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= ;(l+l> mac . 39 3—10 311 3.13
Signaling with R2 pulses represents an example of orthogonal
signaling. Therefore, for coherent detection, we can use Equation (3.71) as
E _ AZT
a—lel
10—3 =Q[.l°£\}OT]=Q(x) Using Table B.l to ﬁnd x, yields x=3.l. Thus, 0.01T
2><10‘8 23.1, T=19.2,us, and R=52,083bits/S 3.14
Signaling with NRZ pulses represents an example of antipodal
signaling. Therefore, for coherent detection, we can use Equation (3.70) as PB=Q _ 2A2 .
1°3=Qllﬂli§eﬁml=wl Using Table B.l to ﬁnd x, yields x=3l Thus, FA (11”0 5? 00.0). = 31, A? = 0.268. Thus, if there were no signal power loss, the minimum power needed would be approximately
260 mW. With a 3dB loss, 538 mW are needed. 312 3.1 5
The power spectral density for a random bipolar (antipodal) sequence in Equation (1.38) is expressed in the form of , 2
TS[51:}§TS ] , where TS is the symbol duration. The total area
3 under the Spectral plot is found by integrating as follows: 2 2
GO ‘ T on '
“4ng df = 222 i [—y—f—l a Letx = 7973, then df =cit/7tTS , and the area is: 2
2T5 J‘m[SiDX] dx : EE :
0 TITS x it
The twosided Nyquist minimum bandwidth extends from
1 l 1 Thus, the onesided (baseband) bandwidth is 2LT— = 555—. S
The sketch below illustrates the rectangular construction having
the same area as the signal power spectral density. The width
(bandwidth) of this rectangle is RS (twosided) and Rﬂ (one—
sided), which is the same as the Nyquist minimum bandwidth for ideal—shaped bipolar pulses. 3.16 The output of an MF is a time series, such as seen in Figure 3.7b
(e.g., a succession of increasing positiveand negative c0rrelations
to an input sine wave). Such an MP output sequence can be
equated to several correlators operating at different starting points
of the input time series. Unlike an MF, a correlator only computes an output Once per Symbol time. A bank of N t 6 correlators is shown in Figure l, where the reference signal for the ﬁrst one is
31(3) , and the reference for each of the others is a symboltimeoffset copy, 510‘ * kT ) of the ﬁrst reference. It is convenient to refer to the reference signals as templates. Since the correlator emulates a
matched ﬁlter, the "matching" is often provided by choosing each of
the S, (I) templates to be a squareroot Nyquist shaped pulse, and thus the overall system transfer function being the product of two root
raised cosine functions, is a raised cosine function. Figure 2 is a
pictorial of the 6 shapedpulse templates, each one occupying 6 symbol times, and each one offset from its staggered neighbor
(above and below) by exactly one symbol time. Each of the template
signals will be orthogonal to one another, provided that the time
offset is chosen to be an integer number of symbols. Each correlator performs productintegration of the received
pulse sequence, r(z‘), by using its respective template. The time
shifted templates account for the staggered time over which each
cerrelator operates. That is, the ﬁrst correlator processes the r0)
waveform over the time intervals 0 to 6, then 6 to 12, and so forth.
The second correlator operates over the intervals 1 to 7, then 7 to
13, and so forth. The sixth correlator operates over the intervals
5 to 11, then 11 to 17, and so forth. In Figure 1, following the bank
of correlators is a commutating switch connecting the correlator
outputs to a sampling switch. Startup consists of loading the
correlators with 6symbol durations of the received waveform,
after which the commutating switch simply "sits" on the Output of
each correlator for one symbol duration before moving on to the
next correlator. Even though a correlator only produces an output 314 at the end of a symbol time, the commutating switch acts to form a
timeseries from the outputs of the staggered correlators. The
output of the commutating switch is a discrete approximation of the
demodulated raisedcosine (smeared) analog waveform seen in
Figure 3.23b. This output is now ready for sampling and detection.
The comrmtating switch itself can be implemented to act as the sampling switch. Recall that the beneﬁcial attribute of a matched ﬁlter or correlator is that it gathers the signal energy that is matched to its template,
yielding some peak amplitude at the end of a symbol time. Each
correlator, operating On the "smeared" signal, gathers the energy that
matches its template over 6symbol times, and when sampled at the
appropriate time, produces an output ready to be detected. REFERENCE 5, (r)
SIGNAL CDMMUTES ONE
POSITION EACH
SYMBOL TIME SAMPLE AND
: '3”
a . . or
: s, (t s:r) : at
117
O ¢ J
‘ 53"
Figure 1 3.16 (cont’d) Figure 2 16 3 3.16 (cont’d) For this example, Figure 3 shows the signal into the
staggered correlators. We see 6 successive Views of the
smeared signal appearing as “snapshots” through a sliding
window (6symbol times in duration). 2r o
_2 .. 21. 2 4 £3 10 12
0 g.— ‘29 2 45—— s s to .112
2 0 E j 2 l ____4 g
l [ # D l
I Figure 3. Time intervals processed by successive correlators 317 3.16 (cont’d) For this example, Figure 4 shows the output of each
successive correlator. We see 6 successive results from
each windowed signal in Figure 3 that has here been
productintegrated with each of the staggered
templates..Note that the signal values at the successive sampling times 6, 7, , 11 correspond to the PAM signal
values that had been sent. Figure 4. Outputs of successive correlators 318 3.17
The overall (channel and system) impulse response is Mr) 2 5(t)+oc8(t—T). We need a compensating (equalizing) ﬁlter
with impulse response 60‘) that forces h(t)*c(t)= 60‘) and zero everywhere else (zeroforcing ﬁlter). The impulse response of the
equalizing ﬁlter can have the following form: C(t) = 005(1) + 0,5(1— T) + (225(15— 21‘) + c350 —3T) + .  where {ck} are the weights or ﬁlter values at times k = 0, l, 2, 3,
After equalizing, the system output is obtained by cenvolving the
overall impulse response with the ﬁlter impulse response, as
follows:
I10) * C(r) : 605(3) + 6,50:  T) + 6250? — 2T) + 6350‘ — 31") +  ..
+OLCOSO‘~T)+(1615(t+—2T)+a625(t3T)+ We solve for the {ck} weights recursively, forcing the output to be
equal to 1 at time t= 0, and to be 0 elsewhere. At t = Contribution to
on ut r 0 C“ 
T [ Cl+0tCO Therefore, the ﬁlter impulse response is:
c(r) = 5(2‘) —o.5(t — T) + 0850— 27‘) —a’5(t — 3T)
And the output is: r(r)=h(t)*c(r)=1x5(t)+0x6(tLZT)+0xB(t3T)0t“ x5(t—4T)
=5(t)—a‘5(t4T) The ﬁlter can be designed as a tapped delay line. The longer it is
(more taps), the more 181 terms can be forced to zero. If or = 1/2,
then the 4tap ﬁlter described above has an impulse response
represented by a 1 lus three Os, and the resulting 181 has a
magnitude of (1/2) = 1/256. Further ISI suppression can be
accomplished with a longer ﬁlter. 0 9‘1. 7‘, IX. 49,
MM“ ' Mu W 44
wziEW (3.871,) . 115:4
71% is a": warm W' e“ 0.91:1. 0.1.371, o.7.07 o
C. = O:3071 9. 9'3»? 0, $513 i
0.1.101 o. 3019‘ 0,31“ 0 320 ’11! WW {50.)} W M
M ,f‘ {4% [may
£5!an 3253, Fan/ﬁe {300} = 014'5) M67? 0.0000)
) Lbaog manna! —o, 5’91) 0, “4.3 WM 1 = 0.180"; M “6? ISI
5‘“ “TJFSI .. 0.42?! 321 3.19 Channel response: [0.01 0.02 0.03 0.10 1.00 0.20 0.10 0.05 0.02]
Matrix description of problem 0.01 0 0 0 0 0 0 0 0 0
0.02 0.01 0 0 0 0 0 0 0 CO 0
0.03 0.02 0.01 0 0 0 0 0 0 01 0
0.10 0.03 002 0.01 0 0 0 0 0 02 0
1.00 0.10 0.03 0.02 0.01 0 0 0 0 C3 0
0.20 1.00 0.10 0.03 0.02 0.01 0 0 0 C4 0
0.10 0.20 1.00 0.10 0.03 0.02 0.01 0 0 C5 0
0.05 0.10 0.20 1.00 0.10 0.03 0.02 0.01 0 0'5 0
0.02 0.05 0.10 0.20 1.00 0.10 0.03 0.02 0.01 07 : 1
0 0.02 0.05 0.10 0.20 1.00 0.10 0.03 0.02 ‘33 0
0 0 0.02 0.05 0.10 0.20 1.00 0.10 0.03 0
0 0 0 0.02 0.05 0.10 0.20 1.00 0.10 0
0 0 0 0 .002 0.05 0.10 0.20 1.00 0
0 0 0 0 0 0.02 0.05 0.10 0.20. 0
0 0 0 0 0 0 0.02 0.05 0.10 0
0 0 o 0 0 0 0 0.02 0.05 0
0 0 0 0 0 o 0 0 0.02 0
Equivalent form xc = z
, 1
Form of Solutlon c=(x7x) x77.
Output of equalized channel: 3011111011 —0.0000 —0.0003 0.0001 0.0003 —0.0000
00 0.0033 0.0000 “0.0000 0.0024 0.9953 0.0111
3; 'glgggg 0.0947 0.0202 —0.0061 0.0063 —0.0024
03 __ 0.1232 —0.0011 0.0001
04 — 1.0521 , . . . . 05 0.2225 Pnor to equalization, the maXImum smgle ISI
2? 23223 magnitude was 0.2, and the sum of all the ISI
C8 0.0039 magnitude contributions was 0.530. After equalization, the maximum single ISI
magnitude is 0.0947 and the sum of all the
ISI magnitude contributions is 0.1450. 322 3.20 G1 02 Signals at points A, B, C, and D have units of volts (which. characterizes most any signal—processing device). If the
transfer function or gain of the multiplier is G1, then its unit are:
r(t) x s,(r) >< G1 = volts (point C) Thus, volts x volts x GI = volts
Units of G1 = 1/volts If the gain of the integrator is 62, then its units are: volts (point C) integrated over T seconds x G; = volts (point D)
Thus, voltseconds x G; = volts
Units of G2 = l/seconds Therefore, the overall gain or transfer function of the
product integrator is 1/voltseconds. We thus can view the overall transformation as an input energy (voltsquared
seconds) times a gain factor of 1/volt—seconds yielding
volts/voltsquaredseconds (i.e., an output voltage
proportional to energy). 3—23 ...
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 Spring '08
 Nagaraj,SV

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