‘Signal Processing and Telecommunications
4.2. AMPLITUDE MODULATION
4.2.1. CONVENTIONAL DOUBLE-SIDEBAND AM WITH CARRIER
These types of modulation are achieved, as the name indicates, by changing or modulating
the amplitude of the cartier signal linearly with the modulating or information signal.
Assume the unmodulated carrier is of the form
SAt)= A,cos(,t) {or A, sin(o,1)} Q
and its amplitude is modulated according to the function
g,(t)=[4+ B.F,()] 8)
where: Sy(t)=> The modulating or information signal; {f,(=1], Le, normalised to
unity amplitude
A= A constant de term
B= amplitude of the modulating signal
Also assume that f,(t) is a cosinusoidal function, defined as:
Salt) = cos(@,t) 4)
In general, f,(t) may be any unity amplitude finite bandwidth Fourier transformable complex
signal (periodic or a-periodic). When f,(t) is complex, the corresponding modulation will be 2-
dimensional (2D), requiring the presence of a quadrature carrier pair, {cos(w,£), -sin(w-)}.
Then the amplitude-modulated (AM) signal @,,(1) is defined as:
b(t) = A. [A+ B.f,(0)].cos(,t)
= A. Alt B/, cos(eo,t)].c0s(c,1) 6
= A.[1+ mcos(,1)].cos(«,t)
2
with: O
1, over-modulation occurs, prohibiting envelope
detection (j.e., cross-over distortion occurs when m>1 or B>A). Hence, under these
circumstances the information f,(t) cannot be recovered by means of envelope detection, but
only through coherent detection. Note that the latter method of demodulation is employed in
the national analog FDMA telephone system, illustrated in Figs. 1 to 3. In this case
synchronous replicas of the carriers (f,, f . . ., fy) have to be recovered before demodulation
can be successfully accomplished. For conventional AM, m=0.8 (i-e., P=80%) is normally not
exceeded in practice (primarily to prevent over-modulation as a result of AGC-fluctuations),
while m=0.3 may be considered to be a common average AM modulation index.
Fig. 5 a and b depict measured AM-DSB with Carrier waveforms and corresponding spectra
in the case of switched AM with modulation index m=1. Note that the sideband amplitudes are
‘50% of the carrier amplitude in this case. Fig. 5 ¢ shows the AM-DSB spectrum (suppressed
cartier) case. Figs. 5 d and e illustrate the modulation spectra for a squarewave modulating
signal (with and without carrier added, respectively), and Fig. 5 f the spectrumwith a triangular
modulating signal with carrier.
10 LPL-07| ‘Signal Processing and Telecommunications
4.2.3 Other Forms of AM
4.2.3.1 AM with Suppressed carrier (AM-SC)
In this case term | of equation (6) is omitted by setting A=0, yielding:
Pas-ont)= B.fy(0). 4,001)
B.cos(0,1).cos{@,t) (10)
AB
[eos(o,- «+ cos(, + 1
2
i.e., only terms Il and Ill of equation (6) are retained (i.e., the LSB and USB of the AM signal,
respectively. Note that since A=0, no carrier is produced). Hence, to suppress the carrier, the
information signal has to be generated with zero de-offset.
3.2.3.2. Synchronous AM Demodulation
Demodulation of an AM-DSB-SC signal is accomplished in a manner similar to modulation,
yielding, after low pass filtering, the original information or modulating signal f,(t) to within a
constant amplitude scaling factor.
y(t) = {A..Bf,(0).cos@,t}. A, cos@,t
AAB. f,(t).cos’ @,t
A.B
=— D1 + cos2@,t
SalO)[1+ 205201] an
A.
= Fn(t)+ OB cos2@,t
u
Recovered information Removed by LPF
The recovered information, f,,(t), is obtained as
~ = = _ AB
Iu()= Bf(Ds B= —y aay
where Bis the constant scale factor. The modulation and demodulation processes are
illustrated in Fig. 6 a and b.
11 LPL-07| Signal Processing and Telecommunications
F (t)=Acosuet
f(t). ——
PN
“M
BAS! SBANDSEIN D.S.B.-0.0. SEIN
(a) D.8.8.-0.0. MODULATE. (SINCHROON)
ONTVANGER
VERSTERKER
3S [2 ie a
£,(t) Ajcosu.t
INFORMASIE, f(t) HERWIN
(b) D.S.B.-0.D. DEMODULASIE. — (SINCHROON)
Fig. 6 Block schematic illustration of AM modulation and demodulation
12 LPL-07| ‘Signal Processing and Telecommunications
4.2.3.3 Power content of an AM signal
The power carried by an AM signal can be obtained from the individual terms (I, Il & Il) of
equation (6). Since the terms are independent (they are disjoint in frequency, and hence
‘orthogonal’), the total power is simply P.=P;+P,+Py, The power contained in the
unmodulated carrier (term |) can be derived from first principles:
Ee Sf (t).dt =
“a
1+ cos2@, (1) |at (13)
IS
v
The integral could also have been performed over a longer period containing several carrier
periods T,, such as over the period T,,=kT, of the information signal, with k any integer.
P= P, -zi[4 em oo.) | a
i= Pa= 7) 2. — Wy
12 pg? Te
=4 a | [t+ cosa(m, - 7, ear a
m0
(2m?
= im
Similarly, for terms Il (LSB) and III (USB) follow that,
Similarly, P,=Pyse=P=P., where P, denotes the sideband power. The total power is thus:
Epo Ft Py t Fin
= P+2P
as)
13 LPL-07| ‘Signal Processing and Telecommunications
When m=1 in (15) (.¢., maximum possible amplitude modulation to still allow envelope
detection without distortion), the total useful information power (%) is
16)
= 33.33%
which illustrates the price to be paid to enable more affordable envelope demodulation and
detection, in stead of using coherent demodulation. Note that in this case two thirds of the
total power is carried by the unmodulated carrier.
14 LPL-07| ‘Signal Processing and Telecommunications
4.3. SIMPLE PRACTICAL MODULATION METHODS
4.3.1. SYNCHRONOUS MODULATION
The synchronous modulator is in essence a mixer (multiplier) which mixes (multiplies) a band-
limited information (‘modulating’) signal f,(t) with a carrier, local oscillator, mixer or heterodyne
signal , f,()=A,cosw,t (depending on the specific application), mixing or frequency translating
(shifting) it to an Intermediate Frequency (IF) or directly to a Radio Frequency (RF), f, [Hz]
This is illustrated in Fig. 7 below.
fg (t) = A,coswyt
MODULASIESEIN VERMENIGVULDIGER GEMODULEERDE SEIN
£,(t) £AO= fy (t)-£, (t)
£,(t) £>> £,
2 e ™naks
Fig. 7 Block schematic representation of synchronous AM-DSB-OD modulator
The modulation (multiplication) process may in practice exhibit distortion effects (e.g.,
harmonic distortion or the generation of unwanted spectral components at multiples of the
carrier frequency) in the large signal case, due to non-linear distortion during amplification or
multiplication. Although unwanted out-of-band spectral components may be removed through
appropriate filtering, inband distortion remains a problem that may lead to irreversible signal
deformation and loss of information.
Another limitation of this modulation method is the requirement of very stable carrier
frequencies f, [Hz] and accurate carrier phase recovery at the receiver, lack of which may lead
tosevere signal attenuation during demodulation, or fluctuation of recovered information power
- a phenomenon called fading
15 LPL-07| ‘Signal Processing and Telecommunications
The simplest form of switched modulation is achieved by forming the product p(t}=9(t).£.t),
where g(t) can take the forms (recall equation (3)):
gi(t)= [1+ mf,()] : AM-DSB withCarrier an
or
g,(t)= B.f,(t) : AM-DSB- SC (18)
Witht,(t}=cos(w,t), the corresponding AM signals are then respectively defined as
Pam-wc(t) = 4.8; (t) cosa,t as)
or
Giu-sc(t) = 4,8, (t) cos@,t (20)
The following AM modulation techniques may be employed to realise the two major forms of
AM defined above:
Switched Modulation
+ Non-Linear Element Modulation
Analogue Modulation
+ Direct Tuned Filter Modulation
Switched modulation will now be investigated in more detail
16 LPL-07| ‘Signal Processing and Telecommunications
43.2. Switched (Chopper) or Modulation
43.2.1 Introduction
We analyse switched modulation by considering a generic expression for AM of the form:
Oy (t)=[A+ BF, 0] [C+ D.p(D] ay
where f,() denotes an arbitrary unity amplitude (j.e., amplitude normalised) modulating or
information signal with maximum frequency f,,=1/T,, [Hz] and f,()=p(t) any suitable periodic
cartier signal (usually a square wave) with period T,=1/,< — Arbitrary de offset added to information signal;
B > — Amplitude of the modulating signal;
©» Arbitrary de added to carrier signal;
D> Cartier amplitude.
Switched modulation is affected by specifically choosing f,(t) a periodic pulse-like signal that,
switched, samples or ‘chops’ f,(t) at a rate of f, [Hz]. The choice of constants A, B, C and D
then determines the type of switched modulator used. For example, choosing
a balanced chopper modulator, and with C0 a unbalanced modulator is obtained. This is
illustrated in Fig. 8 below with f,(t) a low frequency cosinusoid and f,(t) a periodic high
frequency square wave.
results in
Note that equation (21) can be written in normalised form as follows:
yy (0) = AD[L+m.f,O)[S + PO]
(22)
=A, [l+m.f,(O]S+ PO]
where: 4’ =AD » Effective switched carrier amplitude
O t
)- [c+ D. ple) 3 C=D=4V
AtB OT nr
mee A hth tl Inf t
t
Fig. 8 Typical AM-balanced/unbalanced chopper modulator waveforms
18 LPL-07| Signal Processing and Telecommunications
ig. 9 a and b depict measured AM-DSB with Carrier waveforms and corresponding spectra
in the case of switched AM with modulation index m=1. Note that the sideband amplitudes are
50% of the carrier amplitude in this case. Fig. 9 ¢ shows the AM-DSB spectrum (suppressed
cartier) case. Figs. 9 d and e illustrate the modulation spectra for a squarewave modulating
signal (with and without carrier added, respectively), and Fig. 9 fthe spectrum with a triangular
modulating signal with carrier.
(b) SPEKTRUN VAN
DRAER (m= 1)
(a) SINUSVORMIGE MODULASIE.
INFORMASIESEINFREKW: 2 kHz
DRAERFREKWENSIE: 455 kiz
f(t) = sinupt
MODULASIE-INDEKS: m= 1 u *
(0) (0)
(m=
F(t) = sinu,t
Fig. 9 Typical unbalanced and balanced chopper modulator waveforms and spectra.
19 LPL-07| ‘Signal Processing and Telecommunications
met_DRAER
SIMMETRIESE (1:1)
VIERKANTSGOLFMODULASIE.
bailon ree
EN
(oo)
VIERKANTSGOLFMODULAS IE.
Lee Ere
(ft) Am
S.B. met DRAER
DRIEHOEKSGOLFMODULAS IE.
Fig. 9 (Continued) Typical unbalanced and balanced chopper modulator waveforms and spectra,
20 LPL-07| ‘Signal Processing and Telecommunications
We now investigate the spectral characteristics of the generic switched AM modulator defined
in equation (22), and investigate the influence of the constants A to D in the frequency
domain.
The Fourier-Transform of (21) is:
o(f)= 4[5 N+ mE NLSIN+ EY]
= AC.5(f)+ ACME (f)+ AFL) + AME SEL) 3)
u u 4 u
do-term original inform carrier term AM-DSB-SC term
The last term in (23) constitutes the wanted AM-DSB-SC term, the second last term the
presence of a carrier component to produce AM-DSB with carrier, whereas the other terms
denote some form of distortion or unwanted spectral component that need to be suppressed
or removed through appropriate filtering. Note that when the first two unwanted terms are
present, the last two terms need to be retained by using an suitable band-pass filter (BPF);
otherwise, a LPF may suffice. The first dc term may be blocked by using a coupling capacitor
(or high-pass filter - HPF). The second term is a scaled repetition (or replica) of the original
information signal, constituting an unwanted term of the switched modulation process.
Note that the carrier term may be removed (suppressed) by choosing constant A equal to zero.
Setting C=0 has the advantage of simultaneously eliminating the first two unwanted spectral
components. The modulation index m may be adjusted through A and B, respectively, with
the condition that 0 sof ™)a(o - n@,) en
or, alternatively (since 6(@) =. d(f)):
P(f)=D e Sal of alr - nf.) (28)
25 LPL-07| ‘Signal Processing and Telecommunications
The spectrum |P(f]| is shown in Fig. 14 (compare this with the two-sided complex Fourier line
spectrum in Fig. 11b and note the correspondence)
G.S.=> ONBALANSWERKING.
Fig. 14. Two-sided Fourier spectrum of the switching waveform p(t).
Now form the product v,(t)=@,u(t)=g(0).p(f) with g(t)=f,(f). Then:
Pau (t) = fu(t)- P(t) > S(@) = 3-[F,,(@)* P(@)|
=h jn@lPo -A)jaa
=4p th in(A)-[Sa(%#)5(- neo, - A)} da a)
=D, Sa(*2)F,(@-no,)
= DY Sa*#)F, (Ff -nf.)=V,(f)
26 LPL-07| ‘Signal Processing and Telecommunications
The spectrum of the switched AM signal is illustrated in Fig. 15 below.
eV hs
== Wy tet
fal | pa
pe aaa aaah b
walt) + vy (2)
w| GreRDIE TERM BESTAAN SLEGS VERLANGDE INFORMASTE WORD
as p(t) h G.S.TERM (a,) BEVAT HERWIN M.B.V. ‘tn B.D.F. MET
> SWE 2 AE, =|
ONBALANSWERKING BANDWYDTE: Af, 2 2f, = 2fyacc
VIR BALANSWERKING WORD DUS
VEREIS DAT a= 0
Fig. 15 Two-sided Fourier spectrum of the switched AM waveform
27 LPL-07| ‘Signal Processing and Telecommunications
4.3.4 AM Phasor Diagrams
The phasor diagram of a carrier modulated signal g(t) is closely related its equivalent low pass
form (also known as the ‘complex envelope’). To illustrate this concept, we begin with the
expression for a conventional AM signal, defined by equation (5). To derive the complex
envelope, equation (5) is first expanded using trigonometric identities, and then written en
complex exponential form, using the identity
cos@= Ref{e”} and/or sin@= Im{e*} 30)
Thus
Pry(t)= A[1+ meose,t] cos@,t
Am G1
= A,cos@.t + AE costa, - @,)t + cos(@, + yt]
Using (25), (26) can be written in the form
Bau ()= RelA {1+ 3 (0 + fel]
2)
= Ref 7,(Ne”*]
where the term in square brackets is the so-called pre-envelope of @4y(t), and f,,(t)
denotes the complex envelope, given by
Fr = m JOpgt — JO gt
f(t) = AfI+ (elt! 4 @ t 3)
F(t) in equation (28) represents the low-frequency (information) content of the AM-DSB with
Carrier signal $,.,(t) with the carrier component removed (or factored out, as illustrated in
equation (27)). Its in general a complex function.
28 LPL-07| ‘Signal Processing and Telecommunications
By plotting the complex envelope as a phasor in a coordinate system in which the axes are
rotating at the carrier angular frequency w, [1/s] in an anti-clockwise direction, the effect of the
modulating signal on the carrier may be revealed graphically. This is illustrated in Fig. 16
below. Note that the carrier phasor remains stationary in this diagram for all , since itis plotted
relative to the axes which rotate anti-clockwise at an angular frequency equal to that of the
cartier. The contribution of the information or modulating signal is portrayed by the phasors
corresponding to the LSB and USB of the AM signal, respectively, represented by the phasors
in rounded brackets in equation (28). A ‘snapshot’ of these phasors are plotted at the time
instant when their angles 6,,=w/,¢ equal 11/4 [rad] relative to the coordinate system in Fig. 16,
taking into account their direction of rotation relative to the anti-clockwise reference rotation
of the coordinate system.
Itis clear from the phasor diagram in Fig. 16 that the resultant phasor (consisting of the sum
of the carrier phasor and those corresponding to the LSB and USB of the AM signal) remains
strictly real and in the direction of the carrier amplitude. The LSB and USB information
phasors clearly only amplitude-modulate the carrier amplitude A, constructively and
destructively as tvaries over one period T,, of the modulating signal, f,(f), within the amplitude
limits 4,[1- m] to 4,[1+ m]. Note specifically that no phase modulation of the carrier is
incurred. The condition for no envelope distortion can also be deduced easily by noting that,
when @,=0 or Tr [rad] and m=1, the resultant phasor (or AM-signal envelope) equals 2A, and
0, respectively, i.e., no cross-over distortion occurs when m<1 (implying that the envelope
remains positive for all t, allowing recovery of information through envelope detection without
distortion).
29 LPL-07| ‘Signal Processing and Telecommunications
=>
SIGNAL:
/ Fin (€) = COTW
off
= — of eT Sees
Fig. 16 Phasor diagram of AB-DSB with Carrier at the time instant t, when w,,¢,=11/4
radians}.
30 LPL-07| ‘Signal Processing and Telecommunications
4.4 PHASOR DIAGRAM OF AM SINGLE-SIDEBAND (AM-SSB)
The general expression for a Single SideBand (SSB) AM signal is:
ssa (t)
+ f,(cosa,t+4f,()sinw,t G4
Both of the terms in eq. (29) represent AM-DSB signals with suppressed carrier in quadrature
with one-another. A complex envelope derived phasor diagram is illustrated in Fig. 17.
Im
\ @Qx (*4]
\
\
\
R an
gh Pht 1 >Re
|
/
CARRIER ic
/
/
/
Figure 17 Phasor diagram of AM-SSB-USB
31 LPL-07|