coseno elevado wits lab
TRANSCRIPT
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Square Root Raised Cosine Filter
Digital Communication, 4th Edition
Chapter 9: Signal Design for Band-Limited Channels
John G. Proakis
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Introduction
We consider the problem of signal design when the channel isband-limited to some specified bandwidth ofWHz.
The channel may be modeled as a linear filter having an equivalentlow-pass frequency response C(f) that is zero for |f| >W.
Our purpose is to design a signal pulse g(t) in a linearly modulatedsignal, represented as
that efficiently utilizes the total available channel bandwidth W.
When the channel is ideal for |f|
W, a signal pulse can be designed thatallows us to transmit at symbol rates comparable to or exceeding thechannel bandwidth W.
When the channel is not ideal, signal transmission at a symbol rate equal toor exceeding Wresults in inter-symbol interference (ISI) among a number ofadjacent symbols.
( ) ( )nn
v t I g t nT =
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For our purposes, a band-limited channel such as a telephone
channel will be characterized as a linear filter having an
equivalent low-pass frequency-response characteristic C(f),
and its equivalent low-pass impulse response c(t).
Then, if a signal of the form
is transmitted over a band-pass telephone channel, the
equivalent low-pass received signal is
wherez(t) denotes the additive noise.
( ) ( ) ( ) ( )
( ) ( ) ( )tztctv
tzdtcvtrl
+=
+=
( ) ( )tfj c
etvts2
Re=
Characterization of Band-LimitedChannels
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Alternatively, the signal term can be represented in thefrequency domain as V(f)C(f), where V(f) = F[v(t)].
If the channel is band-limited to WHz, then C(f) = 0 for |f| >W.
As a consequence, any frequency components in V(f) above
|f| = Wwill not be passed by the channel, so we limit thebandwidth of the transmitted signal to WHz.
Within the bandwidth of the channel, we may express C(f) as
where |C(f)|: amplitude response.
(f): phase response.
The envelope delay characteristic:
( ) ( ) ( )fjefCfC =
( )( )
df
fdf
2
1=
Characterization of Band-LimitedChannels
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A channel is said to be nondistorting orideal if the amplituderesponse |C(f)| is constant for all |f| Wand(f) is a linearfunction of frequency, i.e.,(f) is a constant for all |f| W.
If |C(f)| is not constant for all |f| W, we say that the channeldistorts the transmitted signal V(f) in amplitude.
If(f) is not constant for all |f| W, we say that the channel
distorts the signal V(f) in delay.As a result of the amplitude and delay distortion caused by thenonideal channel frequency-response C(f), a succession of
pulses transmitted through the channel at rates comparable tothe bandwidth Ware smeared to the point that they are nolonger distinguishable as well-defined pulses at the receivingterminal. Instead, they overlap, and thus, we have inter-symbol
interference (ISI).
Characterization of Band-LimitedChannels
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Fig. (a) is a band-limited pulse having zeros periodically spaced
in time at T, 2T, etc.
If information is conveyed by the pulse amplitude, as in PAM, for
example, then one can transmit a sequence of pulses, each of
which has a peak at the periodic zeros of the other pulses.
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However, transmission of the pulse through a channel modeled as
having a linear envelope delay(f) [quadratic phase(f)]
results in the received pulse shown in Fig. (b), where the zero-
crossings that are no longer periodically spaced.
Characterization of Band-LimitedChannels
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A sequence of successive pulses would no longer be
distinguishable. Thus, the channel delay distortion results in ISI.
It is possible to compensate for the nonideal frequency-response
of the channel by use of afilterorequalizerat the demodulator.
Fig. (c) illustrates the output of a linear equalizer that
compensates for the linear distortion in the channel.
Characterization of Band-LimitedChannels
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The equivalent low-pass transmitted signal for several different
types of digital modulation techniques has the common form
where {In}: discrete information-bearing sequence of symbols.
g(t): a pulse with band-limited frequency-
response G(f), i.e., G(f) = 0 for |f| > W.This signal is transmitted over a channel having a frequency
response C(f), also limited to |f| W.
The received signal can be represented as
where andz(t) is the AWGN.
( ) ( ) ( )tznTthItrn
nl +=
=0
( ) ( )
=
=0n
n nTtgItv
( ) ( ) ( )
= dtcgth
Signal Design for Band-LimitedChannels
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Suppose that the received signal is passed first through a filter and
then sampled at a rate 1/Tsamples/s, the optimum filter from the
point of view of signal detection is one matched to the received
pulse. That is, the frequency response of the receiving filter is
H*(f).
We denote the output of the receiving filter as
where
x(t): the pulse representing the response of the receiving
filter to the input pulse h(t).
v(t): response of the receiving filter to the noisez(t).
( ) ( ) ( )
=
+=0n
n tvnTtxIty
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Ify(t) is sampled at times t= kT+0, k = 0, 1,, we have
or, equivalently,
where0: transmission delay through the channel.
The sample values can be expressed as
( ) ( ) ( )00
00 +++=+
=
kTvnTkTxIykTyn
nk
0
, 0,1,...k n k n k n
y I x v k
=
= + =
0
00
1, 0,1,...
k k n k n k
n
n k
y x I I x v k x
=
= + + =
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We regardx0 as an arbitrary scale factor, which we arbitrarily set
equal to unity for convenience, then
where
Ik: the desired information symbol at the k-thsampling instant.
: ISI
vk: additive Gaussian noise variable at the k-th
sampling instant.
k
knn
nknkk vxIIy ++=
=
0
=
knn
nknxI
0
Signal Design for Band-LimitedChannels
l D f B d L d
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The amount of ISI and noise in a digital communication system
can be viewed on an oscilloscope.
For PAM signals, we can display the received signaly(t) on the
vertical input with the horizontal sweep rate set at 1/T.
The resulting oscilloscope display is called an eye pattern.
Eye patterns for binary and quaternary amplitude-shift keying
(or PAM):
The effect of ISI is to cause the eye to close.Thereby, reducing the margin for additive noise to cause errors.
Signal Design for Band-LimitedChannels
Si l D i f B d Li i d
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Effect of ISI on eye opening:
ISI distorts the position of the zero-crossings and causes a
reduction in the eye opening.
Thus, it causes the system to be more sensitive to a
synchronization error.
Signal Design for Band-LimitedChannels
Si l D i f B d Li it d
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For PSK and QAM, it is customary to display the eye pattern as
a two-dimensional scatter diagram illustrating the sampled values
{yk} that represent the decision variables at the sampling instants.
Two-dimensional digital eye patterns.
Signal Design for Band-LimitedChannels
D i f B d Li i d i l f
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Assuming that the band-limited channel has ideal frequency-
response, i.e., C(f) = 1 for |f| W, then the pulsex(t) has a
spectral characteristicX(f) = |G(f)|2, where
We are interested in determining the spectral properties of the
pulsex(t), that results in no inter-symbol interference.
Since
the condition for no ISI is
k
kn
n
nknkk vxIIy ++=
=
0
( )( )
( )
1 0
0 0k
kx t kT x
k
== =
( ) ( )=W
W
ftjdfefXtx
2
()
Design of Band-Limited Signals forNo ISI The Nyquist Criterion
D i f B d Li i d Si l f
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Nyquist pulse-shaping criterion (Nyquist condition for
zero ISI)
The necessary and sufficient condition forx(t) to satisfy
is that its Fourier transformX(f) satisfy
( )( )
( )
1 0
0 0
nx nT
n
==
( ) TTmfXm
=+
=
Design of Band-Limited Signals forNo ISI The Nyquist Criterion
D i f B d Li it d Si l f
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Proof:
In general,x(t) is the inverse Fourier transform ofX(f). Hence,
At the sampling instant t= nT,
( ) ( )
= dfefXtx ftj 2
( ) ( )
= dfefXnTx fnTj 2
Design of Band-Limited Signals forNo ISI The Nyquist Criterion
D i f B d Li it d Si l f
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Breaking up the integral into integrals covering the finite
range of 1/T, thus, we obtain
where we defineB(f) as
( ) ( )( )( )
( )
( )
( )
2 1 2
22 1 2
1 22 '
1 2
1 22
1 2
1 22
1 2
' '
m T
j fnT m T
m
Tj f nT
Tm
Tj fnT
Tm
Tj fnT
T
x nT X f e df
X f m T e df
X f m T e df
B f e df
+
=
=
=
=
= +
= +
=
( ) ( )
=
+=m
TmfXfB
'
2 1 2 1: ,
2 2
1 1' : ,2 2
'
mf f
Tm m
fT T
f T T
df df
= +
+
=
Design of Band-Limited Signals forNo ISI The Nyquist Criterion
(1)
D si f B d Li it d Si ls f
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ObviouslyB(f) is a periodic function with period 1/T, and,
therefore, it can be expanded in terms of its Fourier series
coefficients {bn} as
where
Comparing (1) and (2), we obtain
( ) 2j nfT nn
B f b e
=
=
( )1 2
2
1 2
TnfT
nT
b T B f e df
=
( )nTTxbn =
(2)
( )( )
( )
Recall that the conditions forno ISI are (from ):
1 0
0 0k
kx t kT x
k
== =
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Design of Band Limited Signals for
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Suppose that the channel has a bandwidth ofW. Then
C(f) 0 for |f| > WandX(f) = 0 for |f| > W.
When T< 1/2W(or 1/T> 2W)
Since consists of nonoverlapping
replicas ofX(f), separated by 1/T, there is no choice for
X(f) to ensureB(f) Tin this case and there is no way
that we can design a system with no ISI.
( ) ( )+
+= TnfXfB
Design of Band-Limited Signals forNo ISI The Nyquist Criterion
Design of Band Limited Signals for
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When T= 1/2W, or 1/T= 2W(the Nyquist rate), the
replications ofX(f), separated by 1/T, are shown below:
In this case, there exists only oneX(f) that results inB(f) = T,
namely,
which corresponds to the pulse
( )( )
0
T f WX f
otherwise
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The smallest value ofTfor which transmission with zero ISI ispossible is T= 1/2W, and for this value,x(t) has to be a sincfunction.
The difficulty with this choice ofx(t) is that it is noncausal andnonrealizable.
A second difficulty with this pulse shape is that its rate of
convergence to zero is slow.The tails ofx(t) decay as 1/t; consequently, a small mistimingerror in sampling the output of the matched filter at thedemodulator results in an infinite series of ISI components.
Such a series is not absolutely summable because of the 1/trate of decay of the pulse, and, hence, the sum of the resultingISI does not converge.
Design of Band-Limited Signals forNo ISI The Nyquist Criterion
Design of Band Limited Signals for
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When T> 1/2W,B(f) consists of overlapping replications of
X(f) separated by 1/T:
In this case, there exist numerous choices forX(f) such that
X(f) such thatB(f) T.
Design of Band-Limited Signals forNo ISI The Nyquist Criterion
Design of Band-Limited Signals for
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A particular pulse spectrum, for the T> 1/2Wcase, that has
desirable spectral properties and has been widely used in
practice is the raised cosine spectrum.
Raised cosine spectrum:
: roll-off factor. (0 1)
( )
10
21 1- 1
1 cos2 2 2T 2
10
2
rc
T f
TT T
X f f f T T
fT
+ = +
+ >
Design of Band-Limited Signals forNo ISI The Nyquist Criterion
Design of Band-Limited Signals for
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The bandwidth occupied by the signal beyond the Nyquistfrequency 1/2Tis called the excess bandwidth and is usuallyexpressed as a percentage of the Nyquist frequency.
= 1/2 => excess bandwidth = 50 %.= 1 => excess bandwidth = 100%.
The pulsex(t), having the raised cosine spectrum, is
x(t) is normalized so thatx(0) = 1.
( )( ) ( )
( ) ( )222
222
41cossin
41
cossin
TtTtTtc
Tt
Tt
Tt
Tttx
=
=
Design of Band-Limited Signals forNo ISI The Nyquist Criterion
Design of Band-Limited Signals for
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Pulses having a raised cosine spectrum:
For= 0, the pulse reduces tox(t) = sinc(t/T), and the
symbol rate 1/T= 2W.
When= 1, the symbol rate is 1/T= W.
Design of Band Limited Signals forNo ISI The Nyquist Criterion
Design of Band-Limited Signals for
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In general, the tails ofx(t) decay as 1/t3 for> 0.
Consequently, a mistiming error in sampling leads to a series
of ISI components that converges to a finite value.
Because of the smooth characteristics of the raised cosine
spectrum, it is possible to design practical filters for the
transmitter and the receiver that approximate the overall desired
frequency response.
In the special case where the channel is ideal, i.e., C(f) = 1,
|f| W, we have
where GT(f) and GR(f) are the frequency responses of the two
filters.
( ) ( ) ( )fGfGfX RTrc =
Design of Band Limited Signals forNo ISI The Nyquist Criterion
Design of Band-Limited Signals for
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If the receiver filter is matched of the transmitter filter, we haveXrc(f)= GT(f) GR(f) = | GT(f)|
2. Ideally,
and GR(f) = , where t0 is some nominal delay that isrequired to ensure physical realizability of the filter.
Thus, the overall raised cosine spectral characteristic is splitevenly between the transmitting filter and the receiving filter.
An additional delay is necessary to ensure the physical realization
of the receiving filter.
( )fGT
( ) ( )02 ftjrcT efXfG
=
Design of Band Limited Signals forNo ISI The Nyquist Criterion
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Implementation ofSquare Root Raise Cosine Filter
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Simplified System Architecture
Transmitter
SRRC-TX DACLPF
DAC SRRC-RX
Receiver
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Design of SRRC
Operation of the SRRC-TX module includes:
Over-sampling
Over-sampling rate is a system design issue (performancevs. cost).
For a over-sampling rate = 4, we pad three zeros between
each sample.Finite-impulse-response (FIR) filter
The FIR filter acts as a low pass filter.
The FIR filter is a square root raised cosine filter with roll-off factor= 0.22.
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Detail of FIR Filter Module
h(0) h(1) h(2)
Input
0
h(n/2)
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Raised Cosine Filter
Raised cosine spectrum:
The pulsex(t), having the raised cosine spectrum, is
( )
10
2
1 1- 11 cos
2 2 2T 21
02
rc
T fT
T TX f f f
T T
fT
+
= +
+ >
( )
( ) ( )
( )
( )2 2 2 2 2 2
sin cos cossin
1 4 1 4
t T t T t T x t c t T
t T t T t T
= =
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Square Root Raised Cosine Filter
The cosine roll-off transfer function can be achieved by using
identical square root raised cosine filter at the
transmitter and receiver.The pulse SRRC(t), having the square root raised cosine
spectrum, is
( )( ) ( )
2
sin 1 4 cos 1
1 4
where is the inverse of chip rate ( 0.2604167 s)
and = 0.22.
C C C
C C
C
t t t
T T TSRRC t
t tT T
T
+ + =
( )rcX f
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Square Root Raised Cosine Filter
Because SRRC(t) is non-causal, it must be truncated, and pulse
shaping filter are typically implemented for6TCabout the t= 0
point for each symbol.
For an over-sampling rate of 4, n is equal to 48.