2.4 LINEAR FILTERING OF RANDOM DATA
2.4.2 Bandlimited Edge-Detected Data
We can now make use of the linear filtering properties of random data signals to find the ESDB of edge-detected data where the pulses are no longer rectangular. Often the data transitions are detected using the circuit of Fig. 2.23. If the lowpass-filter in Fig. 2.23 has a half-cosine impulse function, then the transitions will be sinusoidal, and of the form
(
;sin(2(BT=2)t) for a negative transition;T
2
tT2 sin(2(BT=2)t) for a positive transition ;T
2
tT2. (2.159) T=times the derivative of the data is equal to zero when there is no transition and, is equal to
(
;cos (BTt) for a negative transition;T
2
tT2 cos (BTt) for a positive transition ;T
2
tT2. (2.160)
Mathematical Preliminaries 69
dc(t, ) ec (t, )
Figure 2.24 NRZ data with sinusoidal transitions and raised cosine pulses at each transi- tion.
τp
1/
t T
τp
Figure 2.25 Pulse shape for edge-detected data normalized to have unit area.
After squaring and multiplying by 2, the result is that the edge-detected data is zero for no transition, and for both positive and negative transitions the signal is
ec(t;) = 2cos2(BTt) (for transitions) (2.161)
= 1 + cos(2BTt) for ;T=2tT=2: (2.162)
The resulting signal gives a raised cosine pulse when a transition occurs, as illustrated in Fig. 2.24. If the data signal were alternating every bit, thenec(t;)would be a single tone at the bit-rate.
Derivation of the Energy Spectral Density Based on Rectangular Pulses Results To find the ESDB ofec(t;)the Fourier transform could be obtained directly from the definition. However, it is simpler to apply the results already obtained for the rectangular edge-detected data. If the fundamental pulse shapeeT(t)from Fig. 2.8, is normalized to have unit area as shown in Fig. 2.25, then the new pulseuT(t)is given
by
uT(t) =
1=pT for0tpT
0 elsewhere, (2.163)
and the ESDB from (2.135) is
SdBeu(f ) = SBe(f) (pT )2 =1
2sinc(fpT )2
"
1 + N X1
M=;1sinc
2
f;M T
NT
#
:
(2.164) Asptends toward zero, the envelope gets broader, until in the limit it approaches a constant of1=4, and the normalized energy spectrum is shown if Fig. 2.26 In the time
70 Chapter 2
1/T 2/T 3/T 4/T -1/T
-2/T -3/T
-4/T 5/T
-5/T f
Se(f) White Noise magnitude (1/4)
1/4 Deterministic Harmonics
magnitude (N/4)
N/4
Figure 2.26 Normalized energy spectrum for edge detected data with pulses of unit area.
domain, aspapproaches zero, then the train of unit area pulsesuT(t)become a train of impulse functions. The signal that we desire can now be represented as a convolution of a kernel raised-cosine pulse with this train of random impulses.
ec(t;) = limp
!0
eu(t;)[1 + cos(2BTt)]rect(t=T ) (2.165)
Defining a normalized transfer functionG(j2f )such that G(j2f ) =4F1
T [1 + cos(2BTt)]rect(t=T); (2.166)
we can easily recognize thatG(j2f)as the Fourier transform of rect(t=T )convolved
with impulses of magnitude 1=T at f = 0, and impulses of magnitude 1=2T at f = BT, so that G(j2f) is simply expressed as the superposition of three sinc functions.
G(j2f ) = X1
m=;1
1 2
jmj
sinc((f;mBT)T ) (2.167)
jG(j2f )j2is plotted if Fig. 2.27a, and is compared to a sinc2function in Fig. 2.27b.
The ESDB for the pulsesec(t;)is then given by
SBec(f) = T2jG(j2f )j2plim
!0
SdBeu(f ); (2.168)
or
SBec(f) = T2
4 jG(j2f )j2
"
1 + X1
M=;1sinc
2
f;MBT BT=N
#
: (2.169)
Mathematical Preliminaries 71
-0.2 0 0.2 0.4 0.6 0.8 1
-5 -4 -3 -2 -1 0 1 2 3 4 5
Normalized Frequency (f / B ) T
Normalized Amplitude
-70 -60 -50 -40 -30 -20 -10 0 10
-5 -4 -3 -2 -1 0 1 2 3 4 5
Normalized Frequency (f / B ) T
Normalized Amplitude (dB)
(a) (b)
Figure 2.27 Squared magnitude response of a filter with a raised-cosine impulse response:
(a) linear plot, (b) magnitude in dB compared to a sinc2function.
As the number of bitsN grows the narrow sinc2pulses can be replaced by impulses with equal area as in (2.135), so that
SBec(f) = T2 4
"
1
X
m=;1
1 2
jmj sinc
f;mBT
BT
#
2
"
1 + 1 T
1
X
M=;1(f;MBT)
#
(2.170) Since the envelope ofSBec(f) isjG(j2f )j2, then this energy spectrum will have the desirable property that all harmonics of the signal at multiples of the bit-rate are nulled. This property results from having a kernel-pulse that is non-zero in the interval
t2[0;T ], whereas the rectangular pulses were only non-zero fort2[0;pT ].
Discrete Power Spectrum for Comparison with Simulation The ESDB from (2.170) can be converted to energy dissipated in a1resistor by integratingSBec(f ) over
the appropriate frequency intervals. Ifec(t;)is input to a spectrum analyzer with bandwidth intervals off = BT=Ns, wherefn= nf, then the average two-sided energy-per-bit of the signal in thenthfrequency bin is
EBec(fn) = T2 4
Z fn+BT=2Ns
fn;BT=2Ns jG(j2fn)j2df +
hT
4jG(j2fn)j2 for modBT(fn) = 0i:
(2.171)
72 Chapter 2
ForNslarge, the integral can be approximated byjG(j2fn)j2f. Therefore,
EBec
nBT
Ns
= 14 T Ns
G
j2nB NsT
2
| {z }
modNs(n)6=0
+14 TjG(0)j2(n) ^ +14 TjG(j2BT)j2^
n Ns ;1
+14 TjG(;j2BT)j2 ^
n Ns+ 1
;
(2.172) where
b(n) =
1 forn = 0
0 forn6= 0. (2.173)
The average power is obtained by dividing the energy by the time intervalT. Consider- ing positive frequencies, and remembering that the dc component doesn’t get doubled, then
Pec
nBT Ns
= 12
1 Ns
G
j2nB NsT
2
| {z }
modNs(n)6=0
+12jG(0)j2b(n)
+jG(j2BT)j2bNns ;1
:
(2.174) (2.174) gives the power inNsequally spaced frequency bins; this can be compared directly with simulation results. First, however, we realize that the dc value due to the deterministic part is1=4jG(0)j2 = (1=2)2, so the dc term can be removed by subtracting1=2from the original signal. It is clear that the average value ofec(t;)is
zero when no pulse occurs, and unity when there is a pulse. Since the probability of a pulse is1=2, then the expected value of the signal is1=2, so that by subtracting1=2
fromec(t;)produces a zero-mean random process. A plot of this signalebc(t;)is
shown in Fig. 2.28a with the random NRZ datadc(t;). After removal of the mean, the power spectrum is shown plotted in Fig. 2.28b. The calculated spectrum forNs= 32
is shown in dashed line and a simulation using 32 samples per bit is plotted with a solid line. The simulated curves shows small variations around the calculated curve.
These variation can be reduced by averaging over even more data segments.