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6.450 Introduction to Digital Communication October 21, 2002
MIT, Fall 2002
Lecture 12: QAM
1 Review
In the previous lecture, we discussed pulse amplitude modulation (PAM) as a very simple
mode of digital communication. The input data sequence arrives at a rate of R b/s and
is converted, b bits at a time, into a sequence of real symbols a
k
chosen from a symbol set
/ = ¦−d(M −1)/2, . . . , −d/2, d/2, . . . , d(M −1)/2¦
of size M = 2
b
. The symbol rate is R
s
= R/b symbols per second (symbol/s), and the
symbol interval is T = 1/R
s
= b/R sec.
A PAM modulator is defined by the symbol interval T and an L
2
waveform (pulse) p(t),
as follows: the discrete-time symbol sequence ¦a
k
¦ modulates the amplitudes of the time
shifts ¦p(t −kT)¦ of p(t) to create the transmitted signal
u(t) =

k
a
k
p(t −kT). (1)
The PAM demodulator at the receiver is defined by the interval T and an L
2
waveform
q(t). Assuming no noise and a known time reference at the receiver, the PAM demodulator
filters the transmitted waveform u(t) by q(t). It then samples the resultant waveform
r(T) =
_
u(τ)q(t − τ) dτ at T-space intervals, getting r(T), r(2T), . . . , . If the sample
r(kT) is equal to the transmitter symbol a
k
for each k, we say the PAM system has no
intersymbol interference.
We saw that there is no intersymbol interference if the cascaded filter g(t) = p(t) ∗ q(t)
has the property that g(0) = 1 and g(kT) = 0 for all integer k ,= 0. We defined g(t) to
be ideal Nyquist if it has this property. We then showed that g(t) is ideal Nyquist if and
only if ˆ g(f) satisfies the Nyquist criterion, which is

m
ˆ g(f+m/T) = T for all f. We saw
that the situation of interest for PAM is to select g(t) so that it is ideal Nyquist,with the
additional conditions that it drops off to 0 rapidly with large (positive or negative) t and
that ˆ g(f) drops off to 0 rapidly for [f[ > 1/T.
We next saw that there is little loss in generality if g(t) is selected so that ˆ g(f) is real and
positive. In this case, if we select p(t) so that ˆ p(f) =
_
ˆ g(f), then the T-spaced shifts of
p(t), i.e., ¦p(t−kT)¦ form an orthonormal set and the modulated PAM waveform is just
an orthonormal expansion.
1
2 Passband and quadrature amplitude modulation
The discussion of PAM was focused on transmitting waveforms in the frequency band
f ≤ W for some nominal bandwidth W. Most communication, however, takes place in
some passband of frequencies both appropriate to the physical medium and allowed by
regulatory agencies. Thus the waveforms we have been considering usually must be trans-
lated up to an appropriate passband. Typically, many such transmissions are multiplexed
together, each in a separate passband, necessitating the need to keep each waveform in a
tightly constrained frequency band.
Modulation of a PAM waveform to a passband can be accomplished simply by multi-
plying the baseband PAM waveform u(t) by a sinuosoid, say cos(2πf
c
t). Let u(t) =

k
a
k
p(t−kT) where p(t) is a waveform with a nominal bandwidth W = 1/(2T). Then
u(t) cos(2πf
c
t) has a nominal bandwidth 2W, going from f
c
−W to f
c
+W (assuming that
f
c
> W). This scheme is called double-side-band amplitude modulation. It is very waste-
ful of bandwidth, since additional data could be sent in the same frequency band by
modulating it by sin(2πf
c
t).
The simplest and most widely employed solution to this problem is that of quadrature
amplitude modulation (QAM). We can think of this as modulating one low-pass wave-
form, u

(t) =

k
a

k
p(t−kT) by cos(2πf
c
t) and another low pass waveform u

(t) =

k
a

k
p(t−kT) by sin(2πf
c
t). It is both more convenient and conventional, however,
to view the real waveforms u

(t) and u

(t) as the real and imaginary components of a
complex waveform u(t) = u

(t) + iu

(t). In this view, we have
u(t) =

k
(a

k
+ ia

k
)p(t−kT)
Letting a
k
= a

k
+ia

k
denote the corresponding complex coefficient, the baseband complex
waveform is then
u(t) =

k
a
k
p(t−kT) (2)
The baseband waveform u(t) is then modulated up to a passband around the carrier
frequency f
c
as u(t)e
i2πfct
. We see that u(t)e
i2πfct
is a complex waveform lying in the
nominal positive passband (−
1
2T
+f
c
,
1
2T
+f
c
). The actual transmitted waveform must be
real, so we add a negative frequency component to get the real passband waveform
x(t) = u(t)e
i2πfct
+ u

(t)e
−i2πfct
(3)
This can also be written in the following equivalent ways:
x(t) = 2'[u(t)e
2πifct
] (4)
= 2'[u(t)] cos(2πf
c
t) −2·[u(t)] sin(2πf
c
t) (5)
The factor of 2 in (4) and (5) is an arbitrary scale factor. Some authors leave it out,
(thus requiring a factor of 1/2 in (3)) and others replace it by

2 (requiring a factor of
2
1/

2 in (3)). The confusion comes when we look at the energy in the waveforms. With
our scaling here, |x|
2
= 2|u|
2
. Using the scale factor

2 solves this problem, but as we
see later, will introduce many other problems. At one level, scaling is a trivial matter,
but since it is done inconsistently in the literature, we should get used to handling it in a
consistent way. The following figure might help to keep these scale factors straight.
−f
c
ˆ x(f)
f
c
0 f W
W/2
-
ˆ u(f)
Figure 1: Fourier transform ˆ x(f) of a passband waveform x(t) corresponding to the
transform ˆ u(f) of a complex baseband waveform u(t)
With passband communication, the bandwidth W is universally defined to be the band-
width of the positive frequency part of the passband, i.e., negative frequencies are not
counted. For QAM, the passband is then (−W/2 + f
c
, W/2 + f
c
). The corresponding
baseband waveform must then be (−W/2, W/2), which is a baseband bandwidth of W/2.
When discussing baseband PAM, we defined the nominal bandwidth W as 1/(2T) where
T is the symbol interval. Here, with QAM and passband communication, W is 1/T and
the corresponding baseband waveform has the nominal bandwidth W/2.
With familiarity, the complex notation in (3) is very convenient to use. It allows us to
look at baseband and passband as almost trivial shifts of each other. This is why the use
of complex baseband waveforms is so common. A complex baseband waveform usually
refers to a pair of waveforms that are going to be quadrature modulated, or, equivalently,
to a pair of baseband waveforms after quadrature demodulation. By using the artifice of
complex notation, we can often avoid thinking about what happens at passband. For
example, a carrier phase shift at passband is simply reflected as multiplication by a
complex scalar at baseband. Unfortunately, the physical phenomena of importance on
channels occur at passband, and we must understand the relationship between passband
and baseband in order to interpret these phenomena at baseband.
A QAM modulator has three parts (see figure 2). The first is that of mapping the
incoming binary digits into complex symbols. The second is mapping the sequence of
complex symbols into a baseband waveform, and the third is modulating the baseband
waveform to passband. The demodulator, as should come as no surprise, performs the
inverse of these operations in reverse order, first mapping the received bandpass waveform
into a baseband waveform, then recovering the sequence of symbols, and then recovering
the binary digits. We now discuss these operations.
3
Input
Binary
-
Symbol
encoder
-
Baseband
modulator
-
Baseband to
passband
?
Channel
Baseband
Demodulator

Passband to
baseband
Output
Binary

Symbol
decoder

Figure 2: QAM modulator and demodulator.
2.1 QAM symbol set
The input data sequence arrives at a rate of R b/s and is converted, b bits at a time, into
a sequence of complex symbols a
k
chosen from a symbol set (alphabet, constellation) /
of size M = [/[ = 2
b
. The symbol rate is thus R
s
= R/b symbols per second, and the
symbol interval is T = 1/R
s
= b/R sec.
In the case of QAM, the transmitted symbols a
k
are complex numbers a
k
∈ C, rather
than real numbers. Alternatively, we may think of each symbol as a real 2-tuple in R
2
.
A standard (M

M

)-QAM signal set, where M = (M

)
2
is the Cartesian product of
two M

-PAM sets; i.e.,
/ = ¦(a

+ ia

) [ a

∈ /

, a

∈ /

¦,
where
/

= ¦−d(M

−1)/2, . . . , −d/2, d/2, . . . , d(M

−1)/2¦.
The signal set / thus consists of a square array of M = (M

)
2
= 2
b
signal points located
symmetrically about the origin, as illustrated below for M = 16.
t t t t
t t t t
t t t t
t t t t
d-
The minimum distance between two-dimensional points is denoted by d. Also the average
energy per two-dimensional symbol, which is denoted E
s
, is simply twice the average
energy per dimension:
E
s
=
d
2
[(M

)
2
−1]
6
.
4
In the case of QAM there are obviously many other ways to arrange the signal points than
on a square grid as above. For example, in an M-PSK (phase-shift keyed) signal set, the
signal points consist of M equally spaced points on a circle centered on the origin. (Thus
4-PSK = 4-QAM.) As another example, as seen earlier with quantization, a hexagonal
grid is better than a square grid for minimizing E
s
for large M and a given minimum
distance d.
When we study noise and detection later, we will see that the probability of detection
error is primarily a function of the minimum distance d and that the error probability
decreases rapidly with increasing d. We will also see that E
s
is linear in the signal power of
the passband waveform. In wireless systems the signal power is limited both to conserve
battery power and to meet regulatory requirements. In wired systems, the power is limited
both to avoid crosstalk between adjacent wires and to avoid non-linear effects. Therefore
it is desirable to choose symbol constellations that minimize E
s
for a given d and M. We
will not spend much time on this minimization other than a couple of exercises.
2.2 QAM baseband modulation and demodulation
A QAM baseband modulator is determined by the symbol interval T and a complex L
2
waveform p(t). The discrete-time complex sequence ¦a
k
¦ modulates the amplitudes of a
sequence of time shifts ¦p(t −kT)¦ of the basic pulse p(t) to create a complex transmitted
signal u(t) as follows:
u(t) =

k∈Z
a
k
p(t −kT). (6)
As in the PAM case, we could choose p(t) to be sinc(t/T), but for the same reasons as
before, we usually prefer p(t) to decay faster than the sinc function with increasing t, and
thus for ˆ p(f) to be a continuous function that goes to zero rapidly but not instantaneously
as f increases beyond 1/(2T). As discussed above, we define W = 1/T for QAM, so that
ˆ p(f) should go to 0 rapidly outside the nominal basebandband [−W/2, W/2].
We assume for the moment that the process of converson to passband, channel trans-
mission, and conversion back to baseband, is ideal, recreating the baseband modulator
output u(t) at the input to the baseband demodulator. The baseband demodulator is
determined by the interval T (the same as at the modulator) and a complex L
2
waveform
q(t). The demodulator filters u(t) by q(t) and samples the output at T-spaced sample
times. Denoting the filtered output by
r(t) =
_

−∞
u(τ)q(t −τ) dτ,
we see that the received samples are r(T), r(2T), . . . . Note that this is the same as the
PAM demodulator except that real symbols and waveforms have been replaced by complex
symbols and waveforms. As before, the output r(t) can be represented as
r(t) =

k
a
k
g(t −kT)
5
where g(t) is the convolution of p(t) and q(t). As before, r(kT) = a
k
if g(t) is ideal
Nyquist, namely if g(0) = 1 and g(kT) = 0 for all all non-zero integer k. The Nyquist
criterion is still valid, since it can be easily seen that neither the statement of the theorem
nor the proof assumed that g(t) is real.
The band edge symmetry condition illustrated in Figure 4 of Lecture 11 depends on g(t)
being real, but g(t) is usually chosen to be real anyway (even if p(t) and q(t) are complex),
and the raised cosine frequency functions are often used for QAM. As with PAM, ˆ g(f) is
usually chosen to be real and positive, as with the raised cosine pulse, and ˆ p(f) is usually
chosen to satisfy [ˆ p(f)[ =
_
ˆ g(f). With this choice, the set of time shifts ¦p(t−kT)¦ form
an orthonormal set of functions.
In summary, QAM baseband modulation is virtually the same as PAM baseband modula-
tion. The symbol set for QAM is of course complex, and the modulating pulse p(t) can be
complex, but the Nyquist results about avoiding intersymbol interference are unchanged.
2.3 Baseband to passband and back
Next, we want to modulate the complex QAM baseband signal u(t), which has a spectrum
nominally limited to [−W/2, W/2], to the real QAM bandpass signal x(t) around some
carrier frequency f
c
. Alternative expressions for x(t) are given by (3), (4). and (5).
We can view this modulation process as a two step process. First u(t) is translated up
in frequency by an amount f
c
, resulting in u(t)e
i2πfct
. Next the negative frequency part
u

(t)e
−i2πfct
is added on to make x(t) real. Most engineers would view the first step as
the essence of modulation, i.e., translating u(t) up in frequency. The negative frequency
part of x(t) must be added, however, to make x(t) real.
From an implementation standpoint, the baseband waveform u(t) usually exists as a real
and imaginary part, and the modulation usually consists of multiplying by in-phase and
out-of-phase carriers as in (5), i.e.,
x(t) = 2'[u(t)] cos(2πf
c
t) −2·[u(t)] sin(2πf
c
t).
There are many other possible implementations, however, such as starting with u(t) given
as a magnitude and phase. Thus the positive frequency expression u(t)e
i2πfct
is primarily
a means of gaining insight rather than a component of implementation.
As discussed in the last subsection, u(t) is nominally constrained to the baseband W/2
where W = 1/T. Let B/2 be the actual baseband bandwidth of u(t) with a rolloff
B/W − 1. We assume in everything to follow that B/2 < f
c
. This implies that the
positive frequency function u(t)e
i2πfct
is actually constrained to positive frequencies, and
thus that the Fourier transform ˆ x(f−f
c
) does not overlap with ˆ x(f+f
c
).
The transformation u(t) → v(t) = u(t)e
2πifct
is clearly reversible, since we can simply
multiply v(t) by e
−2πifct
(“demodulation”) to recover u(t). Thus if the positive frequency
waveform v(t) can be retrieved at the receiver, it can be easily demodulated into u(t).
Retrieving v(t) involves the usual assumptions of no noise, appropriate amplitude scaling,
6
and known time-reference. It also requires the receiver to know the correct carrier fre-
quency and phase. Finding this correct frequency and phase is called “carrier recovery,”
and is quite easy in digital QAM systems, as we shall see shortly
1
.
The transformation u(t) → v(t) = u(t)e
2πifct
also preserves inner products. That is,
¸u
1
(t), u
2
(t)) = ¸v
1
(t), v
2
(t)) where v
j
(t) = u
j
(t)e
i2πfct
. This means that distances
(norms) are preserved and energy differences are preserved. It also means that if
u(t) =

k
a
k
p(t−kT, where ¦p(t−kT)¦ is an orthonormal set, then the positive frequency
version v(t) = u(t)e
i2πfct
satisfies v(t) =

a
k
w(t−kT) where w(t−kT) = p(t−kT)e
i2πfct
and ¦w(t−kT¦ is an orthonormal set.
The second step in the conversion from baseband to passband is the addition of the
negative frequency component v

(t) = u

(t)e
−2πifct
to the positive frequency component
v(t). Assuming f
c
> B/2, this operation is also reversible. Mathematically, we can
retrieve v(t) from x(t) simply by filtering x(t) by a complex filter h(t) such that
ˆ
h(f) = 0
for f < 0 and
ˆ
h(f) = 1 for f > 0. This filter is called a Hilbert filter. We note that h(t)
is not an L
2
function, but it can be converted to L
2
by making
ˆ
h(f) have the value 0
except in the positive passband (−B/2 +f
c
, B/2 +f
c
) where it has the value 1. Figure 3
illustrates the sequence of operations from u(t) to x(t) and back again.
u(t)
- n
@
?
e
2πifct
-
v(t)
2'¦ ¦
-
x(t)
Hilbert
filter
-
v(t)
n
@
?
e
−2πifct
-
u(t)
. ¸¸ .
transmitter
. ¸¸ .
receiver
Figure 3: Baseband to passband and back.
An alternative view of converting from passband to baseband is to represent x(t) by
2'[u(t)] cos(2πf
c
t) − 2·[u(t)] sin(2πf
c
t). Then we see that '[u(t)] can be retrieved by
multiplying x(t) by cos(2πf
c
t) and low pass filtering. Similarly, ·[u(t)] can be retrieved
by multiplying x(t) by sin(2πf
c
t) and low pass filtering. From an implementation point of
view, this is usually a more attractive point of view than the Hilbert filter, since most of
the processing is done at baseband. The Hilbert filter is nice conceptually, though, since
it corresponds to the inverse of first forming the positive frequency function v(t) and then
taking the real part.
We have seen that u(t) and its positive frequency function v(t) have the same energy.
From figure 1, it can easily be seen that x(t) has twice the energy of u(t), and thus
|x| =

2|u|. We must emphasize here, however, that the set of baseband waveforms
u are in a complex vector space, and the set of corresponding passband waveforms x are
in a real vector space. The transformation from u (or from v) to x is not linear and the
inner products in one space are complex, whereas those in the other are real. In short,
1
A mathematical purist would argue that f
c
should be known at the receiver just like the assumption
that T and p(t) are known, and that the phase of f
c
is then known from the time-reference. In practice,
however, the carrier recovery problem is separable from the time-reference and arises from the lack of
perfect stability in the oscillators used to generate the carrier.
7
the vector space properties of the space of complex functions baseband limited to B/2
and the vector space properties of real functions limited to [f[ ∈ (−B/2+f
v
, B/2+f
c
are
quite different.
Fortunately, there is a very nice property that can be used to relate an arbitrary set
of orthonormal functions, say θ
k
, in the above complex baseband space to a related
orthogonal set in the corresponding real passband space. For each complex baseband
function θ
k
, define two real passband functions ψ
k,1
(t) and ψ
k,2
(t) as
ψ
k,1
(t) = '¦2θ
k
(t) exp[2πif
c
t]¦
ψ
k,2
(t) = ·¦2θ
k
(t) exp[2πif
c
t]¦
We now show that the set of waveforms ¦ψ
k,1
, ψ
k,2
¦ are real orthogonal waveforms each
of energy 2. To do this, we look at their Fourier transforms. The Fourier transform of
θ
k
(t) exp[2πif
c
t] is
ˆ
θ
k
(f−f
c
). Thus
ˆ
ψ
k,1
(f) =
ˆ
θ
k
(f−f
c
) +
ˆ
θ

k
(f
c
−f) (7)
ˆ
ψ
k,2
(f) =
ˆ
θ
k
(f−f
c
) −
ˆ
θ

k
(f
c
−f)
i
(8)
Using the assumption that B/2 < f
c
, it can be seen from (7) and (8) that
_

−∞
ψ
k,2
(t)ψ
k,1
(t) dt =
_

−∞
ˆ
ψ
k,2
(f)ψ

k,1
(f) df
= i
_
0
−∞
[
ˆ
θ
k
(f
c
−f)[
2
df −i
_

0
[
ˆ
θ
k
(f
c
−f)[
2
df
The limits on each integral above cover the range over which the integrad is non-zero,
so each integral is 1 and the sum is 0. It can be seen by a similar argument that
_
ψ
k,l
(t)ψ
j,l
(t) dt = 0 for all j ,= k and all l, l

= 1, 2. Thus ¦ψ
k,l
¦, for integer k and
l = 1, 2 is a set of orthogonal functions each of energy 2.
If the baseband orthonormal functions are all real, then the above passband functions
simplify to ψ
k,1
(t) = 2θ
k
(t) cos(2πf
c
t) and ψ
k,2
= 2θ
k
(t) sin(2πf
c
t).
3 Degrees of freedom
We found that with PAM, we could choose real symbols separated by T and transmit
them in a bandwidth W = 1/(2T). Thus, over a long interval T
0
, and in a baseband
bandwidth W, we can transmit 2WT
0
real symbols using PAM.
Using QAM, with T-spaced symbols, we need a passband bandwidth of W = 1/T. How-
ever, each symbol is complex and consists of a real and imaginary part, each of which can
be independently selected. Thus, again, over a long interval of time T
0
, we can select an
arbitrary set of 2WT
0
real symbols.
8
The above argument seems a little flaky since W is defined differently for PAM and QAM.
To obtain a more reasonable comparison, consider an overall large baseband bandwidth
W
0
broken into m passbands each of bandwidth W
0
/m. Using QAM in each band, we
can transmit 2W
0
T
0
real symbols in a long interval T
0
. With PAM used over the entire
band W
0
, we again send 2W
0
T
0
real symbols in a duration T
0
. We see that in principle,
QAM and baseband PAM are equivalent in terms of the number of degrees of freedom
that can be used to transmit real symbols. As pointed out earlier, however, PAM when
modulated up to passband uses only half the available degrees of freedom.
Recall that when we were looking at T-spaced truncated sinusoids and T-spaced sinc
weighted sinusoids, we argued that the class of real waveforms occupying a time interval
(0, T
0
) and a frequency interval (−W
0
, W
0
) have about 2T
0
W
0
degrees of freedom for large
W
0
, T
0
. What we see now is that baseband PAM and passband QAM each employ about
2T
0
W
0
degrees of freedom. In other words, these simple techniques essentially use all the
degrees of freedom available in the given bands.
9

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