M-ary Pulse Amplitude modulation (M-PAM)
and
M-ary Quadrature Amplitude
Modulation (M-QAM)
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M-PAM
• M-ary PAM is a one-dimensional signaling
scheme described mathematically by
𝑠𝑖 𝑡 = 𝐴𝑖 cos 2𝜋𝑓𝑐 𝑡
=
2𝐸𝑖
cos 2𝜋𝑓𝑐 𝑡
𝑇
=
2𝐸𝑜
ai cos 2𝜋𝑓𝑐 𝑡
𝑇
= 𝑎𝑖
2
𝑖 = 1,2, … 𝑀
𝐸𝑜 𝜓(𝑡)
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Where the
𝜓 𝑡 =
2
cos 2𝜋𝑓𝑐 𝑡
𝑇𝑏
is the basis function and
𝑎𝑖 = (2𝑖 − 1 − 𝑀)
and
Eo is the energy of the signal with lowest amplitude
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• The average symbol energy:
𝐸𝑎𝑣
(𝑀2 − 1)
=
𝐸𝑜
3
• The probability of symbol error on AWGN
channel:
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4-PAM
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Example: 4-PAM
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Example: 4-PAM
M=4
𝑎𝑖 = (2𝑖 − 1 − 𝑀)
a1=-3,
𝐸𝑎𝑣 =
a2=-1,
a3=+1,
𝐸1 + 𝐸2 + 𝐸3 + 𝐸4 9𝐸𝑜 + 𝐸𝑜 + 𝐸𝑜 + 9𝐸𝑜
=
= 5𝐸𝑜
4
4
𝐸𝑎𝑣
(𝑀2 − 1)
42 − 1
=
𝐸𝑜 =
𝐸𝑜 = 5𝐸𝑜
3
3
“00”
“01”
s1
7
a4=+3
 3 Eo
“11”
s3
s2

Eo
0
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“10”
s4
3 Eo
 1 (t )
comments
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The signal space representation of binary PAM, 4-PAM and
8-PAM constellations for Eo=1
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The signal space representation of binary PAM, 4-PAM and 8-PAM constellations.
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Comments
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Symbol error probability for 2, 4 and 8PAM as a function of SNR per bit.
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M-ary Quadrature Amplitude Modulation
M-QAM
• Quadrature amplitude modulation (QAM) is a
popular scheme for high-rate, high bandwidth
efficiency systems.
• QAM is a combination of both amplitude and phase
modulation. Mathematically, M-ary QAM is described
by
𝑠𝑚𝑛 𝑡 = 𝐴𝑚 cos 2𝜋𝑓𝑐 𝑡 + 𝜃𝑛
𝑚 = 1,2, … , 𝑀1
𝑛 = 1,2, … , 𝑀2
The combined amplitude and phase modulation results in the
simultaneous transmission of log2 M1 M2 bits/symbol
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Digital Modulation
Techniques
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Examples of combined PAM-PSK signal space diagrams.
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8-QAM signal (2 amplitudes and 4 phases)
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• The transmitted M-QAM signal is defined by:
• The signal can be expressed using the two basis functions as
𝑠 𝑡 =
𝐸𝑜 𝑎𝑘 𝜓1 𝑡 + 𝐸𝑜 𝑏𝑘 𝜓2 (𝑡)
• The signal consists of two phase-quadrature carriers with each
one being modulated by a set of discrete amplitudes, hence
the name quadrature amplitude modulation.
• The signal-space representation of QAM signals is shown in
Figure for various values of M which are powers of 2, that is,
M = 2k, k = 2; 3; …..
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• For even values of k, the constellations are
square (4-QAM, 16-QAM, 64-QAM,..)
• for odd values of k the constellations have a
cross shape and are thus called cross
constellations. (32-QAM, 128 QAM, ..)
• For square constellations, QAM corresponds
to the independent amplitude modulation (MPAM) of an in-phase carrier (i.e., the cosine
carrier) and a quadrature carrier (i.e., the sine
carrier).
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Signal-space representation of various QAM constellations.
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32-Cross QAM (in red)
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Illustrating how a square QAM constellation can be
expanded to form a QAM cross-constellation.
4
4
4
Square 16-QAM
4
Square 16-QAM expanded to
32-cross QAM (n=5)
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M-QAM square constellation
• With an even number of bits per symbol, we
may write
• M-ary QAM square constellation can be
viewed as the Cartesian product of a onedimensional L-ary PAM constellation with
itself.
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• In the case of a QAM square constellation, the
pairs of coordinates form a square matirx, as
shown by
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Example: square 16-QAM
• M=16, L=4
• Thus the square constellation is the Cartesian
product of the 4-PAM constellation with itself.
• ak and bk take values from the set {-1,+1, -3,
+3}
• The matrix of the product
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Comments
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“00”
s1
3
Eo

Eo
Eo

3 Eo
Eo
 3 Eo
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
Eo
“11”
Gray coded 16-QAM
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“10”
s4
s3
0
Eo
4-PAM
3 Eo
 3 Eo
Eo
“01”
s2
3
Eo
 1 (t )
Gray Coded 16-QAM with Eo=1
“0000”
s1
“1000”
s5
“0001”
s2
“1001”
s6
“0011” “0010”
3
s3
s4
“1011” “1010”
1
s7
s8
3
-3
-1
1
s9
s 10
s
s
s 14
s
s
“1100”
s 13
“0100”
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 2 (t )
12
11
-1
“1101” “1111” “1110”
16
15
-3
“0101” “0111” “0110”
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 1 (t )
Performance of square QAM in Additive
Gaussian Noise
• The probability of symbol error of M-QAM
with square constellation is given by
• Where Eav is the average symbol energy given
by
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Example: Calculate the average symbol energy for square 16-QAM
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Symbol error probability as a function
of SNR per bit (Eb/No)for 4, 16, and
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64-QAM.
ρ
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Comparison between M-PAM and M-QAM
Prob. Of Symbol Error M-PAM
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Prob. Of Symbol Error M-QAM
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Comparison between M-QAM and M-PSK
Prob. Of Symbol Error M-PSK
Prob. Of Symbol Error M-QAM
Eb/No dB
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Eb/No dB
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Performance comparison of M-PAM, M-PSK and M-QAM
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Comments
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Performance Comparison of M-PSK and M-QAM
• For M-PSK: approximate Pe
• For M-QAM: approximate Pe
3𝐸𝑎𝑣
𝑃𝑒 ≈ 4𝑄(
(𝑀 − 1)𝑁𝑜
• Comparing the arguments of Q(.) for the two modulations we
calculate the advantage in signal-to-noise ratio of M-QAM
over MPSK (to achieve same error performance) as
𝐸𝑃𝑆𝐾
3/(𝑀 − 1)
𝑅𝑀 =
=
𝜋
2
𝐸𝑄𝐴𝑀
2 sin
𝑀
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SNR Advantage of M-QAM over M-PSK for different M
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