2.3 FOURIER TRANSFORM FREQUENCY DOMAIN
2.3.3 Power-Spectral Densities of Various PCM Signals
A general expression for the power spectra of signals modulated by random, and pseudorandom data sequences can be derived by representing the random data sequence as a Markov process with a known transition probability matrix. This analysis was first reported in 1961 by Titsworth and Welch in a Jet Propulsions Laboratory Technical Report [6], and was summarized later in a book by Lindsey and Simon [11, sec. 1-5].
The general expression is rather complicated, and requires several definitions that will not be discussed here. For the special case of binary, symmetric, equally likely signals, (s1(t) =;s0(t) = s(t)) the general expression for the PSD reduces to to the simple result
P (f) = 1 TjFs(j2f )j2; (2.137)
whereFs(j2f )is the Fourier transform of the data pulses(t). When the data pulse is rectangular,
s(t) =rect(t=T ); (2.138)
the Fourier transform is given by
Fs(j2f ) = Tsinc(fT ): (2.139)
Therefore, the PSD, using the method of Titsworth and Welch, is
P (f ) = Tsinc2(fT ); (2.140)
which is the same result that we derived by applying the definition of the Fourier transform directly to the signal. In addition to NRZ data, Lindsey and Simon give results for various pulse-code-modulation (PCM) formats. These formats are illustrated in Fig. 1.5 of [11], and are summarized here in Fig. 2.16. Although we will be dealing with NRZ data in the remainder of this dissertation, before moving on, it is instructive to consider the spectra of other data formats.
Return-to-Zero (RZ) Signaling Format PSD The RZ format has a dc value, and also has spectral lines at harmonics of the bit-rate. For equiprobable data, the PSD as given in (1-23) of [11] is
P (f ) = 116 (f)
| {z }
(dc value)
+ 116nX=;11
n6=0
2 n
2 (f ;nBT)
| {z }
(clock tone harmonics)
+ T
16sinc2(fT=2)
| {z }
(continuous spectrum)
:
(2.141)
Mathematical Preliminaries 61
-1.5 -1 -0.5 0 0.5 1 1.5
0 2 4 6 8 10 12 14 16 18 20
-1.5 -1 -0.5 0 0.5 1 1.5
0 2 4 6 8 10 12 14 16 18 20
-1.5 -1 -0.5 0 0.5 1 1.5
0 2 4 6 8 10 12 14 16 18 20
-1.5 -1 -0.5 0 0.5 1 1.5
0 2 4 6 8 10 12 14 16 18 20
NRZ one: is +1 zero: is -1
RZ
one: is pulse of duration T/2 zero: is no pulse
Manchester
one: is positive transition in center of bit-interval zero: is negative transition in center of bit-interval (derived by multiplying NRZ with the clock) Milller
one: is transition in center of bit-interval zero: is no transition, unless followed by another zero, in which case a transition is placed at the end of the bit-interval
Figure 2.16 Various pulse-code-modulation (PCM) formats for transmission of binary data.
Non-Return-to-Zero (NRZ) Signaling Format PSD We have already shown that the PSD for NRZ data is given by
P (f ) = Tsinc2(fT ): (2.142)
Bi-Phase or Manchester Coding PSD Bi-phase, or Manchester coded waveforms are obtained by dithering an NRZ bit-stream with the system clock, and [11] gives the PSD in (1-25) as
P (f) = Tsinc2(fT=2)sin2(fT=2): (2.143)
Delay Modulation or Miller Coding PSD The PSD for delay modulation is given in (1-31) in [11]. If we define a parametersuch that
4= fT; (2.144)
and two vectorsaandbas
a =
2
6
6
6
6
6
6
6
6
6
6
6
6
4
23
;2
;22
;12 12 5 2
;8 2
3
7
7
7
7
7
7
7
7
7
7
7
7
5
; b =
2
6
6
6
6
6
6
6
6
6
6
6
6
4
cos(0) cos() cos(2) cos(3) cos(4) cos(5) cos(6) cos(7) cos(8)
3
7
7
7
7
7
7
7
7
7
7
7
7
5
; (2.145)
62 Chapter 2
-100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0
0 1 2 3 4 5 6 7 8 9 10
-100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0
0 1 2 3 4 5 6 7 8 9 10
-100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0
0 1 2 3 4 5 6 7 8 9 10
-100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0
0 1 2 3 4 5 6 7 8 9 10
NRZ
RZ
Manchester
Miller
Figure 2.17 Power spectral Densities for RZ, NRZ, Manchester coded, and Miller coded, binary signaling formats.
then the PSD can be written as
P (f ) = T
22(17 + 8cos(8))[ab]: (2.146)
Comparison of Spectra for Various PCM Formats The power spectra for the above PCM signaling formats are plotted in Fig. 2.17. We notice that the PSD for RZ data has the same functional form as for NRZ data, except that the bandwidth is doubled, and there are spectral lines in the RZ spectrum. The spectral lines arise because the random phase reversals that we saw in NRZ data are no longer present. Since the RZ data is always forced to return to zero, there is no ambiguity about the starting point of a transition. In other words, falling edges only occur at the start of a bit period, and rising edges only occur in the middle of a bit-period. Since spectral lines are present in RZ data, we could extract the clock directly from the data signal without using edge-detection circuits. However, the penalty in terms of increased bandwidth required, is most often too high a cost to pay for this convenience.
Manchester coded data also has its power spread over a larger bandwidth than NRZ data. However, due to the presence of at least one transition per bit-period, there is little dc energy in this signal. This can be important for practical circuit design. For example, when the data is detected with an optical transducer, there will be indeterminate dc offsets. Further, there will be an unknown dark current present in the photodiode
Mathematical Preliminaries 63
detector, also giving rise to an unknown dc value in the final data steam. Often the data processing circuitry that follows the optical transducer requires a well defined dc value, necessitating a restoration of the dc value of the data. A common technique for restoring the dc value is to average the data, compare it to a reference, and add the difference back to the data. The problem with this technique is that it performs a highpass function on the data, and any dc components of the data will be filtered out. This is a serious problem in dealing with NRZ data which has most of its energy concentrated at low frequencies. However, with Manchester coded data, the problem is averted.
Miller Coding (delay modulation) offers desirable time-domain and frequency domain properties. In the time domain there is an average of one transition per bit-period as opposed to 1/2 for NRZ data. We will see in chapter 4 that the accuracy of the recovered clock is proportional to the square-root of the average number of transitions per bit-period. Miller coding also has desirable frequency-domain properties. As in the case of Manchester coding, the power at dc is also zero, so that we can avoid problems with restoring the dc value. The primary benefit is that most of the power is concentrated in a much narrower frequency band than for RZ, NRZ, or Manchester coding. This means that a narrowband filter can pass the majority of the signal power, while reducing the contribution of additive broadband noise in comparison with the other signaling formats.