4
Amplitude/Linear Modulation
4.1. A sinusoidal carrier signal A cos(2πfc t+φ) has three basic parameters:
amplitude, frequency, and phase. Varying these parameters in proportion
to the baseband signal results in amplitude modulation (AM), frequency16
modulation (FM), and phase modulation (PM), respectively. Collectively,
these techniques are called continuous-wave (CW) modulation [13, p
111][3, p 162].
Definition 4.2. Amplitude modulation is characterized by the fact that
the amplitude A of the carrier A cos(2πfc t + φ) is varied in proportion to
the baseband (message) signal m(t).
• Because the amplitude is time-varying, we may write the modulated
carrier as
A(t) cos(2πfc t + φ)
• Because the amplitude is linearly related to the message signal, this
technique is also called linear modulation.
4.3. Linear modulations:
(a) Double-sideband amplitude modulation
(i) Double-sideband-suppressed-carrier (DSB-SC or DSSC or simply
DSB) modulation
(ii) Standard amplitude modulation (AM)
(b) Suppressed-sideband amplitude modulation
(i) Single-sideband modulation (SSB)
(ii) Vestigial-sideband modulation (VSB)
16
Technically, the variation of “frequency” is not as straightforward as the description here seems to
suggest. For a sinusoidal carrier, a general modulated carrier can be represented mathematically as
x(t) = A(t) cos (2πfc t + φ(t)) .
Frequency modulation, as we shall see later, is resulted from letting the time derivative of φ(t) be linearly
related to the modulating signal. [14, p 112]
35
4.1
Double-sideband suppressed carrier (DSB-SC) modulation
Definition 4.4. In double-sideband-suppressed-carrier (DSB-SC or
DSSC or simply DSB) modulation, the modulated signal is
x(t) = Ac cos (2πfc t) × m(t).
We have seen that the multiplication by a sinusoid gives two shifted and
scaled replicas of the original signal spectrum:
Ac
Ac
M (f − fc ) + M (f + fc ) .
2
2
√
• When we set Ac = 2, we get the “simple” modulator discussed in
Example 3.12.
X(f ) =
• We need fc > B to avoid spectral overlapping. In practice, fc B.
4.5. Synchronous/coherent detection by the product demodulator:
The incoming modulated signal is first multiplied with a locally generated
sinusoid with the same phase and frequency (from a local oscillator (LO))
and then lowpass-filtered, the filter bandwidth being the same as the message bandwidth B or somewhat larger.
4.6. A DSB-SC modem with no channel impairment is shown in Figure 12.
×
Channel
y
×
v
LPF
Message
(modulating signal)
2 cos  2 f c t 
2 cos  2 f c t 
Demodulator
Modulator
2
Figure 12: DSB-SC modem with no channel impairment
36
2
Once again, recall that
√
1
X (f ) = 2
(M (f − fc ) + M (f + fc ))
2
1
= √ (M (f − fc ) + M (f + fc )) .
2
Similarly,
√
√
v (t) = y (t) × 2 cos (2πfc t) = 2x (t) cos (2πfc t)
1
V (f ) = √ (X (f − fc ) + X (f + fc ))
2
Alternatively, we can work in the time domain and utilize the trig. identity from Example 2.4:
√ √
√
2m (t) cos (2πfc t) cos (2πfc t)
v (t) = 2x (t) cos (2πfc t) = 2
= 2m (t) cos2 (2πfc t) = m (t) (cos (2 (2πfc t)) + 1)
= m (t) + m (t) cos (2π (2fc ) t)
Key equation for DSB-SC modem:







√

√
LPF
m (t) × 2 cos (2πfc t) ×
2 cos (2πfc t)
= m (t) .




|
{z
}


x(t)
37
(31)
4.7. Implementation issues:
(a) Problem 1: Modulator construction
(b) Problem 2: Synchronization between the two (local) carriers/oscillators
(c) Problem 3: Spectral inefficiency
4.8. Spectral inefficiency/redundancy: When m(t) is real-valued, its
spectrum M (f ) has conjugate symmetry. With such message, the corresponding modulated signal’s spectrum X(f ) will also inherit the symmetry
but now centered at fc (instead of at 0). The portion that lies above fc is
known as the upper sideband (USB) and the portion that lies below fc
is known as the lower sideband (LSB). Similarly, the spectrum centered
at −fc has upper and lower sidebands. Hence, this is a modulation scheme
with double sidebands. Both sidebands contain the same (and complete)
information about the message.
4.9. Synchronization: Observe that (31) requires that we can generate
cos (ωc t) both at the transmitter and receiver. This can be difficult in practice. Suppose that the frequency at the receiver is off, say by ∆f , and that
the phase is off by θ. The effect of these frequency and phase offsets can be
quantified by calculating
n
√
o
√
LPF m (t) 2 cos (2πfc t)
2 cos (2π (fc + ∆f ) t + θ) ,
which gives
m (t) cos (2π (∆f ) t + θ) .
Of course, we want ∆ω = 0 and θ = 0; that is the receiver must generate
a carrier in phase and frequency synchronism with the incoming carrier.
38
4.10. Effect of time delay:
Suppose the propagation time is τ , then we have
√
y (t) = x (t − τ ) = 2m (t − τ ) cos (2πfc (t − τ ))
√
= 2m (t − τ ) cos (2πfc t − 2πfc τ )
√
= 2m (t − τ ) cos (2πfc t − φτ ) .
Consequently,
√
v (t) = y (t) × 2 cos (2πfc t)
√
√
= 2m (t − τ ) cos (2πfc t − φτ ) × 2 cos (2πfc t)
= m (t − τ ) 2 cos (2πfc t − φτ ) cos (2πfc t) .
Applying the product-to-sum formula, we then have
v (t) = m (t − τ ) (cos (2π (2fc ) t − φτ ) + cos (φτ )) .
In conclusion, we have seen that the principle of the DSB-SC modem is
based on a simple key equation (31). However, as mentioned in 4.7, there
are several implementation issues that we need to address. Some solutions
are provided in the subsections to follow. However, the analysis will require
some knowledge of Fourier series which is reviewed in the next subsection.
39
4.2
Fourier Series
Definition 4.11. Exponential Fourier series: Let the (real or complex)
signal r (t) be a periodic signal with period T0 . Suppose the following Dirichlet conditions are satisfied
(a) r (t) is absolutely integrable over its period; i.e.,
RT0
|r (t)|dt < ∞.
0
(b) The number of maxima and minima of r (t) in each period is finite.
(c) The number of discontinuities of r (t) in each period is finite.
Then r (t) can be expanded in terms of the complex exponential signals
∞
ejnω0 t n=−∞ as
r̃ (t) =
∞
X
cn e
jnω0 t
= c0 +
n=−∞
∞
X
ck ejkω0 t + c−k e−jkω0 t
(32)
k=1
where
ω0 = 2πf0 =
2π
,
T0
α+T0
1
ck =
T0
Z
r (t) e−jkω0 t dt,
(33)
α
for some arbitrary α. In which case,
r (t) ,
if r (t) is continuous at t
r̃ (t) = r(t+ )+r(t− )
, if r (t) is not continuous at t
2
We give some remarks here.
• The parameter α in the limits of the integration (33) is arbitrary. It
can be chosen to simplify computation of the integral. Some references
40
simply write ck =
1
T0
R
r (t) e−jkω0 t dt to emphasize that we only need
T0
to integrate over one period of the signal; the starting point is not
important.
R
• The coefficients ck = T10 r (t) e−jkω0 t dt are called the (k th ) Fourier
T0
(series) coefficients of (the signal) r (t). These are, in general, complex numbers.
R
• c0 = T10 r (t) dt = average or DC value of r(t)
T0
• The quantity f0 = T10 is called the fundamental frequency of the
signal r (t). The nth multiple of the fundamental frequency (for positive
n’s) is called the nth harmonic.
• ck ejkω0 t + c−k e−jkω0 t = the k th harmonic component of r (t).
k = 1 ⇒ fundamental component of r (t).
4.12. Getting the Fourier coefficients from the Fourier transform:
Consider a restricted version rT0 (t) of r(t) where we only consider r(t) for
F
−−
*
−
− RT0 (f ). Then,
one specific period. Suppose rT0 (t) )
F −1
ck =
1
RT (kf0 ).
T0 0
So, the Fourier coefficients are simply scaled samples of the Fourier
transform.
Example 4.13. Find the Fourier series expansion for the train of impulses
∞
P
δ (T0 ) (t) =
δ (t − nT0 ) drawn in Figure 13.
n=−∞
T0
2T0
Figure 13: Train of impulses
41
t
4.14. The Fourier series in Example 4.13 gives an interesting Fourier transform pair:
A special case when T0 = 1 is quite easy to remember:
We can use the scaling property of the delta function to generalize the special
case:
Example 4.15. Find the Fourier coefficients of the square pulse periodic
signal [5, p 57]shown in Figure 14. Note that multiplication by this signal
is equivalent to a switching (ON-OFF) operation.
1
2T0
T0
T0
4
T0
4
T0
t
2T0
Figure 14: Square pulse periodic signal
4.16. Parseval’s Identity: Pr =
1
1
T0
R
|r (t)|2 dt =
∞
P
k=−∞
T0
42
|ck |2
4.17. Fourier series expansion for real valued function: Suppose
r (t) in the previous section is real-valued; that is r∗ = r. Then, we have
c−k = c∗k and we provide here three alternative ways to represent the Fourier
series expansion:
r̃ (t) =
∞
X
cn e
jnω0 t
∞
X
= c0 +
n=−∞
= c0 +
ck ejkω0 t + c−k e−jkω0 t
(34)
k=1
∞
X
(ak cos (kω0 t)) +
k=1
∞
X
= c0 + 2
∞
X
(bk sin (kω0 t))
(35)
k=1
|ck | cos (kω0 t + ∠ck )
(36)
k=1
where the corresponding coefficients are obtained from
α+T0
1
ck =
T0
Z
r (t) e−jkω0 t dt =
1
(ak − jbk )
2
(37)
α
2
ak = 2Re {ck } =
T0
Z
r (t) cos (kω0 t) dt
(38)
T0
Z
2
bk = −2Im {ck } =
r (t) sin (kω0 t) dt
T0
T0
q
2 |ck | = a2k + b2k
bk
∠ck = − arctan
ak
a0
c0 =
2
The Parseval’s identity can then be expressed as
Z
∞
∞
X
X
1
2
2
2
Pr =
|r (t)| dt =
|ck | = c0 + 2
|ck |2
T0
T0
k=−∞
43
k=1
(39)
(40)
(41)
(42)
4.18. To go from (34) to (35) and (36), note that when we replace c−k by
c∗k , we have
ck ejkω0 t + c−k e−jkω0 t = ck ejkω0 t + c∗k e−jkω0 t
∗
= ck ejkω0 t + ck ejkω0 t
= 2 Re ck ejkω0 t .
• Expression (36) then follows directly from the phasor concept:
Re ck ejkω0 t = |ck | cos (kω0 t + ∠ck ) .
• To get (35), substitute ck by Re {ck } + j Im {ck }
and ejkω0 t by cos (kω0 t) + j sin (kω0 t).
Example 4.19. For the train of impulses in Example 4.13,
δ
(T0 )
(t) =
∞
X
k=−∞
∞
∞
1 X jkω0 t
1
2 X
δ (t − kT0 ) =
e
=
+
cos kω0 t
T0
T0 T0
k=−∞
(43)
k=1
Example 4.20. For the rectangular pulse train in Example 4.15,
1
1
1
1 2
cos ω0 t − cos 3ω0 t + cos 5ω0 t − cos 7ω0 t + . . .
1 [cos ω0 t ≥ 0] = +
2 π
3
5
7
(44)
Example 4.21. Bipolar square pulse periodic signal [5, p 59]:
4
1
1
1
sgn(cos ω0 t) =
cos ω0 t − cos 3ω0 t + cos 5ω0 t − cos 7ω0 t + . . .
π
3
5
7
1
−T0
t
T0
-1
Figure 15: Bipolar square pulse periodic signal
1
−T0
T0
44
t
4.3
Classical DSB-SC Modulators
To produce the modulated signal Ac cos(2πfc t)m(t), we may use the following methods which generate the modulated signal along with other signals
which can be eliminated by a bandpass filter restricting frequency contents
to around fc .
4.22. Multiplier Modulators [5, p 184] or Product Modulator[3, p
180]: Here modulation is achieved directly by multiplying m(t) by cos(2πfc t)
using an analog multiplier whose output is proportional to the product of
two input signals.
• Such a multiplier may be obtained from
(a) a variable-gain amplifier in which the gain parameter (such as the
the β of a transistor) is controlled by one of the signals, say, m(t).
When the signal cos(2πfc t) is applied at the input of this amplifier,
the output is then proportional to m(t) cos(2πfc t).
(b) two logarithmic and an antilogarithmic amplifiers with outputs
proportional to the log and antilog of their inputs, respectively.
◦ Key equation:
A × B = e(ln A+ln B) .
4.23. Square Modulator: When it is easier to build a squarer than a
multiplier, use
(m (t) + c cos (ωc t))2 = m2 (t) + 2c m (t) cos (ωc t) + c2 cos2 (ωc t)
c2 c2
2
= m (t) + 2c m (t) cos (ωc t) + + cos (2ωc t) .
2
2
45
3
ωc
2t
• Alternative, can use m(t) + c cos
.
4.24. Multiply m(t) by “any” periodic and even signal r(t) whose period
is Tc = 2π
ωc . Because r(t) is an even function, we know that
r (t) = c0 +
∞
X
ak cos (kωc t) for some c0 , a1 , a2 , . . ..
k=1
Therefore,
m(t)r (t) = c0 m(t) +
∞
X
ak m(t) cos (kωc t).
k=1
See also [4, p 157]. In general, for this scheme to work, we need
m (t )
M (ω )
A
2π B
×
m ( t ) cos (ωct )
BPF
r (t )
F {m × r}(ω )
1
Aa
c0 A 2 1
−2ωc
−ωc
ωc
2π B
1
Aa2
2
2ωc
ωc − 2π B
BPF
Figure 16: Modulation of m(t) via even and periodic r(t)
• a1 6= 0; that is Tc is the “least” period of r;
• fc > 2B (to prevent overlapping).
Note that if r(t) is not even, then by (36), the outputted modulated
signal is of the form a1 m(t) cos(ωc t + φ1 ).
46
4.25. Switching modulator : Set r(t) to be the square pulse train given
by (44):
r (t) = 1 [cos ω0 t ≥ 0]
1 2
1
1
1
= +
cos ω0 t − cos 3ω0 t + cos 5ω0 t − cos 7ω0 t + . . . .
2 π
3
5
7
Multiplying this r(t) to the signal m(t) is equivalent to switching m(t) on
and off periodically.
It is equivalent to periodically turning the switch on (letting m(t) pass
through) for half a period Tc = f1c .
186
AMPLITUDE MODULATIONS AND DEMODULATIONS
Figure 4.4
Switching
modulator for
DSB-SC.
m(t )
~
0
I
(a)
J~
w(l)
nnnnnnnnnnnnn
(b)
Figure
17: pulse
Switching
modulator
DSB-SC
[4, whose
Figure
4.4].series was found
The square
train w(t)
in Fig. 4.4b for
is a periodic
signal
Fourier
earlier in Example 2.8 [Eq. (2.86)] as
4.26. Switching Demodulator :
ll' (l) =
~2 + ~ (cos We t - ~ cos 3wct + ~ cos Seve! -
·· ·)
1
LPF{m(t) cos(ωc t) × 1[cos(ωc t) ≥ 0]} = m(t)
π
T he signalm (t)lr (t) is given by
J[
.)
(4 .S )
)
(45)
[4, p 162]. Note that this technique still requires the switching to be in sync
l
[ m (t) cos eve! - -m
I (!) cos 3wct + -::111
l (1) cos Seve ! - · · · ] (4.6)
m (t)w(t) = -m(t)
+as-2 in
with the incoming
cosine
the basic 3DSB-SC.
2
)
J[
The signal m(l) ll '(t) co nsists not onl y of the component 111(1) bu t a lso of an infinite
number of modul ated signals with angul
47 ar frequ encies r»c, 3wc, Seve, .. .. Therefore, the
spectrum of m (1)1-v(t) con sists of multiple copies of the message spectrum M (f), shifted to
0, ±fc, ±~fc , ±Sf~- , .. . (with decreas ing re lative weights), as show n in Fig. 4.4c.
For modulation , we are interested in extracting the modu lated co mponent m.(t) cos We i
on ly. To separate this component from the rest of the crowd, we pass the signal m(t)w(t) through
4.4
Energy and Power
Definition 4.27. For a signal g(t), the instantaneous power p(t) dissipated
in the 1-Ω resister is pg (t) = |g(t)|2 regardless of whether g(t) represents a
voltage or a current. To emphasize the fact that this power is based upon
unity resistance, it is often referred to as the normalized power.
Definition 4.28. The total (normalized) energy of a signal g(t) is given
by
Z +∞
Z +∞
Z T
Eg =
pg (t)dt =
|g(t)|2 dt = lim
|g(t)|2 dt.
−∞
T →∞
−∞
−T
4.29. By the Parseval’s theorem discussed in 2.39, we have
Z ∞
Z ∞
2
Eg =
|g(t)| dt =
|G(f )|2 df.
−∞
−∞
Definition 4.30. The average (normalized) power of a signal g(t) is given
by
ZT /2
Z T
1
1
2
Pg = lim
|g(t)|2 dt.
|g (t)| dt = lim
T →∞ T
T →∞ 2T −T
−T /2
Definition 4.31. To simplify the notation, there are two operators that
used angle brackets to define two frequently-used integrals:
(a) The “time-average” operator:
Z
Z T
1
1 T /2
g (t)dt = lim
g (t)dt
hgi ≡ hg (t)i ≡ lim
T →∞ T −T /2
T →∞ 2T −T
(46)
(b) The inner-product operator:
Z
∞
hg1 , g2 i ≡ hg1 (t) , g2 (t)i =
−∞
4.32. Using the above definition, we may write
• Eg = hg, gi = hG, Gi where G = F {g}
D E
• Pg = |g|2
48
g1 (t)g2∗ (t)dt
(47)
• Parseval’s theorem: hg1 , g2 i = hG1 , G2 i
where G1 = F {g1 } and G2 = F {g2 }
4.33. Time-Averaging over Periodic Signal: For periodic signal g(t) with
period T0 , the time-average operation in (46) can be simplified to
Z
1
hgi =
g (t)dt
T0
T0
where the integration is performed over a period of g.
Example 4.34. hcos (2πf0 t + θ)i =
Similarly, hsin (2πf0 t + θ)i =
Example 4.35. cos2 (2πf0 t + θ) =
Example 4.36. ej(2πf0 t+θ) = hcos (2πf0 t + θ) + j sin (2πf0 t + θ)i
Example 4.37. Suppose g(t) = cej2πf0 t for some (possibly complex-valued)
constant c and (real-valued) frequency f0 . Find Pg .
4.38. When the signal g(t) can be expressed in the form g(t) =
P
ck ej2πfk t
k
and the fk are distinct, then its (average) power can be calculated from
X
Pg =
|ck |2
k
49
Example 4.39. Suppose g(t) = 2ej6πt + 3ej8πt . Find Pg .
Example 4.40. Suppose g(t) = 2ej6πt + 3ej6πt . Find Pg .
Example 4.41. Suppose g(t) = cos (2πf0 t + θ). Find Pg .
Here, there are several ways to calculate Pg . We can simply use Example 4.35. Alternatively, we can first decompose the cosine into complex
exponential functions using the Euler’s formula:
4.42. The (average) power of a sinusoidal signal g(t) = A cos(2πf0 t + θ) is
1 2
|A| ,
f0 6= 0,
Pg = 2 2 2
|A| cos θ, f0 = 0.
This property means any sinusoid with nonzero frequency can be written in
the form
p
g (t) = 2Pg cos (2πf0 t + θ) .
4.43. Extension of 4.42: Consider sinusoids Ak cos (2πfk t + θk ) whose frequencies are positive and distinct. The (average) power of their sum
X
g(t) =
Ak cos (2πfk t + θk )
k
is
Pg =
1X
|Ak |2 .
2
k
50
√ √ Example 4.44. Suppose g (t) = 2 cos 2π 3t + 4 cos 2π 5t . Find Pg .
4.45. For periodic signal g(t) with period T0 , there is also no need to carry
out the limiting operation to find its (average) power Pg . We only need to
find an average carried out over a single period:
Z
1
Pg =
|g (t)|2 dt.
T0
T0
(a) When the corresponding Fourier series expansion g(t) =
∞
P
cn ejnω0 t
n=−∞
is known,
∞
X
Pg =
|ck |2
k=−∞
(b) When the signal g(t) is real-valued and its (compact) trigonometric
∞
P
Fourier series expansion g(t) = c0 + 2
|ck | cos (kω0 t + ∠ck ) is known,
k=1
Pg =
c20
+2
∞
X
|ck |2
k=1
Definition 4.46. Based on Definitions 4.28 and 4.30, we can define three
distinct classes of signals:
(a) If Eg is finite and nonzero, g is referred to as an energy signal.
(b) If Pg is finite and nonzero, g is referred to as a power signal.
(c) Some signals17 are neither energy nor power signals.
• Note that the power signal has infinite energy and an energy signal has
zero average power; thus the two categories are mutually exclusive.
Example 4.47. Rectangular pulse
17
Consider g(t) = t−1/4 1[t0 ,∞) (t), with t0 > 0.
51
Example 4.48. Sinc pulse
Example 4.49. For α > 0, g(t) = Ae−αt 1[0,∞) (t) is an energy signal with
Eg = |A|2 /2α.
Example 4.50. The rotating phasor signal g(t) = Aej(2πf0 t+θ) is a power
signal with Pg = |A|2 .
Example 4.51. The sinusoidal signal g(t) = A cos(2πf0 t + θ) is a power
signal with Pg = |A|2 /2.
4.52. Consider the transmitted signal
x(t) = m(t) cos(2πfc t + θ)
in DSB-SC modulation. Suppose M (f − fc ) and M (f + fc ) do not overlap
(in the frequency domain).
(a) If m(t) is a power signal with power Pm , then the average transmitted
power is
1
Px = Pm .
2
(b) If m(t) is an energy signal with energy Em , then the transmitted energy
is
1
Ex = Em .
2
• Q: Why is the power (or energy) reduced?
√
• Remark: When x(t) = 2m(t) cos(2πfc t + θ) (with no overlapping
between M (f − fc ) and M (f + fc )), we have Px = Pm .
52
4.5
Amplitude modulation: AM
4.53. DSB-SC amplitude modulation (which is summarized in Figure 18)
is easy to understand and analyze in both time and frequency domains.
However, analytical simplicity is not always accompanied by an equivalent
simplicity in practical implementation.
‫݂ ܯ‬
݉ ‫ݐ‬
×
ܺ ݂
‫ݐ ݔ‬
Message
(modulating signal)
݂
2 cos  2 f c t 
െ݂௖
݂௖
݂
Modulator
Figure 18: DSB-SC modulation.
Problem: The (coherent) demodulation of DSB-SC signal requires the
receiver to possess a carrier signal that is synchronized with the incoming
carrier. This requirement is not easy to achieve in practice because the
modulated signal may have traveled hundreds of miles and could even suffer
from some unknown frequency shift.
1
4.54. If a carrier component is transmitted along with the DSB signal,
demodulation can be simplified.
(a) The received carrier component can be extracted using a narrowband
bandpass filter and can be used as the demodulation carrier. (There is
no need to generate a carrier at the receiver.)
(b) If the carrier amplitude is sufficiently large, the need for generating a
demodulation carrier can be completely avoided.
• This will be the focus of this section.
53
Definition 4.55. For AM, the transmitted signal is typically defined as
xAM (t) = (A + m (t)) cos (2πfc t) = A cos (2πfc t) + m (t) cos (2πfc t)
|
{z
} |
{z
}
carrier
sidebands
4.56. Spectrum of xAM (t):
• Basically the same as that of DSB-SC signal except for the two additional impulses (discrete spectral component) at the carrier frequency
±fc .
◦ This is why we say the DSB-SC system is a suppressed carrier
system.
Definition 4.57. Consider a signal A(t) cos(2πfc t). If A(t) varies slowly in
comparison with the sinusoidal carrier cos(2πfc t), then the envelope E(t)
of A(t) cos(2πfc t) is |A(t)|.
4.58. Envelope of AM signal : For AM signal, A(t) = A + m(t) and
E(t) = |A + m(t)| .
See Figure 19.
(a) If ∀t, A(t) > 0, then E(t) = A(t) = A + m(t)
• The envelope has the same shape as m(t).
• We can detect the desired signal m(t) by detecting the envelope
(envelope detection).
(b) If ∃t, A(t) < 0, then E(t) 6= A(t).
• The envelope shape differs from the shape of m(t) because the
negative part of A + m(t) is rectified.
◦ This is referred to as phase reversal and envelope distortion.
54
192
AMPLITUDE MODULATIONS AND DEMODULATIONS
Figure 4.8
AM signal and
its enve lope.
m(1)
~np
A+
m(l)
>0
far all
1
A
tA
t
1~
(b)
+ m(1) ':{> 0
far all
1
}L\------t
(~
1--
En velope
~A+
Envelope
m(l)
lA
+ m(t)l
/
I
I
I
(e)
(d)
19:beAM
signal
its envelope
[5, Fig
4.8] ation of rpAM (t) in
andFigure
m(t) cannot
recovered
fromand
the envelope.
Consequently,
demodul
Fig. 4.8d amounts to sim ple envelope detection. Thus, the condition for envelope detection
of an AM signal is
Definition 4.59. The positive constant
A + m (t)
(4.9a)
for all t
::: 0
(t)|
max
(envelope of the sidebands) max. In|m
thi s case there m
is no
p need
t
tIf m(t) ::: 0 for all/ , then A = 0 already sati sfies condition (4 .9a)
= , and such a
µ ≡ to add any carrier becau se the enve lope of the DSB-S C=signal m(t) cos Wet is rn(t)
max
of the
carrier)
|A|
DSB-SC(envelope
signal ca n be detected
by envelope
detection . In the max
following
discuss ion weA
ass ume
t
t
that 111(t) t_ 0 for all t; that is, m(t) can be negative over some range oft.
is called the modulation
index.
iVIessage Signals
m (t) with Ze1·o Offset:
Let ±mp be the maximum and the minimum
va lues of m(t) , res pecti vely (Fig . 4. 8) . This means that m(t) ::: - mp. Hence, the condition of
envelope detection (4.9a) is equi valent to
• mp ≡ max |m (t)|
t
(4.9b)
◦ By the way mp is defined, the message m(t) is between ±mp .
Thus, the minimum carrier amplitude required for the viability of envelope detection is mp·
This is quite clear from Fig. 4.8. We define the modulation index fJ- as
• The quantity µ × 100% is often referred to as the percent modulation.
fJ-
55
mp
= -
A
(4.10a)
Example 4.60. Consider a sinusoidal (pure-tone) message m(t) = Am cos(2πfm t).
Suppose A = 1. Then, µ = Am . Figure 20 shows the effect of changing the
value modulation index on the modulated signal.
50% Modulation
1.5
0.5
0
−0.5
−1.5
Time
Envelope
Modulated Signal
100% Modulation
2
0
−2
Time
150% Modulation
2.5
0.5
0
−0.5
−2.5
1
Time
Figure 20: Modulated signal in standard AM with sinusoidal message
4.61. It should be noted that the ratio that defines the modulation index
compares the maximum of the two envelopes. In other references, the notation for the AM signal may be different but the idea (and the corresponding
motivation) that defines the modulation index remains the same.
• In [3, p 163], it is assumed that m(t) is already scaled or normalized to
have a magnitude not exceeding unity (|m(t)| ≤ 1) [3, p 163]. There,
xAM (t) = Ac (1 + µm (t)) cos (2πfc t) = Ac cos (2πfc t) + Ac µm (t) cos (2πfc t) .
|
{z
} |
{z
}
carrier
56
sidebands
◦ mp = 1
◦ The modulation index is then
max |Ac µm (t)|
max (envelope of the sidebands)
t
max (envelope of the carrier)
t
=
=
max |Ac |
t
t
|Ac µ|
= µ.
|Ac |
• In [14, p 116],
m (t)
m (t)
cos (2πfc t) = Ac cos (2πfc t) + Ac µ
cos (2πfc t)
xAM (t) = Ac 1 + µ
|
{z
}
mp
mp
.
|
{z
}
carrier
sidebands
◦ The modulation index is then
max (envelope of the sidebands)
t
max (envelope of the carrier)
t
=
max Ac µ m(t)
mp t
max |Ac |
m
=
t
|Ac | µ mpp
|Ac |
= µ.
4.62. Power of the transmitted signals.
(a) In DSB-SC system, recall, from 4.52, that, when
x(t) = m(t) cos(2πfc t)
with fc sufficiently large, we have
1
Px = Pm .
2
Therefore, all transmitted power are in the sidebands which contain
message information.
(b) In AM system,
xAM (t) = A cos (2πfc t) + m (t) cos (2πfc t) .
{z
} |
{z
}
|
carrier
sidebands
If we assume that the average of m(t) is 0 (no DC component), then the
spectrum of the sidebands m(t) cos(2πfc t+θ) and the carrier A cos(2πfc t+
θ) are non-overlapping in the frequency domain. Hence, when fc is sufficiently large
1
1
Px = A2 + Pm .
2
2
57
• Efficiency:
• For high power efficiency, we want small
m2p
µ2 Pm .
◦ By definition, |m(t)| ≤ mp . Therefore,
m2p
Pm
≥ 1.
◦ Want µ to be large. However, when µ > 1, we have phase
reversal. So, the largest value of µ is 1.
◦ The best power efficiency we can achieved is then 50%.
• Conclusion: at least 50% (and often close to 2/3[3, p. 176]) of
the total transmitted power resides in the carrier part which is
independent of m(t) and thus conveys no message information.
4.63. An AM signal can be demodulated using the same coherent demodulation technique that was used for DSB. However, the use of coherent
demodulation negates the advantage of AM
• Note that, conceptually, the received AM signal is the same as DSBSC signal except that the m(t) in the DSB-SC signal is replaced by
A(t) = A + m(t). We also assume that A is large enough so that
A(t) ≥ 0.
• Recall the key equation of switching demodulator (45):
LPF{A(t) cos(2πfc t) × 1[cos(2πfc t) ≥ 0]} =
1
A(t)
π
(48)
We noted before that this technique requires the switching to be in
sync with the incoming cosine.
58
4.64. Demodulation of AM Signals via rectifier detector: The receiver
will first recover A + m(t) and then remove A.
• When ∀t, A(t) ≥ 0, we can replace the switching demodulator by
the rectifier demodulator/detector . In which case, we suppress
the negative part of A(t) cos(2πfc t) using a diode (half-wave rectifier:
HWR).
◦ Here, we define a HWR to be a memoryless device whose inputoutput relationship is described by a function fHWR (·):
x, x ≥ 0,
fHWR (x) =
0, x < 0.
• This is mathematically equivalent to a switching demodulator in (45)
and (48).
• It is in effect synchronous detection performed without using a local
carrier [4, p 167].
• This method needs A(t) ≥ 0 so that the sign of A(t) cos(2πfc t) will be
the same as the sign of cos(2πfc t).
• The dc term
m(t)/π.
A
π
may be blocked by a capacitor to give the desired output
59
6
AMPLITUDE MODULATIONS AND DEMODULATIONS
gure 4.10
/[A
VR(t)
[a+ m(t)] cos wet
ctifier detector
AM.
-_f
+ m(t)]
I
"
rr
[A
+ 111(1)]
'
-·
'
[A
I
-;-[A
+ m(1)]
~
/
+ m(l)]
Low-pass
cos wet
filter
Figure 21: Rectifier detector for AM [5, Fig. 4.10].
signal is multiplied by w(t). Hence, the half-wave rectified output vR(t) is
VR(t) ={[A+ m(t)] COS Wet) w(t)
Figure 4.11
Envelope
detector for AM.
=[A+ m(t)] cos Wet [
4 .4 Bandw idth-Efficient Amplitude Modulati ons
197
~ + ~ (cos (Vet- ~cos 3wet + ~cos Swet- · · ·)]
c
AM signal
l
= -[A+ m(t)] +other terms of higher frequencies
(4.12)
(4.13)
(4.14)
][
When vR(t) is applied to a low-pass filter of cutoff
(a) B Hz, the output is [A+ m(t)]jn, and all the
other term s of frequencies higher than B Hz are suppressed. The de
termde tector
Ajnoutput
may be blocked
Envelope
by a capac itor (Fig. 4.10) toRCgive
the
desired
output
m(t)
j
n.
The
outp
ut
can
be doubled by
too large
\
using a full-wave
rectifi er.
····· K' f<K~
-- ~. .
Envelop~.--· ... ·· · ' ( I"' I""
It is interesting to note that because of the multip lication with ll '(l), rectifier detection is in
i" " !'--The high carrier content
effect synchronous detection performed without using a local carrier.
in AM ensures that its zero crossings are periodic and the informatio n abo ut frequency and
phase of the carrier at the transmitter is built in to the AM signal itself.
, .,
~-~ .
,. ·<
K
~·· · ·~"
W'~
Envelope Detector: fn an enve lope detector, the o utput of the detector follows the
envelope of the modulated signal. The simpl e circuit show n in Fig. 4. lla functions as an
envelope detector. On the positive cycle
.... of the input signa l, the input grows and may exceed
...
-·· '
the charged vo ltage on the capacity vc(t), turning on the diode and allow ing the capac itor C
..···
to charge up to the peak voltage
of the input signal cycle. As the input
signal fall s below this
·...
·· .. ..
peak value, it falls quickly below the capacitor (b)
voltage (which is very nearly
······ the peak voltage),
thus caus ing the diode to open. The capacitor no w di scharges through the resi stor R at a slow
rate (with a time constant RC). During the next positive cycle, the same drama repeats . As the
Figure 22:
Envelope
detector nfor
AM
Fig.
4.11].rippl e signal of
a declining envelope.
Capacitor
d ischarge
positi
ve [5,
peaks
ca uses
input signal
rises above the capacitor
voltage, betwee
the diode
conducts
again.aThe
capacitor again
freque ncy We in the output. Thi s rip ple can be reduced by choosing a larger time constant
charges to RC
thesopeak
value of this (new) cycle. The capacitor discharges slowly during the cutoff
that the capac ito r disc harges ve ry littl e between the positive peaks (RC » I /eve) . If
period. RC were made too large, however, it wo uld be imposs ible for the capac itor voltage to follow
60charges
a fast
declining
e nvelope
b). Because
the maxup
imum
AM voltage
envelope dec
During
each
positive
cycle,(Fig.
the4.11
capacitor
to rate
the of
peak
of line
the input
do minated
the ba ndw
idththe
B of
the positive
message cycle,
sig nal mas
(r )shown
, the desin
ignFig.
criterion
of Thus,
RC
signal andisthen
decaysbyslowly
until
next
4 . ll b.
the
should be
output voltage vc(t), close ly follows the (rising) envelope of the input AM signal. Equally
important, the slow capacity discharge via the resistor R a ll ows the
capacity vo ltage to follow
I
4.65. Demodulation of AM signal via envelope detector :
• Design criterion of RC:
2πB 1
2πfc .
RC
• The envelope detector output is A + m(t) with a ripple of frequency fc .
• The dc term can be blocked out by a capacitor or a simple RC high-pass
filter.
• The ripple may be reduced further by another (low-pass) RC filter.
4.66. AM Trade-offs:
(a) Disadvantages:
• Higher power and hence higher cost required at the transmitter
• The carrier component is wasted power as far as information transfer is concerned.
• Bad for power-limited applications.
(b) Advantages:
• Coherent reference is not needed for demodulation.
• Demodulator (receiver) becomes simple and inexpensive.
• For broadcast system such as commercial radio (with a huge number of receivers for each transmitter),
◦ any cost saving at the receiver is multiplied by the number of
receiver units.
◦ it is more economical to have one expensive high-power transmitter and simpler, less expensive receivers.
(c) Conclusion: Broadcasting systems tend to favor the trade-off by migrating cost from the (many) receivers to the (fewer) transmitters.
4.67. References: [3, p 198–199], [5, Section 4.3] and [13, Section 3.1.2].
61
4.6
Quadrature Amplitude Modulation (QAM)
4.68. We are now going to define a quantity called the “bandwidth” of a
signal. Unfortunately, in practice, there isn’t just one definition of bandwidth.
Definition 4.69. The bandwidth (BW) of a signal is usually calculated
from the differences between two frequencies (called the bandwidth limits).
Let’s consider the following definitions of bandwidth for real-valued signals
[3, p 173]
(a) Absolute bandwidth: Use the highest frequency and the lowest frequency in the positive-f part of the signal’s nonzero magnitude spectrum.
• This uses the frequency range where 100% of the energy is confined.
• We can speak of absolute bandwidth if we have ideal filters and
unlimited time signals.
(b) 3-dB bandwidth (half-power bandwidth): Use the frequencies
where the signal power starts to decrease by 3 dB (1/2).
√
• The magnitude is reduced by a factor of 1/ 2.
(c) Null-to-null bandwidth: Use the signal spectrum’s first set of zero
crossings.
(d) Occupied bandwidth: Consider the frequency range in which X%
(for example, 99%) of the energy is contained in the signal’s bandwidth.
(e) Relative power spectrum bandwidth: the level of power outside
the bandwidth limits is reduced to some value relative to its maximum
level.
• Usually specified in negative decibels (dB).
• For example, consider a 200-kHz-BW broadcast signal with a maximum carrier power of 1000 watts and relative power spectrum
bandwidth of -40 dB (i.e., 1/10,000). We would expect the station’s power emission to not exceed 0.1 W outside of fc ± 100 kHz.
62
Example 4.70. Message bandwidth and the transmitted signal bandwidth
4.71. Rough Approximation: If g1 (t) and g2 (t) have bandwidths B1 and
B2 Hz, respectively, the bandwidth of g1 (t)g2 (t) is B1 + B2 Hz.
This result follows from the application of the width property18 of convolution19 to the convolution-in-frequency property.
Consequently, if the bandwidth of g(t) is B Hz, then the bandwidth of
g 2 (t) is 2B Hz, and the bandwidth of g n (t) is nB Hz. We mentioned this
property in 2.38.
4.72. BW Inefficiency in DSB-SC system: Recall that for real-valued baseband signal m(t), the conjugate symmetry property from 2.28 says that
M (−f ) = (M (f ))∗ .
The DSB spectrum has two sidebands: the upper sideband (USB) and the
lower sideband (LSB), both containing complete information about the baseband signal m(t). As a result, DSB signals occupy twice the bandwidth
required for the baseband.
4.73. To improve the spectral efficiency of amplitude modulation, there
exist two basic schemes to either utilize or remove the spectral redundancy:
(a) Single-sideband (SSB) modulation, which removes either the LSB or
the USB so that for one message signal m(t), there is only a bandwidth
of B Hz.
(b) Quadrature amplitude modulation (QAM), which utilizes spectral redundancy by sending two messages over the same bandwidth of 2B
Hz.
We will only discussed QAM here. SSB discussion can be found in [3, Sec
4.4], [13, Section 3.1.3] and [4, Section 4.5].
18
19
This property states that the width of x ∗ y is the sum of the widths of x and y.
The width property of convolution does not hold in some pathological cases. See [4, p 98].
63
Definition 4.74. In quadrature amplitude modulation (QAM ) or
quadrature multiplexing , two baseband real-valued signals m1 (t) and
m2 (t) are transmitted simultaneously via the corresponding QAM signal:
√
√
xQAM (t) = m1 (t) 2 cos (2πfc t) + m2 (t) 2 sin (2πfc t) .
Transmitter (modulator) m1  t  Receiver (demodulator)  
2 cos  2 f c t   2 2 sin  2 f c t  m2  t  v1  t  H LP  f  m̂1  t  H LP  f  m̂2  t  2 cos  2 f c t  xQAM  t  h t  y t   2 Channel 2 sin  2 f c t   
v2  t  Figure 23: QAM Scheme
• QAM operates by transmitting two DSB signals via carriers of the same
frequency but in phase quadrature.
• Both modulated signals simultaneously occupy the same frequency
band.
• The “cos” (upper) channel is also known as the in-phase (I ) channel
and the “sin” (lower) channel is the quadrature (Q) channel.
4.75. Demodulation: The two baseband signals can be separated at the
receiver by synchronous detection:
n
o
√
LPF xQAM (t) 2 cos (2πfc t) = m1 (t)
n
o
√
LPF xQAM (t) 2 sin (2πfc t) = m2 (t)
64
• m1 (t) and m2 (t) can be separately demodulated.
4.76. Sinusoidal form (envelope-and-phase description [3, p. 165]):
√
xQAM (t) = 2E(t) cos(2πfc t + θ(t)),
where
q
envelope: E(t) = m21 (t) + m22 (t)
m
(t)
2
phase: θ(t) = − tan−1
m1 (t)
• The envelope is defined as nonnegative. Negative “amplitudes” can be
absorbed in the phase by adding ±180◦ .
4.77. Complex form:
xQAM (t) =
√
2Re (m(t)) ej2πfc t
where20 m(t) = m1 (t) − jm2 (t).
• We refer to m(t) as the complex envelope (or complex baseband
signal ) and the signals m1 (t) and m2 (t) are known as the in-phase
and quadrature(-phase) components of xQAM (t).
• The term “quadrature component” refers to the fact that it is in phase
quadrature (π/2 out of phase) with respect to the in-phase component.
• Key equation:






 n
√ j2πf t o √ −j2πf t 
c
LPF
Re m (t) × 2e c
= m (t) .
×
2e




|
{z
}


x(t)
20
If we use − sin(2πfc t) instead of sin(2πfc t) for m2 (t) to modulate,
√
√
xQAM (t) = m1 (t) 2 cos (2πfc t) − m2 (t) 2 sin (2πfc t)
√
= 2 Re m (t) ej2πfc t
where
m(t) = m1 (t) + jm2 (t).
65
4.78. Three equivalent ways of saying exactly the same thing:
(a) the complex-valued envelope m(t) complex-modulates the complex carrier ej2πfc t ,
• So, now you can understand what we mean when we say that a
complex-valued signal is transmitted.
(b) the real-valued amplitude E(t) and phase θ(t) real-modulate the amplitude and phase of the real carrier cos(2πfc t),
(c) the in-phase signal m1 (t) and quadrature signal m2 (t) real-modulate
the real in-phase carrier cos(2πfc t) and the real quadrature carrier
sin(2πfc t).
4.79. References: [3, p 164–166, 302–303], [13, Sect. 2.9.4], [4, Sect. 4.4],
and [8, Sect. 1.4.1]
4.80. Question: In engineering and applied science, measured signals are
real. Why should real measurable effects be represented by complex signals?
Answer: One complex signal (or channel) can carry information about
two real signals (or two real channels), and the algebra and geometry of
analyzing these two real signals as if they were one complex signal brings
economies and insights that would not otherwise emerge. [8, p. 3 ]
66
4.7
Suppressed-Sideband Amplitude Modulation
4.81. The upper and lower sidebands of DSB are uniquely related by symmetry about the carrier frequency, so either one contains all the message
information. Hence, transmission bandwidth can be cut in half if one sideband is suppressed along with the carrier.
Definition 4.82. Conceptually, in single-sideband modulation (SSB),
a sideband filter suppresses one sideband before transmission. [3, p 185–186]
(a) If the filter removes the lower sideband, the output spectrum consists
of the upper sideband alone.
(b) If the filter removes the upper sideband, the output spectrum consists
of the lower sideband alone.
Definition 4.83. In vestigial-sideband modulation (VSB), one sideband is passed almost completely while just a trace, or vestige, of the other
sideband is included. [3, p 191–192]
4.84. In (analog) television video transmission, an AM wave is applied to
a vestigial sideband filter. This modulation scheme is called VSB plus
carrier (VSB + C). [3, p 193]
• The unsuppressed carrier allows for envelope detection, as in AM
◦ Distortionless envelope modulation actually requires symmetric sidebands, but VSB + C can deliver a fair approximation.
67
Sirindhorn International Institute of Technology
Thammasat University
School of Information, Computer and Communication Technology
ECS332 2015/1
5
Part II.3
Dr.Prapun
Angle Modulation: FM and PM
5.1. We mentioned in 4.1 that a sinusoidal carrier signal
A cos(2πfc t + φ)
has three basic parameters: amplitude, frequency, and phase. Varying these
parameters in proportion to the baseband signal results in amplitude modulation (AM), frequency modulation (FM), and phase modulation (PM),
respectively.
5.2. As usual, we will again assume that the baseband signal m(t) is bandlimited to B; that is, |M (f )| = 0 for |f | > B.
As in the AM section, we will also assume that
|m(t)| ≤ mp .
In other words, m(t) is bounded between −mp and mp .
68
Definition 5.3. Phase modulation (PM ):
xPM (t) = A cos (2πfc t + φ + kp m (t))
Definition 5.4. The main characteristic21 of frequency modulation (FM)
is that the carrier frequency f (t) would be varied with time so that
f (t) = fc + km(t),
(49)
where k is an arbitrary constant.
• The arbitrary constant k is sometimes denoted by kf to distinguish it
from a similar constant in PM.
Example 5.5. With a sinusoidal message signal in Figure 24a, the frequency
deviation of the FM modulator output in Figure 24d is proportional to
m(t). Thus, the (instantaneous) frequency of the FM modulator output is
maximum when m(t) is maximum
and minimum when
m(t) is minimum.
4.1 Phase and Frequency Modulation Defined
159
Figure 24: Different modulations of sinusoidal message signal. (a) Message signal. (b)
Unmodulated carrier. (c) Output of phase
modulator (d) Output of frequency modulator [14, Fig 4.2 p 159 ]
(a)
(b)
(c)
(d)
Figure 4.2
The phase deviation of the PM output is proportional to m(t). However,
because the phase is varied continuously, it is not straightforward (yet) to
Angle modulation with sinusoidal messsage signal. (a) Message signal. (b) Unmodulated carrier. (c)
Output of phase modulator with 𝑚(𝑡). (d) Output of frequency modulator with 𝑚(𝑡).
quadrature with the carrier component, whereas for AM they are not. This will be illustrated in
21
Treat
Example
4.1. this as a practical definition. The more rigorous definition will be provided in 5.15.
The generation of narrowband angle modulation is easily accomplished using the method
shown in Figure 4.3. The switch allows for the generation of either narrowband FM or narrow-
69
(.)dt
m(t)
2 π fd
FM
(t)
Ac
×
Σ
xc(t)
see how Figure 24c is related to m(t). In Example 5.18, we will come back
to this example and re-analyze the PM output.
158
Example 5.6. Figure 25 illustrates the outputs of PM and FM modulators
Chapter 4when
∙ Angle Modulation
and Multiplexing
the message
is a unit-step function.
Figure
4.125:
Figure
m(t)
1
t
t0
(a)
Comparison of PM and FM mod-
Comparison of PM and FM modulator
ulator outputs for a unit-step input. (a) Mesoutputs for a unit-step input.
sage
signal.
Unmodulated carrier. (c)
(a)
Message
signal. (b)
(b) Unmodulated
carrier.
Phase modulator
output (d) Frequency modPhase(c)modulator
output
1
=
𝜋).
(d)
Frequency
modulator
(𝑘
𝑝
ulator
output. [14, Fig 4.1 p 158]
2
output.
t
t0
(b)
t
t0
(c)
t
Frequency = fc
t0
(d)
Frequency = fc + fd
where Re(⋅) implies that the real part of the argument is to be taken. Expanding 𝑒𝑗𝜙(𝑡) in a
power
yieldsPM modulator output,
• series
For the
{ [
]
}
𝜙2 (𝑡)
𝑗2𝜋𝑓𝑐 𝑡
(𝑡) = Re 𝐴𝑐 1 + 𝑗𝜙(𝑡) −
−⋯ 𝑒
◦ the𝑥𝑐(instantaneous)
frequency
is fc for both t < t0(4.11)
and
2!
t > t0
If the peak phase deviation is small, so that the maximum value of |𝜙(𝑡)| is much less than
◦ the phase of the unmodulated carrier is advanced by kp = π2 radians
unity, the modulated carrier can be approximated as
for t > t0 giving rise𝑗2𝜋𝑓
to𝑐 𝑡 a signal𝑗2𝜋𝑓
that
is discontinuous at t = t0 .
𝑐 𝑡]
+ 𝐴𝑐 𝜙(𝑡)𝑗𝑒
𝑥𝑐 (𝑡) ≅ Re[𝐴𝑐 𝑒
Taking
the real
partFM
yieldsmodulator
• For
the
output,
𝑥𝑐 (𝑡) ≅ 𝐴𝑐 cos(2𝜋𝑓𝑐 𝑡) − 𝐴𝑐 𝜙(𝑡) sin(2𝜋𝑓𝑐 𝑡)
(4.12)
◦ the frequency is fx for t < t0 , and the frequency is fc + fd for t > t0
The form of (4.12) is reminiscent of AM. The modulator output contains a carrier com◦ phase-shifted carrier. The
ponent and
term phase
in whichis,
a function
of 𝑚(𝑡)
multiplies a 90at
◦ athe
however,
continuous
t = t0 .
first term yields a carrier component. The second term generates a pair of sidebands. Thus,
if 𝜙(𝑡) has a bandwidth 𝑊 , the bandwidth of a narrowband angle modulator output is 2𝑊 .
70
The important difference between AM and angle modulation
is that the sidebands are produced by multiplication of the message-bearing signal, 𝜙 (𝑡), with a carrier that is in phase
CISE 5.1–1
CHAPTER 5
•
Angle CW Modulation
Modulating
signal
AM
FM
PM
Figure 5.1–2
Illustrative AM, FM, and PM waveforms.
Figure 26: Illustrative AM, FM, and PM waveforms. [3, Fig 5.1-2 p 212]
carrier amplitude, we modulate the frequency by swinging it over a range of, say,
Example
Figure
26 illustrates
the
outputs
of regardless
AM, FM,ofand
PM mod50 Hz, 5.7.
then the
transmission
bandwidth
will
be 100 Hz
the message
bandwidth.
we’ll
soon see,
argument has
a serious
flaw, for it ignores the disulators
whenAs
the
message
is this
a triangular
(ramp)
pulse.
tinction between instantaneous and spectral frequency. Carson (1922) recognized
To
about FM, we notion
will first
to know
it actually
the understand
fallacy of themore
bandwidth-reduction
andneed
cleared
the air what
on that
score.
means
to
vary
the
frequency
of
a
sinusoid.
Unfortunately, he and many others also felt that exponential modulation had no
advantages over linear modulation with respect to noise. It took some time to overthis belief but, thanks to Armstrong (1936), the merits of exponential modula5.1come
Instantaneous
Frequency
tion were finally appreciated. Before we can understand them quantitatively, we
must address
theThe
problem
of spectral analysis.
Definition
5.8.
generalized
sinusoidal signal is a signal of the form
=analogy
A cos (θ(t))
Suppose FM had been defined inx(t)
direct
to AM by writing xc(t) Ac cos vc(t)(50)
t
called
vc[1 mx(t)].
Demonstrate theangle.
physical impossibility of this definition by
withθ(t)
vc(t) is
where
the generalized
finding f(t) when x(t) cos vmt.
• The generalized angle for conventional sinusoid is 2πfc t + φ.
• In [3, p 208], θ(t) of the form 2πfc t + φ(t) is called the total instantaneous angle.
Narrowband PM and FM
Definition
5.9.
If θ(t)
in (50) contains
the starts
message
m(t), we
Our spectral
analysis
of exponential
modulation
with information
the quadrature-carrier
have
a process
may be termed angle modulation.
version
of Eq. that
(1), namely
xci 1t2 cos vct xcq
1t2 sinisvconstant.
• The amplitude of xan
wave
c 1t2angle-modulated
ct
(9)
•where
Another name for this process is exponential modulation.
1 2
xci 1t 2 Ac cos f1t 2 Ac c 1 f 1t2 p d
2!
71
(10)
◦ The motivation for this name is clear when we write x(t) as ARe ejθ(t) .
◦ It also emphasizes the nonlinear relationship between x(t) and
m(t).
• Since exponential modulation is a nonlinear process, the modulated
wave x(t) does not resemble the message waveform m(t).
5.10. Suppose we want the frequency fc of a carrier A cos(2πfc t) to vary
with time as in (49). It is tempting to consider the signal
A cos(2πg(t)t),
(51)
where g(t) is the desired frequency at time t.
Example 5.11. Consider the generalized sinusoid signal of the form 51
above with g(t) = t2 . We want to find its frequency at t = 2.
(a) Suppose we guess that its frequency at time t should be g(t). Then,
at time t = 2, its frequency should be t2 = 4. However, when compared with cos
(2π(4)t) in Figure 27a, around t = 2, the “frequency”
of cos(2π t2 t) is quite different from the 4-Hz cosine approximation.
Therefore, 4 Hz is too low to be the frequency of cos(2π t2 t) around
t = 2.
(a)
(b)
Figure 27: Approximating the frequency of cos(2π (t2 ) t) by (a) cos (2π(4)t) and (b)
cos (2π(12)t).
72
1
(b) Alternatively, around t = 2, Figure 27b shows that cos (2π(12)t) seems
to provide a good approximation. So, 12 Hz would be a better answer.
Definition 5.12. For generalized sinusoid A cos(θ(t)), the instantaneous
frequency 22 at time t is given by
1 d
θ(t).
2π dt
Example 5.13. For the signal cos(2π t2 t) in Example 5.11,
θ (t) = 2π t2 t
f (t) =
(52)
and the instantaneous frequency is
f (t) =
1 d
1 d
θ (t) =
2π t2 t = 3t2 .
2π dt
2π dt
In particular, f (2) = 3 × 22 = 12.
5.14. The instantaneous frequency formula (52) implies
Z t
Z t
θ(t) = 2π
f (τ )dτ = θ(t0 ) + 2π
f (τ )dτ.
−∞
5.2
(53)
t0
FM and PM
Definition 5.15. Frequency modulation (FM ):


Zt
xFM (t) = A cos 2πfc t + φ + 2πkf
m (τ )dτ  .
(54)
−∞
The instantaneous frequency is given by
f (t) = fc + kf m (t) .
22
Although f (t) is measured in hertz, it should not be equated with spectral frequency. Spectral frequency
f is the independent variable of the frequency domain, whereas instantaneous frequency f (t) is a timedependent property of waveforms with exponential modulation.
73
5.16. Phase modulation (PM ): The phase-modulated signal is defined
in Definition 5.3 to be
xPM (t) = A cos (2πfc t + φ + kp m (t))
Its instantaneous frequency is
(55)
Therefore, the instantaneous frequency of the output of the PM modulator is
• maximum when the slope of m(t) is maximum and
• minimum when the slope of m(t) is minimum.
Example 5.17. Sketch FM and PM waves for the modulating signal m(t)
shown in 28a.
Figure 28: FM and PM waveforms generated from the same message.
This “indirect” method of sketching xP M (t) (using ṁ(t) to frequencymodulate a carrier) works as long as m(t) is a continuous signal. If m(t)
is discontinuous, this indirect method fails at points of discontinuities. In
such a case, a direct approach should be used to specify the sudden phase
changes.
74
(b)
Figure 29: xPM (t) and the corresponding m(t).
(c)
Example 5.18. Consider xPM (t) in Example 5.5. It is copied here in Figure
29 along with the corresponding message m(t) which generates it.
5.19. Relationship between
FM and PM:
(d)
Figure
4.2
• Equation
(54) implies that one can produce frequency-modulated signal
Angle modulation with sinusoidal messsage signal. (a) Message signal. (b) Unmodulated carrier. (c)
phasewith
modulator.
Outputfrom
of phaseamodulator
𝑚(𝑡). (d) Output of frequency modulator with 𝑚(𝑡).
• Equation
(55)component,
implies whereas
that one
produce
phase-modulated
quadrature
with the carrier
for AMcan
they are
not. This will
be illustrated in
Example
4.1. a frequency modulator.
from
The generation of narrowband angle modulation is easily accomplished using the method
shown
in Figure
The switch allowsabove
for the generation
of either narrowband
FM or narrow• The
two4.3.observations
are summarized
in Figure
30.
(.)dt
2 π fd
Frequency
modulator
t
m(t)
m (t )
∫
∫ m (τ )dτ
−∞
FM
(t)
×
PM
Phase
Modulator
−sin ω c t
kp
Figure 4.3
Carrier
oscillator
d
()
( ) of narrowband
Generation
angle modulation.
m′ t
m t
dt
Σ
xFM ( t )
cos ωct
90° phase
shifter
Frequency
Modulator
signal
Figure 30: With the help
of integrating and difAc
ferentiating
networks, a
xc(t)
phase modulator can produce frequency modulation and vice versa [4, Fig
5.2 p 255].
xPM ( t )
Phase modulator
• By looking at an angle-modulated signal x(t), there is no way of telling
whether it is FM or PM.
◦ Compare Figure 24c and 24d in Example 5.5.
◦ In fact, it is meaning less to ask an angle-modulated wave whether
it is FM or PM. It is analogous to asking a married man with
children whether he is a father or a son. [5, p 255]
75
5.20. Generalized angle modulation (or exponential modulation):
x(t) = A cos (2πfc t + θ0 + (m ∗ h)(t))
where h is causal.
(a) Frequency modulation (FM ): h(t) = 2πkf 1[t ≥ 0]
(b) Phase modulation (PM ): h(t) = kp δ(t).
5.21. So far, we have spoken rather loosely of amplitude and phase modulation. If we modulate two real signals a(t) and φ(t) onto a cosine to produce
the real signal x(t) = a(t) cos(ωc t + φ(t)), then this language seems unambiguous: we would say the respective signals amplitude- and phase-modulate
the cosine. But is it really unambiguous?
The following example suggests that the question deserves thought.
Example 5.22. [8, p 15] Let’s look at a “purely amplitude-modulated”
signal
x1 (t) = a(t) cos(ωc t).
Assuming that a(t) is bounded such that 0 ≤ a(t) ≤ A, there is a welldefined function
1
θ(t) = cos−1
x1 (t) − ωc t.
A
Observe that the signal
x2 (t) = A cos (ωc t + θ(t))
is exactly the same as x1 (t) but x2 (t) looks like a “purely phase-modulated”
signal.
5.23. Example 5.22 shows that, for a given real signal x(t), the factorization
x(t) = a(t) cos(ωc t + φ(t)) is not unique. In fact, there is an infinite number
of ways for x(t) to be factored into “amplitude” and “phase”.
76
5.3
Bandwidth of FM Signals
5.24. FM: The “Holy Grail” Technique for BW Saving?
In the 1920s, the idea of frequency modulation (FM) was naively proposed
very early as a method to conserve the radio spectrum. The argument was
presented as follows:
• If m(t) is bounded between −mp and mp , then the maximum and minimum values of the (instantaneous) carrier frequency would be fc + kmp
and fc − kmp , respectively. (Think of this as a delta function shifting
to various location between fc + kmp and fc − kmp in the frequency
domain.)
• Hence, the spectral components would remain within this band with a
bandwidth 2kmp centered at fc .
• Conclusion: By using an arbitrarily small k, we could make the information bandwidth arbitrarily small (much smaller than the bandwidth
of m(t).
In 1922, Carson argued that this is an ill-considered plan. We will illustrate
his reasoning later. In fact, experimental results shows that
As a result of his observation, FM temporarily fell out of favor.
5.25. Armstrong (1936) reawakened interest in FM when he realized it
had a much different property that was desirable. When the kf is large, the
inverse mapping from the modulated waveform xFM (t) back to the signal
m(t) is much less sensitive to additive noise in the received signal than is
the case for amplitude modulation. FM then came to be preferred to AM
because of its higher fidelity. [1, p 5-6]
77
Finding the “bandwidth” of FM Signals turns out to be a difficult task.
Here we present a few approximation techniques.
5.26. First, from 5.20, we see that both FM and PM can be viewed as
x(t) = A cos (2πfc t + θ0 + φ(t))
(56)
where φ(t) = (m ∗ h)(t) if h(t) is selected properly.
The Fourier transform of φ(t) is Φ(f ) = M (f ) ∗ H(f ). So, if M (f ) is
band-limited to B, we know that Φ(f ) is also band-limited to B as well.
Now, let us rewrite (56) as
n
o
n
o
j(2πfc t+θ0 +φ(t))
j(2πfc t+θ0 ) jφ(t)
x(t) = A Re e
= A Re e
e
Recall that if φ(t) is band-limited to B, then φn (t) is band-limited to nB.
So, we can make a rough sketch of |X(f )| as follows
So,we conclude that the absolute bandwidth would be infinite.
5.27. When φ(t) is small,
78
• The “approximated” expression of x(t) is similar to AM.
◦ The modulator output contains a carrier component and a term in
which a function of m(t) multiplies a 90◦ phase-shifted carrier.
◦ The first term yields a carrier component. The second term generates a pair of sidebands. Thus, if φ(t) has a bandwidth B, the
bandwidth of x(t) is 2B.
• The important difference between AM and angle modulation is that
the sidebands are produced by multiplication of the message-bearing
signal, φ(t), with a carrier that is in phase quadrature with the carrier
component, whereas for AM they are not.
• For larger values of |φ(t)| the terms φ2 (t), φ3 (t), . . . cannot be ignored
and will increase the bandwidth of x(t).
• Recall, from (29) that
F
−−
*
−
−
g(t) cos(2πfc t + φ) )
−1
F
1 jφ
e G(f − fc ) + e−jφ G(f + fc ) .
2
Therefore, when
x (t) ≈ A cos (2πfc t + θ0 ) − Aφ (t) cos (2πfc t + θ0 − 90◦ ) ,
we have
A jθ0
−jθ0
j(θ0 −90◦ )
−j(θ0 −90◦ )
X (f ) ≈
e δ(f − fc ) + e
δ(f + fc ) − e
Φ(f − fc ) − e
Φ(f + fc )
2
A jθ0
e δ(f − fc ) + e−jθ0 δ(f + fc ) + jejθ0 Φ(f − fc ) − je−jθ0 Φ(f + fc ) .
=
2
5.28. For potentially wideband m(t), here, we present a technique to
roughly estimate the bandwidth of xF M (t).
To do this, we consider m(t) that is a piecewise constant function (also
known as step function or staircase function); this implies that the instantaneous frequency f (t) = fc + kf m(t) of xFM (t) is also piecewise constant.
79
For example, we can consider the transmitted signal xFM(t) constructed
from five different tones. Its instantaneous frequency is increased from f1
to f5 .
Assume that each tone lasts Ts = R1s [s] where Rs is called the “(symbol)
rate” of the data transmission. Increasing the value of Rs reduces the time
to complete the transmission.
Recall that the Fourier transform of a cosine contains simply (two shifted
and scaled) delta functions at the (plus and minus) frequency of the cosine.
However, recall also that when we consider the cosine pulse, which is timelimited, its Fourier transform contains (two) sinc functions. In particular,
the cosine pulse
cos (2πf0 t) , t1 ≤ t < t2 ,
p (t) =
0,
otherwise,
can be viewed as the pure cosine function cos (2πf0 t) multiplied by a rectangular pulse r (t) = 1 [t1 ≤ t < t2 ]. By (28), we know that multiplication
by cos (2πf0 t) will shift the spectrum R(f ) of the rectangular pulse to ±fc
and scaled its values by a factor of 21 :
1
1
P (f ) = R (f − f0 ) + R (f + f0 )
2
2
where the Fourier transform23 R(f ) of the rectangular pulse is given by
R (f ) = (t2 − t1 ) e−jπf (t1 +t2 ) sinc (πf (t2 − t1 )) .
See Figure 31 for an example.
23
To get this, first consider the rectangular
width
pulse oft2 −t
t2 − t1 centered at t = 0. From (13), the
1
1
corresponding Fourier transform is 2 t2 −t
sinc
2π
f . Finally, by time-shifting the rectangular
2
2
t2 −t1
t2 +t1
pulse in the time domain by
, we simply multiply the Fourier transform by e−2πf ( 2 ) in the
2
frequency domain.
80
Cos Pulse
cos 2
100
0,
,
0.5
0.6,
otherwise.
1
Figure 31: Cosine pulse
and its spectrum which
contains two sinc functions at ± freqeuncy of
the cosine (which is 100
Hz in the figure). When
the pulse only lasts for
a short time period, the
sinc pulses in the frequency domain are wide.
x(t)
0.5
0
-0.5
-1
0
0.1
0.2
0.3
0.4
0.5
t [s]
0.6
0.7
0.8
0.9
1
0.05
|X(f)|
0.04
0.03
0.02
0.01
0
-200
-150
-100
-50
0
f [Hz]
50
100
150
200
When m(t) is piecewise constant, xFM (t) is a sum of cosine pulses. Therefore, its spectrum X(f ) will be the sum of the sinc functions centered at the
frequencies of the pulses as shown in Figure 32.
100 Hz
cos 2
200 Hz
cos 2
300 Hz
cos 2
400 Hz
cos 2
500 Hz
cos 2
1
0.5
0
-0.5
-1
0
0.05
0.1
0.15
0.2
0.25
Seconds
0.03
Magnitude
1
0.02
0.01
0
-1000
-800
-600
-400
-200
0
200
Frequency [Hz]
400
600
800
1000
1
Figure 32: A digital version of FM: xFM (t) and the corresponding XFM (f ).
• X(f ) extends to ±∞. It is not band-limited.
• One may approximate its bandwidth by assuming that “most” of the
energy in the sinc function is contained in its main lobe which is at
± T1s = ±Rs from its peak. Therefore, the bandwidth of xFM (t) becomes
81
Sirindhorn International Institute of Technology
Thammasat University
School of Information, Computer and Communication Technology
ECS332 2015/1
6
Part III.1
Dr.Prapun
Sampling, Reconstruction, and Pulse Modulation
6.1
Sampling
Definition 6.1. Sampling is the process of taking a (sufficient) number of
discrete values of points on a waveform that will define the shape of wave
form.
• The signal is sampled at a uniform rate, once every Ts seconds.
m[n] = m(nTs ) = m(t)|t=nTs .
• We refer to Ts as the sampling period, and to its reciprocal fs = 1/Ts
as the sampling rate.
• The reverse process is called “reconstruction”.
6.2. Sampling = loss of information? If not, how can we recover the original
waveform back.
• The more samples you take, the more accurately you can define a waveform.
• Obviously, if the sampling rate is too low, you may experience distortion
(aliasing).
82
• The sampling theorem, to be discussed in the section, says that when
the waveform is band-limited, if the sampling rate is fast enough, we can
reconstruct the waveform back and hence there is no loss of information.
◦ This allows us to replace a continuous time signal by a discrete
sequence of numbers.
◦ Processing a continuous time signal is therefore equivalent to processing a discrete sequence of numbers.
◦ In the field of communication, the transmission of a continuous
time message reduces to the transmission of a sequence of numbers.
Example 6.3. Mathematical functions are frequently displayed as continuous curves, even though a finite number of discrete points was used to
construct the graphs. If these points, or samples, have sufficiently close
spacing, a smooth curve drawn through them allows us to interpolate intermediate values to any reasonable degree of accuracy. It can therefore be
said that the continuous curve is adequately described by the sample points
alone.
Example 6.4. Plot y = x2 .
Example 6.5. Plot g(t) = sin(100πt).
(See slides.)
Theorem 6.6. Sampling Theorem: In order to (correctly and completely) represent an analog signal, the sampling frequency, fs , must be
at least twice the highest frequency component of the analog signal.
Example 6.7. In example 6.5, the frequency of the sine wave is 50 Hz.
Therefore, we need the sampling frequency to be at least 100.
Example 6.8. Suppose the sampling frequency is 200 samples/sec. The
analog signal should not have the frequency higher than 100 Hz.
(See slides)
83
Definition 6.9.
(a) Given a sampling frequency, fs , the Nyquist frequency is defined as
fs /2.
(b) Given the highest (positive-)frequency component fmax of an analog
signal,
(i) the Nyquist sampling rate is 2fmax and
(ii) the Nyquist sampling interval is 1/(2fmax ).
6.10. Much more can be said about the result of performing the sampling
process on a signal. Here we will use g(t) to denote the signal under consideration. You may replace g(t) below by m(t) if you want to think of it as
an analog message to be transmitted by a communication system. We use
g(t) here because the results provided here work in broader setting as well.
Definition 6.11. In ideal sampling, the (ideal instantaneous) sampled
signal is represented by a train of impulses whose area equal the instantaneous sampled values of the signal
gδ (t) =
∞
X
g [n]δ (t − nTs ) .
n=−∞
6.12. The Fourier transform Gδ (f ) of gδ (t) can be found by first rewriting
gδ (t) as
gδ (t) =
∞
X
g (nTs )δ (t − nTs ) =
n=−∞
= g (t)
∞
X
g (t)δ (t − nTs )
n=−∞
∞
X
δ (t − nTs ).
n=−∞
Multiplication in the time domain corresponds to convolution in the frequency domain. Therefore,
( ∞
)
X
Gδ (f ) = F {gδ (t)} = G (f ) ∗ F
δ (t − nTs ) .
n=−∞
84
For the last term, the Fourier transform can be found by applying what we
found in Example 4.1324 :
∞
∞
X
X
F
−−
*
−
− fs
δ (f − kfs ).
δ (t − nTs ) )
−1
F
n=−∞
k=−∞
This gives
∞
X
Gδ (f ) = G (f ) ∗ fs
δ (f − kfs ) = fs
∞
X
G (f ) ∗ δ (f − kfs ).
k=−∞
k=−∞
Hence, we conclude that
gδ (t) =
∞
X
n=−∞
F
−
*
)
−
− Gδ (f ) = fs
g [n]δ (t − nTs ) −
−1
F
∞
X
G (f − kfs ).
(57)
k=−∞
6.13. As usual, we will assume that the signal g(t) is band-limited to B
Hz ((G(f ) = 0 for |f | > B)). In which case, the Fourier transform of the
sampled signal is given by
6.14. Remarks:
(a) Gδ (f ) is “periodic” (in the frequency domain) with “period” fs .
• So, it is sufficient to look at Gδ (f ) between ± f2s
(b) The MATLAB script plotspect that we have been using to visualize
magnitude spectrum also relies on sampled signal. Its frequency domain
plot is between ± f2s .
(c) Although this sampling technique is “ideal” because it involves the use
of the δ-function. We can extract many useful conclusions.
(d) One can also study the discrete-time Fourier transform (DTFT) to look
at the frequency representation of the sampled signal.
24
We also considered an easy-to-remember pair and discuss how to extend it to the general case in 4.14.
85
6.2
Reconstruction
6.15. From (57), we see that when the sampling frequency fs is large
enough, the replicas of G(f ) will not overlap in the frequency domain. In
such case, the original G(f ) is still intact and we can use a low-pass filter
with gain Ts to recover g(t) back from gδ (t).
6.16. To prevent aliasing (the corruption of the original signal because its
replicas overlaps in the frequency domain), we need
Theorem 6.17. A low-pass signal g whose spectrum is band-limited to
B Hz (G(f ) = 0 for |f | > B) can be reconstructed (interpolated) exactly
(without any error) from its sample taken uniformly at a rate (sampling
frequency/rate) fs > 2B Hz (samples per second).[5, p 302]
6.18. Ideal Reconstruction: Continue from 6.15. Assuming that fs >
2B, the low-pass filter that we should use to extract g(t) from Gδ (t) should
be

|f | ≤ B,

B < |f | < fs − B,
HLP (f ) =

|f | ≥ fs − B,
In particular, for “brick-wall” LPF, the cutoff frequency fcutoff should be
between B and fs − B.
6.19. Reconstruction Equation: Suppose we use
quency for our “brick-wall” LPF in 6.18,
fs
2
as the cutoff fre-
The impulse response of the LPF is hLP (t) = sinc 2π f2s t = sinc(πfs t).
86
The output of the LPF is
∞
X
gr (t) = gδ (t) ∗ hLP (t) =
!
g [n]δ (t − nTs )
∗ hLP (t)
n=−∞
=
∞
X
g [n]hLP (t − nTs ) =
∞
X
g [n] sinc (πfs (t − nTs )) .
n=−∞
n=−∞
When fs > 2B, this output will be exactly the same as g(t):
∞
X
g (t) =
g [n] sinc (πfs (t − nTs ))
(58)
n=−∞
• This formula allows perfect reconstruction the original continuous-time
function from the samples.
• At the sampling instants t = nTs , all sinc functions are zero at these
times save one, and that one yields g(nTs ) which is the correct values.
• Note that at time t between the sampling instants, g(t) is interpolated
by summing the contributions from all the sinc functions.
• The LPF is often called an interpolation filter, and its impulse response
is called the interpolation function.
Example 6.20.
Figure 33: Application of the reconstruction
equation
g[n] = g(nTs)
8
6
4
2
0
1
2
3
4
5
6
7
8
5
6
7
8
n
8
6
gr(t)
4
2
0
-2
1
2
3
4
t [T ]
s
87
Theorem 6.21. Sampling theorem for uniform periodic sampling: If
a signal g(t) contains no frequency components for |f | ≥ B, it is completely
described by instantaneous sample values uniformly spaced in time with
1
sampling period Ts ≤ 2B
. In which case, g(t) can be exactly reconstructed
from its samples (. . . , g[−2], g[−1], g[0], g[1], g[2], . . .) by the reconstruction
equation (58).
Example 6.22. We now return to the sampling of the cosine function (sinusoid).
Figure 34: Reconstruction of the
signal g(t) = cos(2π(2)t) by its
samples g[n]. The upper plot uses
Ts = 0.4. The lower plot uses
Ts = 0.2.
2
g[n]
g(t)
1
0
-1
-2
-1.5
-1
-0.5
0
t
0.5
1
1.5
2
1
g[n]
g(t)
0.5
0
-0.5
-1
-2
1
-1.5
-1
-0.5
0
t
0.5
1
1.5
2
6.23. Remarks:
• Need a lot of g[n] for the reconstruction.
• Practical signals are time-limited.
◦ Filter the message as much as possible before sampling.
6.24. The possibility of fs = 2B:
• If the spectrum G(f ) has no impulse (or its derivatives) at the highest
frequency B, then the overlap is still zero as long as the sampling rate
is greater than or equal to the Nyquist rate, that is, fs ≥ 2B.
• If G(f ) contains an impulse at the highest frequency ±B, then fs = 2B
would cause overlap. In such case, the sampling rate fs must be greater
than 2B Hz.
88
Theorem 6.21. Sampling theorem for uniform periodic sampling: If
a signal g(t) contains no frequency components for |f | ≥ B, it is completely
described by instantaneous sample values uniformly spaced in time with
1
sampling period Ts ≤ 2B
. In which case, g(t) can be exactly reconstructed
from its samples (. . . , g[−2], g[−1], g[0], g[1], g[2], . . .) by the reconstruction
equation (58).
Example 6.22. We now return to the sampling of the cosine function (sinusoid).
Figure 34: Reconstruction of the
signal g(t) = cos(2π(2)t) by its
samples g[n]. The upper plot uses
Ts = 0.4. The lower plot uses
Ts = 0.2.
2
g[n]
g(t)
1
0
-1
-2
-1.5
-1
-0.5
0
t
0.5
1
1.5
2
1
g[n]
g(t)
0.5
0
-0.5
-1
-2
1
-1.5
-1
-0.5
0
t
0.5
1
1.5
2
6.23. Remarks:
• Need a lot of g[n] for the reconstruction.
• Practical signals are time-limited.
◦ Filter the message as much as possible before sampling.
6.24. The possibility of fs = 2B:
• If the spectrum G(f ) has no impulse (or its derivatives) at the highest
frequency B, then the overlap is still zero as long as the sampling rate
is greater than or equal to the Nyquist rate, that is, fs ≥ 2B.
• If G(f ) contains an impulse at the highest frequency ±B, then fs = 2B
would cause overlap. In such case, the sampling rate fs must be greater
than 2B Hz.
88
Example 6.25. Consider a sinusoid g(t) = sin (2π(B)t). This signal is
bandlimited to B Hz, but all its samples are zero when uniformly taken at
a rate fs = 2B, and g(t) cannot be recovered from its (Nyquist) samples.
Thus, for sinusoids, the condition of fs > 2B must be satisfied.
Let’s check with our formula (57) for Gδ (f ). First, recall that
ejx − e−jx
1
1
sin x =
= ejx − e−jx .
2j
2j
2j
Therefore,
g (t) = sin (2π (B) t) =
1
1
1
1 j2π(B)t
e
− e−j2π(B)t = ej2π(B)t − ej2π(−B)t
2j
2j
2j
2j
and
Note that G(f ) is pure imaginary. So, it is more suitable to look at the
plot of its imaginary part. (We do not look at its magnitude plot because
the information about the sign is lost. We also do not consider the real part
because we know that it is 0.)
89
6.26. A maximum of 2B independent pieces (samples/symbols) of information per second can be transmitted, errorfree, over a noiseless channel of
bandwidth B Hz [4, p 260].
• Start with 2B pieces of information per second. Denote the sequence
of such information by mn .
1
• Construct a signal m(t) whose (Nyquist) sample values m[n] = m n 2B
agrees with mn by the reconstruction equation (58).
6.27. A bandpass signal whose spectrum exists over a frequency band
fc − B2 < |f | < fc + B2 has a bandwidth B Hz. Such a signal is also
uniquely determined by samples taken at above the Nyquist frequency 2B.
The sampling theorem is generally more complex in such case. It uses two
interlaced sampling trains, each at a rate of fs > B samples per second
(known as second-order sampling). [5, p 304]
90
6.3
Analog Pulse Modulation
In Section 6.1 we saw that continuous bandlimited signals can be represented
by a sequence of discrete samples. Moreover, in Section 6.2, we saw that
the continuous signal can be reconstructed if the sampling rate is sufficiently
high.
6.28. Because the sequence m[n] completely contains the information about
m(t), in this section, instead of trying to send m(t), we consider transmitting
the message in the form of pulse modulation.
Definition 6.29. In analog pulse modulation, some attribute of a pulse
varies continuously in one-to-one correspondence with a sample value.
• Example of a pulse:
• Three attributes can be readily varied: amplitude, width, and position.
• These lead to pulse-amplitude modulation (PAM), pulse-width modulation (PWM), and pulse-position modulation (PPM) as illustrated in
Figure 35.
Definition 6.30. Unmodulated pulse train:
∞
P
p (t − nTs )
n=−∞
Definition 6.31. In Pulse-Amplitude Modulation (PAM), the sample
values modulate the amplitude of a pulse train:
xPAM (t) =
∞
X
m [n] p (t − nTs )
n=−∞
91
Example 6.32.
6.33. One advantage of using pulse modulation is that it permits the simultaneous transmission of several signals on a time-sharing basis.
• When a pulse-modulated signal occupies only a part of the channel
time, we can transmit several pulse-modulated signals on the same
channel by interweaving them.
• One User: TDM (time division multiplexing).
◦ Transmit/multiplex multiple streams of information simultaneously.
• Multiple Users: TDMA (time division multiple access).
6.34. Frequency-Domain Analysis of PAM:
xPAM (t) =
∞
X
m [n] p (t − nTs ) =
m [n] p (t) ∗ δ (t − nTs )
n=−∞
n=−∞
= p (t) ∗
∞
X
∞
X
!
m [n] δ (t − nTs )
= p (t) ∗ mδ (t)
n=−∞
Therefore,
XPAM (f ) = P (f ) Mδ (f ) .
6.35. Figure 35 compares different types of analog pulse modulation.
Definition 6.36. Pulse-Width Modulation (PWM): A PWM waveform
consists of a sequence of pulses with the width of the nth pulse is proportional to the value of m[n].
92
3. 5
Anal og Pul se Mo dul a ti o n
183
Figure
Figure
3.56 35: Illustration
Illustration
PPM. of PAM, PWM, and
of PAM, PWM, and
PPM.
PWM
signal
PPM
signal
0
T,
2T,
9Ts
• Seldom used in modern communications systems.
attribu te of a pu lse ca n take o n a certa in va lue from a set of allowabl e va lu es. In thi s section we
exam in e analog pul se mod ul ation . In the fo ll ow ing secti o n we exa mi ne a co up le of examp les of
dig ital pul se mod ulation.
As me ntioned, analog pulse modularion res ults whe n some attribute of a pu lse va ries
conti nuou sly in o ne-to-o ne cotTespondence w it h a sa mp le value . T hree attributes can be
readi ly varied: a mplitude, wid th , and position . T hese lead to pu lse ampl itude mod ul ation
(PAM), pul se-w idt h mod ul ati o n (PWM), and pulse-positio n mod ul atio n (PPM) as ill ustrated
in Figure 3.56.
• Used extensively for DC motor control in which motor speed is proportional to the width of the pulses . Since the pulses have equal amplitude,
the energy in a given pulse is proportional to the pulse width.
Definition 6.37. Pulse-Position Modulation (PPM): A PPM signal
3.5
.1 Pulse-Amplitude
M odu lation
consists
of a sequence
of pulses in which the pulse displacement from a specAified
PAM wavefo
consists of a seque
of fl at-topped pul sesto
des ignat
sample va lues.
The
timerm reference
isnce
proportional
theingsample
values
of the informationamp li tude of each pulse corresponds to the value of the message s ig nal m( t ) a t the leading edge
signal.
ofbearing
the pulse. The
essential di fference betwee n PAM a nd the sampli ng ope rati o n discussed in the
previous chap te r is that in PAM we allow the sampling pulse to have fi nite w idth . The fin itewidth pulse can be generated from the imp ul se-train sampling fun ctio n by passing the impulsetrain samples thro ugh a holding circ uit as shown in Figure 3.57. The impulse response of the
ideal hold ing circuit is given by
• Have a number of applications in the area of ultra-wideband communications.
6.38. Pulse-modulation scheme are really baseband coding schemes, and
they yield baseband signal.
93
7
Inter-symbol Interference and Pulse Shaping
7.1. Recall
that, in Pulse-Amplitude
Modulation (PAM), we start with
PAM:
Pulse
Amplitude Modulation
a sequence of numbers
· · · , m[−3], m[−2], m[−1], m[0], m[1], m[2], m[3], · · ·
as shown in Figure 36.
m[-4]
-4T
m[-2]
-3T
-2T
m[-1]
m[0]
m[1]
-T
T
m[3]
2T
3T
m[2]
m[-3]
Figure 36: Sequence of
Numbers for PAM
m[4]
t
4T
• These numbers may come from sampling a continuous-time signal m(t).
Alternatively, it may directly represent (digital) information that intrinsically available in discrete-time.
1
• Because the m[n] may not come from sampling, we call each m[n] a
symbol.
We use these numbers to modify (modulate) the height (amplitude) of a
pulse train. A single pulse is denoted by p(t). This pulse occurs every T
seconds.
PAM: Pulse Amplitude Modulation
Pulse
p t 
-.5T
m[-4]
-4T
m[-2]
-3T
-2T
m[-1]
Figure 37: PAM
1
t
.5T
m[0]
-T
m[1]
T
m[-3]
m[3]
2T
3T
m[2]
4T
t
The PAM signal is then
x (t) =
∞
X
m [n] p (t − nT )
n=−∞
1
94
This signal is transmitted via the communication channel which usually
corrupts it. At the receiver, the received signal is y(t). A more advanced
receiver would try to first cancel the effect of the channel. However, for
simplicity, let’s assume that our receiver simply samples y(t) every T seconds
to get
y[n] = y(t)|t=nT
and we will take this to be the estimate m̃[n] of our m[n].
• If m[n] is the sampled version of m(t), then at the receiver, after we
recover m[n], we can reconstruct m(t) by using the reconstructing equation (58).
Because our assumed receiver is so simple, we are going to also assume25
that y(t) = x(t).
7.2. In this section, our goal is to design a “good” pulse p(t) that satisfies
two important properties
(a) m̃[n] = m[n] for all n. Under our assumptions above, this means we
want x[n] ≡ x(nT ) = m[n] for all n.
(b) P (f ) is band-limited and hence X(f ) is band-limited.
We will first give examples of “poor” p(t).
Example 7.3. Let’s consider the rectangular pulse used in Figure 37:
p(t) = 1[|t| ≤ T /2].
(a) m̃[n] = m[n] for all n.
(b) The Fourier transform of the rectangular pulse is a sinc function. So,
it is not band-limited.
Example 7.4 (Slide). Let’s try a wider rectangular pulse:
p(t) = 1[|t| ≤ 1.5T ].
Here, we face a problem called inter-symbol interference (ISI) in our
sequence m̃[n] at the receiver. The pulses are too wide; they interfere with
other pulses at the sampling time instants (decision making instants), making m̃[n] 6= m[n].
25
Alternatively, we may assume that there is an earlier part of the receiver that (perfectly) eliminates
the effect of the channel for us.
95
Example 7.5 (slide). p(t) = 1[|t| ≤ T /4].
• When the pulse p(t) is narrower than T , we know that the pulses in
PAM signal will not overlap and therefore we won’t have any ISI problem.
Example 7.6 (slides).
p t 
3
4
1
p t 
3
4
t
1
p t 
-T
T
t -2T
1
-T
T
2T
t
Figure 38: Examples of pulses that do not cause ISI.
• Even when the pulses are wider than T , if they do not interfere with
other pulses at the sampling time instants (decision making instants),
we can still have no ISI.
7.7. We can now conclude that a “good” pulse satisfying condition (a)
in 7.2 must not cause inter-symbol interference (ISI): at the receiver,
the nth symbol m̃[n] should not be affected by the preceding or succeeding
transmitted symbol m[k], k 6= n. This requirement means that a “good”
pulse should have the following property:
1, n = 0,
p (t)|t=nT =
(59)
0, n 6= 0.
Combining this with condition (b) in 7.2, we then want “band-limited pulses
specially shaped to avoid ISI (by satisfying (59))” [3, p 506].
7.8. An obvious choice for such p(t) would be the sinc function that we
used in the reconstruction equation (58):
96
Recall Figure 33, repeated here (with modified labels) as Figure 39.
Figure 39: Using the sinc pulse in
PAM
8
m[n]
6
4
2
0
1
2
3
4
5
6
7
8
5
6
7
8
n
8
6
x(t)
4
2
0
-2
1
2
3
4
t [T ]
s
Practically, there are problems that force us to seek better pulse shape.
(a) Infinite duration
(b) Steep slope at each 0-intercept.
(c) max {x (t)} could be a lot larger than max {m [n]}.
t
n
Figure 40: Using the sinc pulse in
PAM can cause high peak.
2
m[n]
1
0
-1
-2
2
4
6
8
10
12
14
16
10
12
14
16
n
3
2
x(t)
1
0
-1
-2
2
4
6
8
t [T ]
s
97
7.9. Because the sinc function may not be a good choice, we now have to
consider other pulses that are band-limited and also satisfy (59). To check
that a signal is band-limited, we need to look in the frequency domain.
However, condition (59) is specified in the time domain. Therefore, we will
try to translate condition (59) into a requirement in the frequency domain.
7.10. Note that condition (59) considers p (t)|t=nT which can be thought of
as the samples p[n] of the pulse p(t) where the sampling period is Ts = T .
Recall, from (57), that
∞
∞
X
X
F
−−
*
−
− Gδ (f ) = fs
G (f − kfs ).
g [n]δ (t − nTs ) )
gδ (t) =
−1
F
n=−∞
k=−∞
Therefore,
∞
X
∞
1 X
k
−−
*
−
− Pδ (f ) =
pδ (t) =
p [n]δ (t − nT ) )
P f−
.
F −1
T
T
n=−∞
F
(60)
k=−∞
On the LHS, by condition (59), the only nonzero term in the sum is the
one with n = 0. Therefore, condition (59) is equivalent to pδ (t) = δ(t).
F
−−
*
−
− 1. Therefore, we must have Pδ (f ) ≡ 1.
However, recall that δ(t) )
F −1
Hence, to check condition (59), we can equivalently check that the RHS of
(60) must be ≡ 1.
Note that Pδ (f ) is “periodic” (in the freq. domain) with “period” T1 .
(Recall that Gδ (f ) is “periodic” (in the freq. domain) with “period” fs .)
Therefore, the checking does not need to be performed across all frequency
1
f . We only need to focus on one period: |f | ≤ 2T
.
This observation is formally stated as the “Nyquist’s criterion” below.
7.11. Nyquist’s (first) Criterion for Zero ISI: A pulse p(t) whose
Fourier transform P (f ) satisfies the criterion
∞
X
k
1
P f−
≡ T,
|f | ≤
(61)
T
2T
k=−∞
has sample values satisfying condition (59):
1, n = 0,
p[n] = p (t)|t=nT =
0, n =
6 0.
98
• Using this pulse, there will be no ISI in the sample values of
y (t) =
∞
X
m [n] p (t − nT )
n=−∞
Definition 7.12. A pulse p(t) is a Nyquist pulse if its Fourier transform
P (f ) satisfies (61) above.
Example 7.13. We know that the sinc pulse we used in Example 7.8 works
(causing no ISI). Let’s check it with the Nyquist’s criterion:
Example 7.14.
Example 7.15.
99
Example 7.16.
Example 7.17. An important family of Nyquist pulses is called the raised
cosine family. Its Fourier transform is given by

0 ≤ |f | ≤ 1−α
 T,
2T
T
πT
1−α
1−α
1+α
PRC (f ; α) =
1
+
cos
|f
|
−
,
≤
|f
|
≤
α
2T
2T
2T
 2
0,
|f | ≥ 1+α
2T
with a parameter α called the roll-off factor.
Figure 41: Raised cosine
pulse (in the frequency
domain)
T
2
f
1
2
1
2
Figure 42: Raised cosine pulse (in the frequency domain) with different values of the rolloff factor
0
0.5
1
T
2
1
2
1
2
3
4
100
1
f
Figure 43: Raised cosine
pulse (in the time domain) with different values of the roll-off factor
1
1
2
0.5
0
t
2
x t  

 m  n p  t  nT 
n 
2
1.5
1
a)
p  t   pRC  t ;0 
0.5
0
-0.5
-1
-1.5
-2
0
0.5
1
1.5
b)
p  t   pRC  t ;0.5 
2
2.5
3
3.5
2
2.5
3
3.5
2
2.5
3
3.5
t
1.5
1
0.5
0
-0.5
-1
-1.5
0
0.5
1
1.5
t
1.5
c)
p  t   pRC  t ;1
1
0.5
0
-0.5
-1
-1.5
0
0.5
1
1.5
t
1
Figure 44: Using the raised cosine pulses in PAM
101
8
8.1
Introduction to Digital Communications
Digitization and PCM
8.1. Generally, analog signals are continuous in time and in range (amplitude); that is, they have values at every time instant, and their values can
be anything within the range. On the other hand, digital signals exist only
at discrete points of time, and their amplitude can take on only finitely (or
countably) many values.
8.2. The analog-to-digital (A/D) converter or ADC enables digital
communication systems to convey analog source signals such as audio and
video.
8.3. Suppose we want to convey an analog message m(t) from a source to
our destination. We now have many options.
(a) Use m(t) to modulate a carrier A cos(2πfc t) using AM, FM, or PM
techniques studied earlier.
(b) Sample the continuous-time message m(t) to get a discrete-time message m[n].
Note that m[n] is a sequence of numbers. (There are uncountably many
possibilities for these numbers).
(i) Send m[n] using analog pulse modulation techniques (PAM, PWM,
PPM).
(ii) Quantize m[n] into mq [n] which has finitely (or countably) many
levels.
i. Send mq [n] using pulse modulation techniques (PAM, PWM,
PPM).
ii. Pulse Code Modulation (PCM): Convert mq [n] into binary
sequence. Then use two basic pulses to represent 1 and 0.
102
Digitization (analog to digital)
Figure 45: An overview
of digitization (sampling
+ quantizing) and PCM
Vertical lines are
used for sampling
111 111
111
110
101
101
100
011
010
100
100
100
011
010
001
001
000
001
000
Horizontal lines are
used for quantization
Time
100111111100001000010100011001
1
Definition 8.4. Through quantization, each sample is approximated, or
“rounded off,” to the nearest quantized level [5, p 320] or quantum level
[3, p 545] (permissible number).
• This process introduces permanent errors that appear at the receiver
as quantization noise in the reconstructed signal.
Example 8.5. Simple quantizer: Suppose amplitudes of the message signal
m(t) lie in the range (−mp , mp ). A simple quantizer may partition the
signal range into L intervals. Each sample amplitude is approximated by
the midpoint of the interval in which the sample value falls. Each sample is
now represented by one of the L numbers.
• Such a signal is known as an L-ary digital signal.
8.6. From practical viewpoint, a binary digital signal (a signal that can
take on only two values) is very desirable because of its simplicity, economy,
and ease of engineering. We can convert an L-ary signal into a binary signal
by using pulse coding.
• A binary digit is called a bit.
• L = 2` levels cam be mapped into (represented by) ` bits.
103
Example 8.7. Suppose L = 8. The binary code can be formed by the
binary representation of the 8 decimal digits from 0 to 7.
Example 8.8. Telephone (speech) signal:
• The components above 3.4 kHz are eliminated by a low-pass filter.
◦ For speech, subjective tests show that signal intelligibility is not
affected if all the components above 3.4 kHz are suppressed.
• The resulting signal is then sampled at a rate of 8,000 samples per
second (8 kHz).
◦ This rate is intentionally kept higher than the Nyquist sampling
rate of 6.8 kHz so that realizable filters can be applied for signal
reconstruction.
• Each sample is finally quantized into 256 levels (L = 256), which requires eight bits to encode each sample (28 = 256).
[5, p 320]
Example 8.9. Compact disc (CD) audio signal:
• High-fidelity: Require the audio signal bandwidth to be 20 kHz.
• The sampling rate of 44.1 kHz is used.
• The signal is quantized into L = 65, 536 of quantization levels, each of
which is represented by 16 bits (16-bit two’s complement integer) to
reduce the quantizing error.
[5, p 321]
104
8.2
Digital PAM Signals
8.10. Digital message representation (at baseband) commonly takes the
form of an amplitude-modulated pulse train. We express such signals by
writing
X
x(t) =
m[n]p(t − nTs )
n
where the modulating amplitude m[n] represents the nth symbol in the
message sequence.
• The amplitudes belong to a set A of M discrete values called the alphabet.
8.11. Note that Ts does not necessarily equal the pulse duration but rather
the pulse-to-pulse interval or the time allotted to one symbol. Thus, the
signaling rate (or symbol rate) is Rs = T1s . measured in symbols per
second, or baud.
• In the special but important case of binary signaling (M = 2), we write
Ts = Tb for the bit duration and the bit rate is Rb = T1b .
8.12. Figure 46 depicts various PAM formats or line codes for the binary
message 10110100, taking rectangular pulses for clarity.
(a) The simple on-off waveform in Figure 46a represents each 0 by an “off”
pulse and each 1 by an “on” pulse.
(i) In the a return-to-zero (RZ) format, the pulse duration is smaller
than Tb after which the signal return to the zero level.
(ii) A nonreturn-to-zero (NRZ) format has “on” pulses for full bit
duration Tb .
(b) The polar signal in Figure 46b has opposite polarity pulses
• Its DC component will be zero if the message contains 1s and 0s
in equal proportion.
(c) Figure 46c, we have bipolar signal where successive 1s are represented
by pulses with alternating polarity.
• Also known as pseudo-trinary or alternate mark inversion (AMI)
105
(d) The split-phase Manchester format in Figure 46d represents 1s with
a positive half-interval pulse followed by a negative half-interval pulse,
and vice versa for the representation of 0s.
• Also called twinned binary.
• Guarantee zero DC component regardless of the message sequence.
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Page 483
(e) Figure 46e shows a quaternary signal derived by grouping
the mesConfirming
Pages
sage bits in blocks of two and using four amplitude levels to prepresent
the four possible combinations 00, 01, 10, and 11.
• Quaternary coding can be generalized to M-ary coding in which
blocks of n message bits are represented by an M-level waveform
11.1 Digital Signals and Systems
483
with M = 2n .
0
1
1
1
0
1
0
0
RZ
NRZ
A
(a)
t
0
Tb
Tb
Figure 46: Line codes with
rectangular pulses: (a) unipolar RZ and NRZ; (b) polar RZ and NRZ; (c) bipolar
NRZ; (d) split-phase Manchester; (e) polar quaternary
NRZ.
A/2
(b)
t
0
– A/2
1
0
1
1
0
1
0
0
A
(c)
t
0
–A
A/2
(d)
t
0
– A/2
1
0
1
1
0
1
0
0
3A/2
A/2
t
0
(e)
– A/2
Ts
– 3A/2
Figure 11.1–1
Binary PAM formats with rectangular pulses: (a) unipolar RZ and NRZ; (b)
106 Manchester; (e) polar
polar RZ and NRZ; (c) bipolar NRZ; (d) split-phase
quaternary NRZ.
Finally, Fig. 11.1–1e shows a quaternary signal derived by grouping the mes-
8.3
Digital PAM with Noise
8.13. In this section, the transmitted signal is still in the form of digital
PAM as in the previous subsection:
X
m[n]p(t − nTs )
x(t) =
n
However, here, we also consider the effect of additive noise. Therefore, the
received signal is
y(t) = x(t) + N (t)
PAM with Noise
where N (t) is a random noise process.
m[0] = 1
1
x(t)
0.5
0
-0.5
-1
m[1] = -1
0
1
2
3
m[2] = -1
4
5
6
7
8
9
5
6
7
8
9
t [T]
y(t)
5
0
-5
0
1
2
3
4
t [T]
Figure 47: Digital PAM with Noise
1
8.14. Note that
• The noise N (t) is random.
• The message m[n] should be random (at least from the perspective of
the receiver; if the receiver had known in advance the value of m[n],
there would have been no point of transmitting m[n]).
107
◦ This makes x(t) and y(t) random.
To emphasize the randomness in the signals under consideration, we
sometimes write M [n], X(t), and Y (t) using capital letters26 .
8.15. Simple receiver: For simplicity, let’s assume that our receiver simply samples y(t) every T seconds to get
y[n] = y(t)|t=nT .
When the alphabet A contains only two symbols of opposite sign (A =
{−a, a}, where a > 0), the decoded value m̂[n] of our m[n] can be found by
a, y [n] ≥ 0,
m̂ [n] =
−a, y [n] < 0.
Here we use “0” as the threshold level/value. Turn out that this middle
point is the optimal threshold to use when the two possible symbol values
from the alphabet are equally likely.
Example 8.16. In Figure 47, A = {−1, 1}.
n
0
1
2
3
4
5
6
7
8
9
m[n] 1
-1
-1
1
-1
1
1
1
1
-1
y[n] 1.69 -0.27 -0.81 4.20 1.58 1.04 -0.34 0.35 2.19 -1.51
m̂[n] 1
-1
-1
1
1
1
-1
1
1
-1
Note that the value of the noise N (t) at t = 4T is too positive. Even when
“-1” was transmitted, the received value is 1.58 which exceeds 0. Therefore,
we get an error at n = 4.
Similarly that the value of the noise N (t) at t = 6T is too negative. Even
when “1” was transmitted, the received value is −0.34 which is lower than
0. Therefore, we get an error at n = 6.
Among the ten symbols sent in this example, there are two symbol errors.
Therefore, the symbol error rate (SER) or symbol error probability is
2/10.
Because there are two symbols in the alphabet, each symbol transmission
conveys 1 bit. Hence, the bit error rate (BER) or bit error probability is
also 2/10.
26
Caution: Here, capital letters represent random variables/processes. In earlier sections, we used capital
letter to represent Fourier transform. However, we won’t talk about Fourier transform here; so confusion
can be avoided.
108
8.17. Additive White Gaussian Noise: At each time instant t, the
noise N (t) is usually modeled by a Gaussian random variable with mean 0
and standard deviation σN ,
1
−1
fN (t) (n) = √
e 2
2πσN
n
σN
2
.
Furthermore, the “white” part means that the noise values at different time
instants are independent.
Definition 8.18. In general, a Gaussian (normal) random variable X
with mean m and standard deviation σ is characterized by its probability
density function (PDF):
2
1
− 21 ( x−m
)
σ
fX (x) = √
e
.
2πσ
To talk about such X, we usually write X ∼ N (m, σ 2 ). Probability involving X can be evaluated by
Z
P [X ∈ A] =
fX (x)dx.
A
In particular,
Z
b
P [X ∈ [a, b]] =
fX (x)dx = FX (b) − FX (a)
a
Rx
where FX (x) = −∞ fX (t)dt is called the cumulative distribution function
(CDF) of X.
We usually express probability involving Gaussian random variable via
the Q function which is defined by
Z∞
Q (z) =
z
x2
1
√ e− 2 dx.
2π
Note that Q(z) is the same as P [Z > z] where Z ∼ N (0, 1); that is Q (z)
is the probability of the “tail” of N (0, 1).
It can be shown that
• Q is a decreasing function
109
N ( 0,1)
1
0.9
0.8
Q(z)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
-3
z
-2
-1
0
1
2
3
z
Figure 48: Q-function
• Q (0) =
1
2
• Q (−z) = 1 − Q (z)
◦ This is useful for converting the argument of the Q function to
positive value.
• For X ∼ N (m, σ 2 ),
c−m
P [X > c] = Q
.
σ
2
8.19. Three important noise probabilities for N ∼ N (0, σN
):
P [N > c] =
, P [N < c] =
, P [a < N < b] =
Note that all strict inequalities above can also be replaced by the ones
that also include equalities because the noise is a continuous random variable
and hence including one particular noise value does not change probability.
8.20. For the simple receiver in 8.15, suppose N (t) ∼ N (0, σ 2 ).
(a) When a “−a” was transmitted, error occurs when N (t) > a
(b) When a “+a” was transmitted, error occurs when N (t) < −a
110
A formula that connects these events with the (combined) error probability is called the Total Probability Theorem: If a (finite or infinitely)
countable collection of events {B1 , B2 , . . .} is a partition of Ω, then
X
P (A) =
P (A|Bi )P (Bi ).
(62)
i
In particular,
P (A) = P (A|B)P (B) + P (A|B c )P (B c ).
Here, we replace event A by the error event E. Event B is defined to be
the event that the transmitted symbol is “a”. The error probability is then
Example 8.21. In a digital PAM system, equally-likely symbols are selected
from an alphabet set A = {−4, 4}. The pulse used in the transmitted signal
is a Nyquist pulse. The additive noise at each particular time instant is
Gaussian with mean 0 and standard deviation 2.
(a) Find the probability that the received signal at a particular time is > 6.
(b) Find the symbol error probability when 0 is used as the threshold level
for the decoding decision at the receiver.
111