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II - Baseband pulse transmission∗
1 Introduction
We discuss how to transmit digital data symbols, which have to be converted into
material form before they are sent or stored. In the sequel, we associate the symbols
with pulses, and a sequence of these pulses are added up to form a pulse train that
carries the entire message. Let the symbol duration be Ts. The reciprocal 1/Ts is
called the baud rate of the transmission. For a symbol representing n bits, the symbol
duration and the bit duration are related by Ts = nTb and the bit rate in bits/second
is 1/Tb = n/Ts. If the pulse train itself or a similar waveform is transmitted, the
communication system is said to be baseband.
The simplest way to generate a pulse train is to assume that the pulses last only
as long as the data symbols and do not overlap with succeeding pulses. The shape
and relations among pulses are called their format. A format is sometimes called a line
code.
The next not so-easy way to generate a pulse train is to let the pulses overlap and
explore the conditions under which the amplitudes of individual pulses may still be
observed from samples of the entire pulse train; these are called Nyquist pulses.
Finally, we consider orthogonal pulses which overlap more seriously, but in a way
that makes all but one pulse in the train invisible to a properly designed detector.
These last two pulse classes introduce two basic detectors: the sampling receiver
and the linear receiver. The former works with Nyquist pulses by simply observing
the data symbol values at the right moments in the pulse train. The latter observes
the entire signal and depends on the orthogonality property to separate out the data
symbols [1].
2 Power spectral density of PAM signals
Consider the pulse amplitude modulation (PAM) signal
∗FDNunes, IST 2013.
16
2 POWER SPECTRAL DENSITY OF PAM SIGNALS 17
x(t) =∞∑
k=−∞Akp(t− kTs) (1)
where Ak is a real discrete random variable that models the digital source and p(t) is
the signaling pulse. Recall that the power spectrum is defined as
Gx(f) = limT0→∞
E{|XT0(f)|2}
where XT0(f) is the Fourier transform of the truncated PAM signal xT0(t) in the interval
of duration T0 = (2L + 1)Ts, i.e.,
xT0(t) =L∑
k=−L
Akp(t− kTs)
We get
Gx(f) = |P (f)|2 limL→∞
1
(2L + 1)Ts
L∑
k=−L
L∑
m=−L
RA(k −m)e−j2πf(k−m)Ts
where P (f) is the Fourier transform of p(t), RA(k−m) = E{AkAm} is the autocorrela-
tion of the time sequence of random variables, Ak, assumed to be wide-sense stationary.
Finally doing L →∞ we obtain
Gx(f) =|P (f)|2
Ts
∞∑
n=−∞RA(n)e−j2πfnTs
If we consider that the data symbols are uncorrelated, that is
RA(n) =
{A2
k, n = 0
AkAk+n, n 6= 0=
{σ2
A + m2A, n = 0
m2A, n 6= 0
we obtain
Gx(f) = σ2A
|P (f)|2Ts
+m2
a
Ts
|P (f)|2∞∑
n=−∞e−j2πfnTs (2)
Using now the Poisson’s sum formula
∞∑
n=−∞e±j2πnf/U = U
∞∑
m=−∞δ(f −mU)
leads to
Gx(f) = σ2A
|P (f)|2Ts
+m2
A
T 2s
∞∑
m=−∞
∣∣∣∣P(
m
Ts
)∣∣∣∣2
δ(f − m
Ts
)(3)
2 POWER SPECTRAL DENSITY OF PAM SIGNALS 18
The first part of the right-hand member is the continuous spectrum and the second
part is the discrete spectrum, which is null if mA = 0 and/or P (m/Ts) = 0, ∀m [2]. In
that case
Gx(f) = σ2A
|P (f)|2Ts
(4)
The previous result can be used to compute the power spectra of line codes. Ex-
amples of line codes are shown in Fig. 1 where NRZ stands for nonreturn-to-zero and
RZ denotes return-to-zero.
1 0 1 1 0 0
polar NRZ
polar RZ
unipolar NRZ
bipolar RZ
Manchester
data =
Tb
Tb
Tb
Tb
TbTb
TbTb
TbTb
bit=0 bit=1
Figure 1: Examples of binary line codes based on the square pulse. The pulses corre-
sponding to bits 0 and 1 appear at left
In the polar NRZ line code, symbols 1 and 0 are represented by transmitting pulses
of amplitudes +A and −A, respectively. This code is relatively easy to generate but its
disadvantage is that its power spectrum is large near zero frequency. In the unipolar
NRZ line code symbol 1 is represented by transmitting a pulse of amplitude A for the
duration of the symbol and symbol 0 is represented by switching off the pulse (on-off
signaling). Disadvantages of the on-off signaling are the waste of power due to the
transmitted DC level and the fact that the power spectrum does not approach zero at
zero frequency. The bipolar RZ line code uses three amplitude levels where positive and
negative pulses are used alternatively for symbol 1, with each pulse having half-symbol
wide, and no pulse is always used for symbol 0. This line code is also called alternate
mark inversion (AMI) signaling. The power spectrum of the transmitted signal has
no DC component and small low-frequency components when symbols 0 and 1 occur
2 POWER SPECTRAL DENSITY OF PAM SIGNALS 19
with equal probabilities. In the Manchester code symbol 1 is represented by a positive
pulse of amplitude A followed by a negative pulse of amplitude −A. For symbol 0,
the polarities of these pulses are reversed. The Manchester code suppresses the DC
component and has relatively small low-frequency components, regardless of the signal
statistics [3].
Example:
Determine the power spectral density (PSD) of the Manchester code for independent
and equally likely bits 0 and 1.
We consider the basic pulse p(t) of Fig. 2.
1
-1
0
Tb
t
p(t)
Figure 2: Basic pulse of the Manchester code
The digital source is formed by symbols Ak = −A and Ak = A. The pdf of the r.v.
Ak is
pAk(a) =
1
2δ(a + A) +
1
2δ(a− A)
with mean and variance
mA = Ak =∫ ∞
−∞apAk
(a) da = 0
σ2A = E{(Ak −mA)2} =
∫ ∞
−∞(a−mA)2pAk
(a) da = A2
The Fourier transform of p(t) is
P (f) =∫ Tb/2
0e−j2πft dt−
∫ Tb
Tb/2e−j2πft dt
= j2
πfe−jπfTb sin2
(πfTb
2
)
3 MATCHED FILTER 20
leading to
|P (f)|2 = T 2b sinc2
(fTb
2
)sin2
(πfTb
2
)
where sinc(x) = sin(πx)/(πx) and finally
Gx(f) = A2Tb sinc2
(fTb
2
)sin2
(πfTb
2
)
The power spectrum has no discrete part as shown in Fig. 3.
−5 0 5−2.5 2.50
0.05
0.1
0.15
0.2
0.25
fTb
× A
2 Tb
Manchester code PSD
Figure 3: Power spectral density of the Manchester code with independent and equally
likely symbols
This spectrum has a null at frequency zero which may be useful for certain channels,
as in digital recording using magnetic tapes.
3 Matched filter
A basic problem that often arises is that of detecting a pulse transmitted over a channel
that is corrupted by channel noise. Consider the receiver model shown in Fig. 4,
involving a linear time-invariant filter of impulse response h(t). The filter input consists
of a pulse signal g(t) corrupted by additive channel noise w(t) with
x(t) = g(t) + w(t), 0 ≤ t ≤ T
3 MATCHED FILTER 21
where T is an arbitrary observation interval and w(t) is the sample function of a zero-
mean white noise process with power spectral density Gw(t) = N0/2. The goal of the
receiver is to detect the pulse signal g(t) in an optimum manner, given the received
signal x(t). Thus, we have to optimize the design of the filter in order to minimize the
effects of noise at the filter output in some statistical sense. Since the filter is linear,
the output may be expressed as
y(t) = go(t) + n(t)
where go(t) and n(t) are the filter responses to the signal and noise, respectively. The
filter output is sampled at the optimal time t = T where the peak pulse signal-to-noise
ratio
η =|go(T )|2E[n2(t)]
is maximized. The quantity |go(T )|2 is the instantaneous power of the output signal
and E[n2(t)] is the average output noise power. We wish to specify the filter’s impulse
response h(t) such that the signal-to-noise ratio η is maximized.
+
white noisew(t)
signal g(t)
linear time-invariantfilter of impulse responseh(t) sample at time
t=T
y(t) y(T)
Figure 4: Linear receiver
The output signal go(t) may be written as the inverse Fourier transform
go(t) =∫ ∞
−∞H(f)G(f)exp(j2πft) df
Thus
|go(T )|2 =∣∣∣∣∫ ∞
−∞H(f)G(f) exp(j2πfT ) dt
∣∣∣∣2
and we may re-write the expression for the peak pulse signal-to-noise ratio as
η =
∣∣∣∫∞−∞ H(f)G(f) exp(j2πfT ) df
∣∣∣2
N0
2
∫∞−∞ |H(f)|2 df
(5)
The problem is to find, for a given G(f), the particular form of the frequency re-
sponse H(f) that makes maximizes η. To solve this problem we resort to the Schwarz’s
inequality
3 MATCHED FILTER 22
∣∣∣∣∫ ∞
−∞H(f)G(f) exp(j2πfT ) df
∣∣∣∣2
≤∫ ∞
−∞|H(f)|2 df
∫ ∞
−∞|G(f) exp(j2πfT )|2 df
The equality holds if, and only, if
H(f) = kG∗(f) exp(−j2πfT ) (6)
where k is an arbitrary constant. Replacing (2) in (5) leads to
ηmax =2
N0
∫ ∞
−∞|G(f)|2 df =
2Eg
N0
where Eg is the energy of signal g(t) and the optimum value of H(f) is
Hopt(f) = kG∗(f) exp(−j2πfT )
The corresponding impulse response of the optimum filter is given by
hopt(t) = kg(T − t)
which is, except for the scaling factor k, the time-reversed and delayed version of the
input signal g(t); that is, the impulse response is “matched” to the input signal [3].
Example: Matched filter for rectangular pulse
Let
g(t) = A∏ (
t− T/2
T
)
The matched filter for additive white noise is
hopt(t) = kg(T − t)
= kA∏ (
T/2− t
T
)= kA
∏ (t− T/2
T
)
and the matched filter output for signal g(t) is
go(t) = g(t) ∗ hopt(t)
=∫ ∞
−∞hopt(λ)g(t− λ) dλ
= kA2∫ ∞
−∞
∏ (λ− T/2
T
) ∏ (λ− t + T/2
T
)dλ
3 MATCHED FILTER 23
or
g0(t) = kA2∫ T
0
∏ (λ− t + T/2
T
)dλ
= kA2T∧ (
t− T
T
)
The matched filter output is shown in Fig. 5. In the presence of noise with PSD
Gw(f) = N0/2 the peak signal-to-noise ratio is
ηmax =2Eg
N0
=2A2T
N0
3.1 integrate-and-dump circuit
For the special case of a rectangular pulse, the matched filter can be implemented
using an integrate-and-dump circuit, which is shown in Fig. 6. The integrator output
is sampled at t = T , where T is the pulse duration. Immediately after t = T , the
integrator is restored to its initial condition; hence the name of the circuit.
0 T
A
t
kA T2
0 T 2T t
g(t)
g (t)o
Figure 5: Rectangular pulse and matched filter response
At the sampling time the integrator output is
y(T ) =∫ T
0[g(t) + w(t)] dt = AT + n
where n is a zero-mean random variable (r.v.) given by
n =∫ T
0w(t) dt
3 MATCHED FILTER 24
+ ∫(.)dty(T)
t=T
0
T
g(t)
w(t)
rectangularpulse
noise
Figure 6: Integrate-and-dump receiver
The variance of n is
σ2n = E
∫ T
0
∫ T
0w(λ)w(α) dλdα
=∫ T
0
∫ T
0E{w(λ)w(α) dλdα
=N0
2
∫ T
0
∫ T
0δ(λ− α) dλdα
yielding
σ2n =
N0T
2Thus, at the sampling time t = T the signal-to-noise ratio is
y2(T )
σ2n
=2A2T
N0
which is equal to ηmax. Therefore, at the sampling time the matched filter and the
integrate-and-dump circuit are equivalent.
3.2 correlator circuit
This result can be readily to non-rectangular pulses. In fact, the matched filter can
be implemented using a correlator circuit, as shown in Fig. 7, where k is an arbitrary
constant. At the sampling time the integrator output is
y(T ) = k∫ T
0g2(t) dt + k
∫ T
0g(t)w(t) dt
= kEg + n
4 NYQUIST PULSES 25
where n is a zero-mean r.v. with variance
σ2n = k2
∫ T
0
∫ T
0g(λ)g(α)E{w(λ)w(α)} dλdα
=N0
2k2
∫ T
0g2(α) dα =
N0
2k2Eg
and the signal-to-noise ratio at the sampling time is
y2(T )
σ2n
=k2E2
g
(N0/2)k2Eg
=2Eg
N0
which proves that the correlator is equivalent to the matched filter at the optimal
sampling time. Note that the integrate-and-dump receiver is a particular case of the
correlator receiver where g(t) is a rectangular pulse.
+ ∫ (.)dty(T)
t=T
0
T
g(t)
w(t)
pulse
noise
+
kg(t)
Figure 7: Correlator receiver
4 Nyquist pulses
Although the previous line codes, based on the square pulse, are simple to implement,
the small duration of the pulses and their discontinuities cause the line codes to have
very large bandwidths. Two ways to reduce the bandwidth of any pulse are to round
off its corners and transitions and to increase the pulse duration. Overlapping pulses
interfere with each other. However, several kinds of interference still allow an effective
detector to be built. The first class of such pulses obeys a zero-crossing criterion called
the Nyquist pulse criterion.
Let the basic pulse p(t) be centered at time zero. A pulse p(t) satisfies the Nyquist
pulse criterion if it passes through 0 at time t = nT , n = ±1,±2, . . . but not a t = 0
p(nT ) =
{p(0), n = 0
0, n 6= 0(7)
4 NYQUIST PULSES 26
For Nyquist pulses the symbol detector (called sample detector) is quite simple: a
sample at time nT directly gives the value of the n.th transmission symbol. Consider
that the PAM signal
y(t) =∞∑
k=−∞Akp(t− kT )
is sampled at t = nT . The result is
y(nT ) =∞∑
k=−∞Akp[(n− k)T ] = Anp(0)
which is proportional to the transmitted symbol An. Assume now that the sampling
time is affected by a synchronization error ε with |ε| < T . Then
y(nT + ε) =k=∞∑
k=−∞Akp[(n− k)T + ε]
= Anp(ε) +∞∑
k=−∞k 6=n
Akp[(n− k)T + ε]
︸ ︷︷ ︸ISI
The term ISI is the intersymbol interference.
We can obtain an equivalent condition to (7) for the elimination of ISI in the
frequency domain. Let P (f) be the Fourier transform of p(t). Then, the Poisson sum
formula yields
∞∑
n=−∞P
(f − n
T
)= T
∞∑
m=−∞p(mT )e−j2πmfT
= Tp(0) (8)
Thus, the pulse p(t) satisfies the Nyquist pulse criterion (7) if and only if (Nyquist’s
first criterion)
∞∑
n=−∞P
(f − n
T
)= Tp(0) = constant (9)
If P (f) is nonzero outside the Nyquist interval−1/(2T ) ≤ f ≤ 1/(2T ), many classes
of pulses satisfy (9). Thus, the Nyquist criterion does not uniquely specify the frequency
response P (f). On the contrary, if P (f) is limited to an interval smaller than Nyquist’s,
it is impossible for (9) to hold [4] and ISI cannot be removed from the received signal.
4 NYQUIST PULSES 27
If P (f) is exactly bandlimited in the Nyquist interval −1/(2T ) ≤ f ≤ 1/(2T ), (9)
requires that
P (f) =
{Tp(0), |f | ≤ 1/(2T )
0, elsewhere
That is, the only pulse satisfying the Nyquist criterion is the pulse with rectangular
spectrum
P (f) = Tp(0)∏ (
f
1/T
)= constant
and in the time domain
p(t) = p(0) sinc(
t
T
)
There are two serious problems with this solution. First, it is not physically real-
izable because of its instantaneous jump to 0 at f = ±1/(2T ). The second problem
comes from the fact that even small synchronization errors of sampling times t = nT
would lead to the appearance of ISI. For this reason, it is convenient to use pulses with
wider bandwidths to reduce sensitivity to inaccuracies in the sampling times. The most
common example in practice is the raised cosine pulse, defined in frequency by
P (f) =
T, |f | ≤ 1−α2T
T2
{1− cos
[πTα
(|f | − 1+α
2T
)]}, 1−α
2T≤ |f | < 1+α
2T
0, |f | ≥ 1+α2T
(10)
The spectra of the raised cosine pulse are plotted in Fig. 8 for different values of α.
The parameter α is called the rolloff factor or excess bandwidth factor, since the band-
width of p(t) is (1 + α)/(2T ), while the narrowest possible bandwidth (corresponding
to p(t) = sinc(t/T )) is 1/(2T ).
In the time domain the pulse is defined by
p(t) = sinc(
t
T
)cos(απt/T )
1− (2αt/T )2
and decays asymptotically as 1/|t|3 for |t| → ∞. The raised cosine pulses are shown in
Fig. 9 for different values of α.
Consider the PAM signal in (1) where the signalling pulse p(t) is raised cosine and
Ak = ±1. Assume that x(t) is sampled at t = ε, where ε is the synchronization error,
with |ε| < T (T is the symbol duration). We obtain
y(ε) = y0(ε) + yISI(ε)
4 NYQUIST PULSES 28
−1 −0.5 0 0.5 10
0.2
0.4
0.6
0.8
1
normalized frequency, fT
spec
trum
of t
he r
aise
d co
sine
pul
se,
× T
α=0α=0.2α=0.5α=1.0
Figure 8: Raised cosine spectra for different values of the rolloff factor
with
y0(ε) = A0 sinc(
ε
T
)cos(απε/T )
1− (2αε/T )2
and
yISI(ε) =∞∑
k=−∞k 6=0
Ak sinc (ε/T − k)cos (απ (ε/T − k))
1− (2α (ε/T − k))2
The signal-to-noise ratio at the sampling time t = ε, due to ISI, is
SNR(ε) =E{y2
0(ε)}E{y2
ISI(ε)}
=sinc2 (ε/T ) cos2(απε/T )
[1− (2αε/T )2]2E{y2ISI(ε)}
(11)
Fig. 10 shows the signal-to-noise ratios versus the rolloff factor α for different
normalized synchronization errors ε/T . The plots were computed by Monte Carlo
simulation assuming that the sequence of symbols {Ak} is random. Notice that, for a
constant value of ε, the signal-to-noise ratio grows with α; thus, the degradation due
to ISI increases as α diminishes.
5 ORTHOGONAL PULSES 29
−6 −4 −2 0 2 4 6−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
normalized time, t/T
rais
ed c
osin
e pu
lse,
p(t
)
α=0α=0.2α=0.5α=1.0
Figure 9: Raised cosine pulses for different values of the rolloff factor
5 Orthogonal pulses
So far, we have designed a class of pulse trains with relatively narrow bandwidth whose
underlying symbols are easy to extract using the Nyquist criterion. The problem is that
a simple receiver can have poor error probability in the presence of channel additive
noise. The key to solving this problem is to make the pulses orthogonal.
A pulse p(t) is orthogonal if
∫ ∞
−∞p(t)p(t− nT ) dt = 0, n = ±1 ± 2, . . .
where T is the symbol interval.
Consequently, a correlation of the pulse train
s(t) =∑m
amp(t−mT )
with p(t− nT ) gives the symbol an
∫ ∞
−∞
[∑m
amp(t−mT )
]p(t− nT ) dt = an
∫ ∞
−∞p2(t− nT ) dt (12)
= anEp
where Ep is the energy of p(t).
5 ORTHOGONAL PULSES 30
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110
15
20
25
30
35
rolloff factor, α
sign
al−
to−
nois
e ra
tio, d
B
ε/T=0.05ε/T=0.1ε/T=0.15
raised−cosine pulse with ISI
Figure 10: Signal-to-noise ratio due to ISI at the sampling time of a raised cosine pulse
versus the rolloff factor for different normalized synchronization errors ε/T
The correlation in (12) may be realized by a simple linear filter. We may rewrite
(12) as
∫s(t)p(t− nT ) dt =
∫s(λ)p[−(nT − λ)] dt
= s(t) ∗ p(−t)∣∣∣t=nT
= anEp
Thus, the desired correlation is the value of the convolution of the pulse train with
a filter with impulse response hr(t) = p(−t) at time t = nT . We can implement this
by applying the train to a filter with transfer function Hr(f) = P ∗(f) and sampling
the output at time nT . The corresponding communication system is sketched in Fig.
11. In the detector, the signal passes through the filter Hr(f) and gets sampled at
time t = nT . If there is no noise, the sample gives the transmission symbol an, thus
eliminating the ISI. Otherwise, the sample is compared to the noise-free symbol values
in a threshold (Vth) comparator. This detector circuit is known as linear receiver.
Note that the Nyquist’s first criterion (equation (9)) has to be verified for the pulse
p(t) ∗ p(−t) whose Fourier transform is P (f)P ∗(f) = |P (f)|2. Thus, the Nyquist
criterion for ISI elimination is now
∞∑
n=−∞
∣∣∣P(f − n
T
) ∣∣∣2
= constant (13)
In the case the pulse p(t) is symmetric the spectrum P (f) is real and condition (13)
5 ORTHOGONAL PULSES 31
may be written as
∞∑
n=−∞P 2
(f − n
T
)= constant
Probably the most common used pulse in sophisticated systems is the root raised-
cosine pulse [1]. The pulse p(t) is symmetric in time, so expression (5) may be applied.
The root raised cosine pulse spectrum, P (f), is such that P 2(f) is expressed by (10).
The root raised cosine pulse satisfies the orthogonality constraint and has the same
excess bandwidth specified by the rolloff factor α.
Σ amδ( t-mT) P(f) +
noise
+
-
Vth
binary decision
comparator
t=nT
Σam p(t-mT)s(t)=
w(t)
x(t)=r(t)
P (f)*
receiver
Figure 11: Communication system for orthogonal pulses and linear receiver
Consider now a more complex communication system displayed in Fig. 12 and
modeled as the cascade of a modulator having the ideal impulse δ(t) as its basic wave-
form, and of a shaping filter with frequency response S(f). The number of symbols
to be transmitted per second is 1/T . The channel is represented by a time-invariant
linear system having known transfer function C(f) and a generator of additive noise
w(t). We aim to design a receiver having the form of a linear filter followed by a
sampler. After linear filtering, the received signal is sampled every T seconds and the
resulting sequence is sent to the detector. The detector makes decisions on a sample-
by-sample basis. The design criterion concerns the elimination of ISI regarding the
cascade Q(f) = S(f)C(f)U(f), leaving open the choice how to partition the overall
transfer function between the transmitter and the receiver, i.e., how to choose S(f)
and U(f) once the product S(f)C(f)U(f) has been specified. Note that the pulse
corresponding to Q(f) must be a Nyquist pulse. The freedom to chose U(f) permits
to impose one further condition, that is, the minimization of the error probability (in
fact, in the absence of ISI, errors are caused only by additive noise).
The average noise power at the receiving filter output is
σ2n =
∫ ∞
−∞Gw(f)|U(f)|2 df
Since the overall channel frequency response is the fixed function Q(f), the signal
power spectral density at the shaping filter output is (see equation (4))
5 ORTHOGONAL PULSES 32
source modulator S(f) C(f) +
w(t)
U(f) detectorsampler
r(t) x(t) kx A k^
Akδ(t-kT)A k
transmitterreceiver
channel
Figure 12: Transmission system for linearly modulated data over a time-dispersive
channel with sampling receiver
σ2A
T|S(f)|2 =
σ2A
T
|Q(f)|2|C(f)U(f)|2
and the corresponding signal power is
P =σ2
A
T
∫ ∞
−∞|Q(f)|2
|C(f)U(f)|2 df (14)
Minimization of σ2n under the constraint (14) can be performed using the Lagrange
multipliers. The minimizing U(f) can be shown to be given by
|U(f)| = |Q(f)|1/2
G1/4w (f)|C(f)|1/2
(15)
and the corresponding shaping filter is obtained through
S(f) =Q(f)
C(f)U(f)(16)
In (15) and (16) it is assumed that Q(f) = 0 at those frequencies for which the
denominators are zero. In the special case of white noise and C(f) constant, we obtain
|U(f)| = γ|Q(f)|1/2
where γ is a nonzero factor. Thus
|U(f)| = γ′|S(f)|1/2|U(f)|1/2
or
|U(f)| = (γ′)2|S(f)|where γ′ is a nonzero factor. So, only one design has to be implemented for the shaping
filter and the receiving filter.
6 ZERO-FORCING EQUALIZATION 33
6 Zero-forcing equalization
The theory previously developed devoted to the design of an optimum receiver in the
presence of channel distortion was based on the assumption of a linear channel and of
the exact knowledge of its impulse response (or transfer function). While the former
assumption is reasonable in many situations, the latter is often unrealistic. For instance,
the channel may be static but selected randomly from an ensemble, as happens with
telephone lines. Or, the channel may be vary randomly with time due to fading. Thus,
the receiver designed to cope with the effects of ISI and additive noise should be self-
optimizing or adaptive. That is, its parameters should be automatically adjusted to
an optimum operating point.
Two philosophies can be adopted to design an adaptive receiver. The first, assumes
that the relevant channel parameters are first estimated, then fed to a detector which
is (approximately) optimum for those parameters. This can be, for example, a Viterbi
detector, which requires the knowledge of the channel impulse response samples. The
other approach consists of using an equalizer to compensate for the unwanted channel
features, and feeds the detector with a sequence of samples that have been cleaned
from ISI.
Fig. 13 shows a transversal equalizer with 2N + 1 taps and total delay 2ND. The
distorted pulse shape p(t) at the input to the equalizer is assumed to have its peak at
t = 0 and ISI on both sides. The equalized output pulse will be
peq(t) =N∑
n=−N
cnp(t− nD −ND)
or doing t = tk ≡ (k + N)D
peq(tk) =N∑
n=−N
cnp(kD − nD) =N∑
n=−N
cnpk−n (17)
with pk−n = p[(k − n)D]. Ideally, we would like the equalizer to eliminate all ISI,
resulting in
peq(tk) =
{1, k = 0
0, k 6= 0
But this cannot be achieved, in general, because the 2N + 1 tap gains are the only
variables at our disposal. We may instead chose the tap gains such that
peq(tk) =
{1, k = 0
0, k = ±1,±2, . . . ,±N(18)
6 ZERO-FORCING EQUALIZATION 34
D D ... D
+ + + ++c c c c c-N -N+1 -N+2 N-1 N
Σ
p (t)eq
p(t)
total delay 2ND
Figure 13: Transversal equalizer with 2N + 1 taps
thereby forcing N zero values on each side of the peak of peq(t). The corresponding
tap gains are computed from (17) and (18) combined in the matrix equation
p0 · · · p−2N...
. . ....
pN−1 · · · p−N−1
pN · · · p−N
pN+1 · · · p−N+1...
. . ....
p2N · · · p0
c−N...
c−1
c0
c1...
cN
=
0...
0
1
0...
0
(19)
Equation (19) describes a zero-forcing equalizer. The equalized pulse will have
a maximum value for k = 0 and is forced to be zero for the N preceding and the
N following decision instants, thus the name zero-forcing equalizer. This equalizer
removes the ISI in 2N sampling instants. For an ideal equalizer N would have to be
very large. Practical equalizers have taps in the range of 3 to several hundreds [5]. This
equalization strategy is optimum in the sense that it minimizes the peak intersymbol
interference, and it has the added advantage of simplicity [6].
Example:
Consider that a three-tap zero forcing equalizer is to be designed for the distorted
pulse p(t) plotted in Fig. 14 (solid line). Inserting the values of pk into (19) with
N = 1, leads to
7 MEAN SQUARE ERROR MINIMIZATION 35
1.0 0.1 0.0
−0.2 1.0 0.1
0.1 −0.2 1.0
c−1
c0
c1
=
0
1
0
Therefore,
c−1 = −0.096, c0 = 0.96, c1 = 0.2
The corresponding samples of peq(t) are plotted in Fig. 14 (dashed line). As ex-
pected, there is one zero on each side of the peak. However, zero forcing has produced
some small ISI at points further out where the unequalized pulse was zero.
−5 −4 −3 −2 −1 0 1 2 3 4 5
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
time, × D
p(t)p
eq(t)
−0.2
1.0
−0.01 0
0.020.06
0.1
0
0.1
1.0
Figure 14: Distorted and equalized pulse
The zero-forcing equalizer is relatively easy to implement because it ignores the
effect of channel noise w(t). A serious consequence of this simplification is that it leads
to overall performance degradation due to noise enhancement at frequencies where
the equalizer gain is large. A more refined approach for the receiver design is to use
the mean-square error criterion, which provides a balanced solution to the problem of
reducing the effects of both channel noise and ISI.
7 Mean square error minimization
We have thus far treated the following two channel conditions separately:
- Channel noise acting alone, which led to the formulation of the matched filter
receiver.
7 MEAN SQUARE ERROR MINIMIZATION 36
- Intersymbol interference (ISI) acting alone, which led to the formulation of the
pulse-shaping transmit filter so as to realize the Nyquist channel.
In a real-life situation, however, channel noise and ISI act together, affecting the
behavior of a data transmission system in a combined manner. In the sequel, consider
again the communication system of Fig. 12 and assume that U(f) must be chosen so as
to minimize the effects of additive noise at the detector output, and hence to minimize
the probability of error for the transmission system under the condition of no ISI.
We shall consider the minimum mean-square error (MMSE) criterion for the system
optimization; this choice allows ISI and noise to be taken jointly into account, and in
most practical situations leads to values of error probability very close to their minimum
[4].
Instead of constraining the noiseless samples to be equal to the transmitted symbols
in Fig. 12, we can take into account the presence of additive noise and try to minimize
the mean-squared difference between the sequence of transmitted symbols {Ak} and
the sampler outputs {xk}. By allowing for a channel delay of D symbol intervals, we
want to determine the shaping filter S(f) and the receiving filter U(f) so that the
mean-square value of
εk ≡ xk − Ak−D
is minimized. This will result in a system that, although not specifically designed for
optimum error performance, should provide a satisfactory performance even in terms
of error probability.
For the special case of a channel bandlimited to the Nyquist interval [−1/(2T ), 1/(2T )],
it can be proved [4] that the transfer function of the optimum receiving filter is given
by
Uopt(f) =σ2
AP ∗(f)
Gw(f) + (σ2A/T )|P (f)|2 e−j2πfDT (20)
where
σ2A
T|P (f)|2
is the power spectral density of the received digital signal (see equation (4)), D is the
channel delay in symbol intervals and P (f) ≡ S(f)C(f). Equation (20) shows that, in
the absence of noise, the optimum receiving filter is simply the inverse of P (f). This
results from having an overall transfer function that is constant in the Nyquist band,
then verifying the Nyquist’s first criterion for zero ISI. However, when Gw(f) 6= 0,
elimination of ISI does not provide the best solution. On the contrary, for spectral
regions where the denominator of the right-hand side of (20) is dominated by Gw(f),
Uopt(f) approaches the matched filter frequency response P ∗(f)/Gw(f).
REFERENCES 37
More generally, for a channel with nonzero transfer function outside the Nyquist
interval, the transfer function of the optimum receiving filter is [4], [3]
Uopt(f) =P ∗(f)
Gw(f)Γ(f) (21)
where Γ(f) is a periodic function with period 1/T given by
Γ(f) =σ2
Ae−j2πfDT
1 + σ2AL(f)
, L(f) =1
T
∞∑
k=−∞
|P (f + k/T )|2Gw(f + k/T )
Note that in (21) P ∗(f)/Gw(f) is the transfer function of a filter matched to the
impulse response p(t) of the cascade of the shaping filter and the channel. Also, Γ(f),
being a periodic transfer function with period 1/T , can be thought of as the transfer
function of a transversal filter whose taps are spaced T seconds apart. Thus, we can
conclude that the optimum receiving filter is the cascade of a matched filter and a
transversal equalizer, as shown in Fig. 15. The former reduces the noise effects and
the latter reduces ISI.
To implement (21) exactly we need an equalizer of infinite length. In practice,
we may approximate the optimum solution by using an equalizer with a finite set of
coefficients ck, −N ≤ k ≤ N , provided N is large enough [3].
p(-t)received signal
...
+ + + + +
Σ
y(nT)
c c c c c-N -N+1 -N+2 N-1 N
T T T
transversal equalizermatched filter
x(t)
y(t)
Figure 15: Optimum linear receiver consisting of the cascade connection of matched
filter and transversal equalizer
References
[1] John B. Anderson, “Digital Transmission Engineering”, IEEE Press, N. York,
1999.
REFERENCES 38
[2] Leon W. Couch II, “Digital and Analog Communication Systems”, Macmillan, N.
York, 1990.
[3] Simon Haykin, “Communication Systems”, 4.th edition, Wiley, N. York, 2001.
[4] Sergio Benedetto and Ezio Biglieri, “Principles of Digital Transmission with Wire-
less Applications”, Kluwer, N. York, 1999.
[5] Kamilo Feher, “Digital Communications: microwave applications”, Prentice-Hall,
Englewood Cliffs, NJ, 1981.
[6] A. Bruce Carlson, “Communication Systems. An Introduction to Signals and Noise
in Electrical Communication”, McGraw-Hill, N. York, NJ, 1986.