to or exceeds 10% of the carrier frequency component, i.e., if
1 . 0 ) ( n J (5.41)
The actual number of significant sidebands for different values of can be found from a plot of Bessel functions such as Figure 2.4.14. It can be seen that Jn() diminishes rapidly for n >,
particularly as becomes large.
The bandwidth for very large can then be approximated by taking the last significant sideband at n = so that the transmission bandwidth BT is given by,
2 2 2 m 2 m m m T n B for large (5.42)
For very small , the only Bessel functions of significant magnitude are J0() and J1().
Therefore, the bandwidth for the narrowband case is, m
T
B 2 for small (5.43)
We now have bounds on the limiting cases. A more general rule to take care of intermediate cases is, given by,
) 1 ( 2 m T B (5.44)
This is known as Carson’s rule.
5.3.5 Commercial FM Transmissions
As noted earlier, narrowband FM is similar to AM. Advantages of using narrowband FM over AM include the possibility of rejection of large noise pulses. Narrowband FM is used primarily in telemetry and mobile and/or wireless communications.
Provided that we are concerned with only the bandwidth, we can apply our knowledge of pure sinusoidal FM to more general waveforms also in the wideband case. For wideband FM we noted that the bandwidth depends mainly on the peak frequency deviation ( or F). This in turn, depends, for a given modulator constant, on the amplitude of the modulating signal. Hence some limit must be placed on the modulating signal to avoid excessive bandwidth, even though the bandwidth of the modulating signal may be well-defined.
For commercial FM broadcasting, carrier frequencies spaced at 200 kHz intervals in the range 88-108 MHz are assigned. The peak frequency deviation is fixed at 75 kHz. The 200 kHz between station assignments, in comparison to 10 kHz for AM broadcasting allows the transmission of high-fidelity program material using wideband FM. Suppose we take the modulating frequency fm to be 15 kHz (typically the maximum audio frequency in FM
broadcasting). Use of Carson’s rule yields a bandwidth of 180 kHz.
5.4 Fundamentals of Digital Modulation Schemes
The modulation schemes described in this section all have digital modulating signals, which modulate either the amplitude, frequency or phase of an analog carrier. As the modulating signal has two or more discrete levels, the modulated signal will also have a set of discrete amplitudes,
22 frequencies or phases. The term keying in these modulation schemes implies that the amplitude, phase or freqeuency of the carrier switches between the allowable values. These modulation schemes are generally known as digital modulation techniques.
Amplitude Shift Keying (ASK)
The modulation varies the amplitude of the carrier. Consider the amplitude modulation of a carrier with a binary digital signal having voltage levels 0 and V Volts. This scheme shown in Figure 5.16 is called Amplitude Shift Keying (ASK) or On-Off Keying (OOK). The information is encoded in the amplitude of the signal. Optical communication systems carry information on light beams in this manner. This is analogous to AM.
Figure 5.16 Amplitude Shift Keying (ASK)
The modulating signal may be a multilevel digital signal, in which case the modulated signal will also have the same number of discrete amplitude levels. For example, a 4-level ASK signal will encode 2 bits of information in one amplitude level.
Frequency Shift Keying (FSK)
In this family of digital modulation schemes, the digital signal modulates the frequency of the carrier, analogous to FM. In the case of a binary digital signal, the carrier will have two discrete frequencies as shown in Figure 5.17. Multilevel digital signals will cause the carrier frequency to switch between several discrete frequencies.
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Phase Shift Keying (PSK)
This is a modulation scheme where a digital signal modulates the phase of an analog carrier, and can be considered to be analogous to PM.
The simplest type of PSK technique is called Binary Phase Shift Keying (BPSK), where the modulating signal is a binary digital signal. This scheme is shown in Figure 2.4.18. The carrier has two phases, one for a '0' level and another for a '1' level. Thus the information is encoded in the phase of the carrier.
Extending this technique, Quarternary Phase Shift Keying (QPSK) has four signal phases, each carrying two bits of binary information. Examples of other higher-level M-PSK techniques are 8-PSK, 16-PSK etc.
Figure 5.18 Phase Shift Keying (PSK) Quadrature Amplitude Modulation (QAM)
Quadrature Amplitude Modulation is a combination of ASK and PSK. Here, the digital signals encode both the carrier amplitude and the phase. The carrier can take on one of several discrete amplitudes and one of several discrete phases in accordance with the combination of information bits modulating the signal. 4-level QAM and QPSK are the same. Other higher-level QAM techniques are 16-QAM, 64-QAM etc.
5.4.1 Signal Constellations for digital modulation techniques
The signal-space diagrams or signal constellations illustrate digital modulation schemes in terms of how they encode information in the carrier phase/amplitude. The signal constellations for some ASK schemes are shown in Figure 5.19, for PSK schemes in Figure 5.20 and QAM schemes in Figure 5.21.
5.5 Fundamentals of Pulse Modulation Schemes
In pulse modulation techniques, the carrier is not sinusoidal, but consists of a train of periodically repetitive rectangular pulses. The modulating signal is analog. The amplitude, width or position of the pulses can be altered by the modulating signal as illustrated in Figure 2.4.22.
24 In Pulse Amplitude Modulation (PAM), the amplitude of the pulses of the carrier is varied in accordance with the amplitude of the modulating signal. In Pulse Width Modulation (PWM), the duration of each pulse is varied in accordance with the modulating signal. In Pulse Position Modulation (PPM), the position in time of each pulse is varied according to the modulating signal. 00 900 1800 2700 A Binary ASK For ‘0’: s1(t) = 0 For ‘1’: s2(t) = A cos ct 4 Level -ASK For ‘00’: s1(t) = 0
For ‘01’: s2(t) = A/3 cos ct For ‘11’: s3(t) = 2A/3 cos ct For ‘10’: s4(t) = A cos ct 00 900 1800 2700 A
25 A 00 900 1800 2700 A
Binary Phase Shift Keying (BPSK)
For ‘0’: s1(t) = A cos ct For ‘1’: s2(t) = A cos (ct + ) -A 00 900 1800 2700 A
Quarternary Phase Shift Keying (QPSK) For ‘00’: s1(t) = A cos ct For ‘01’: s2(t) = A cos (ct + /2) For ‘11’: s3(t) = A cos (ct + ) For ‘10’: s4(t) = A cos (ct + 3/2) -A -A A 00 900 1800 2700 A
8-Level Phase Shift Keying (8-PSK)
-A -A 000 001 011 111 101 100 110 010 00 01 11 10 0 1
Figure 5.20 Signal Constellations for PSK
00 900
1800
2700
4-Level QAM (4-QAM) 00 01 11 10 A 00 900 1800 2700 A
16-Level QAM (16-QAM) -A
-A
00 01
11 10
26
