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Analysis of a digital PSK modulator
Microelectron. Reliab.,Vol. 20. pp. 509-512 © PergamonPress Ltd. 1980.Printed in Great Britain
0026--2714/80~801-0509$02.00/0
ANALYSIS OF A DIGITAL PSK MODULATOR S. M. B o z i c a n d C. TAMVACLIS Department of Electronic and Electrical Engineering, University of Birmingham, P.O. Box 363, Birmingham B15 2TT (Received for publication 14 November 1979)
Abstract--This is a time and frequency domain study of an exclusive-OR gate used as a digital modulator. The output spectrum obtained has been partitioned into the desired PSK waveform, and the undesired part to be removed by the following filter. A simpler scheme of PSK generation has al~o been briefly discussed.
INTRODUCTION
I
The microelectronic realizations of digital modulators follow the general trend of digital processing in data communications. Papers in the technical literature deal with various theoretical and practical areas of digital filters and modulators [I, 2]. However, there are some aspects of theoretical analysis which do not appear to be sufficiently clear. This paper attempts to clarify certain points of analysis using the exclusive-OR gate as a modulator [2-1. The approach is such that it links logic function expression with the analogue system functional analysis. The result obtained for the frequency spectrum of the shape factor, at the modulator output, is split into two parts. One part produces the desired sinewave carrier modulation of data bits, and the other one, which is undesirable, defines the transfer function (and impulse response) required of the following filter. This filter is not discussed but it would be a binary transversal filter [31, designed to have a required impulse response within a given time interval. An alternative and simpler version of PSK modulator is also briefly discussed. It offers a solution particularly suitable for binary logic circuits [4].
B = ~(1
+
u),
(2)
where u is a square-wave function oscillating between + 1 and - 1 . The exclusive-OR output can then be written as
A" B
+
A" B
=
1 1~ akp(t - kD)" ~(
+
u)
~akp(t kD)'~(I+ =). -
It can be easily shown that the above reduces to
A~B=~[1-~xkp(t-kD)'u(t)~
(3)
where xk = ak -- ak is a sequence of bipolar numbers (+1, - 1 ) . The first term is d.c. on the account of ak + ak = 1 and ~.p(t - kD) = 1, because p(t) = 1 in each data interval. Ignoring d.c. term, we see that the modulation term can be written as
(4)
re(t) = x(t) " u(t), where
x(t) = ~,xkp(t
-
kD),
(5)
k EXCLUSIVE-OR AS PSK MODULATOR
The PSK modulator, Fig. 1, consists of the exclusiveOR circuit followed by the binary transversal filter (BTF). We concentrate in this paper on the analysis of the exclusive-OR, fed by the data (A) and carrier (B) signals. Both of these are in binary (0, 1) form, and one can easily obtain the output waveform, as is shown in Fig. 1. For a more detailed analysis we describe the general data signal as
C
A=o (t)
=
a(t)= ~ a k p ( t -
kD),
0
A@B
~I I
(l+u) I
_D 2
A
~ BOR =~
0 D 2
,i, ~'~T~'-
(1)
k
o
where p(t) is a square wave within each data interval D, and ak a random sequence of binary (0, 1) values carrying the information. The carrier is a periodic binary (0, 1)waveform, and as such can be described by:
I
A~B
nom m m m n n Fig. 1.
5O9
,i
I
510 and
S.M. Bozac and C. TAMVACLIS
x(t)
u(t) can be expressed either as u2.+ x cos (2n + 1)o9~t
,/ p(t)
I
(6)
n= 0,1,2,3,...
or as
2 -I
u2,+1 sin (2n + l)m~t, n=0,1,2,3 ....
depending on whether u(t) is centred at zero or shifted by T/4, (see Fig. 1 for T, in waveform B) of the centre, respectively. In either case the Fourier transform of the product in equation (4) can then be written in the following way
M(co) = Fire(t)] = ~ u2.+ , F[x(t) cos (2n +
1)coot]
or
1 M(~) = ~ u 2 . + ,
X[¢o - (2n + l)m¢]
+~u2.+,X[to+(2n+
~o t
X(Co)=P(oJ)'[~kxke-~° ~ where P(o~) is the shape factor (SF) and the factor within the brackets is the discrimination factor (DF). Therefore, equation (7) can be rewritten as
where r is an integer number [21. With reference to the waveforms in Fig. 1, one can show that r = D/T. This means there must be an integer number of carrier pulses within the data pulse, i.e. the transitions of the data signal coincide with the transitions of the carrier. Furthermore, we have e.i% ko = e - Jo~,ko
for to, given by equation (10) Therefore, if the carrier frequency is chosen to be as in equation (10), the output spectrum of the exclusiveOR gate, equation (8), can be more simply written as M(co)= t l~~ u.2 . + l P [ C o - ( 2 n + 1)oJc] +
1 M(co) = ~ u 2 . + 1P[t~ - (2n + 1)co~]
=~ u~.+,P[o~+
1
SF
(8)
THE CHOICE OF CARRIER FREQUENCY
The ideal PSK modulation we want to produce is
s(t)=[~xkp(t-kD)]costo~t, with the frequency spectrum given by
S(to) = ½P(a~-- co~)[~..xk e- ;'°'-°~,k°] Lk JDF +½P(og+
One could calculate the modulated waveform shape factor from the expression shown in equation (11). In that case we would use the input data shape factor and perform operations indicated in SF part of equation (11). However, it seems easier to derive this quantity from the modulated waveform if(t) as is illustrated in Fig. 2. Then for r carrier pulses within one datum pulse one can write for the shape factor
p'(t) =
(9)
The output of the exclusive-OR gate given by equation (8) will have the same discrimination factor as the ideal PSK of equation (9) if the carrier frequency is chosen to be R
_IDF" (11)
CALCULATION AND PARTITIONING OF THE SHAPE FACTOR
O..)c)[~Xke-J(t°+%)kD]DF.
coc = r/~,
L
The first factor in the above equation is the shape factor (SF) which at the output of the exclusive-OR gate would correspond to the modulated waveform denoted by p'(t) in Fig. 2. The frequency spectrum for p'(t) is derived and examined in the following section.
+ (2n + l)r~,]
X [~"~.Xk e -j[t°+(2n+ llc°clkD] I_ /DF.
(2n + 1)co,]
g n
x
+ ~u2.+,P[~
pl(t)
Fig. 2.
1)co,]. (7)
In the above, X(og) is the Fourier transform of equation (5) given by
i
(10)
~ ( - 1)k rect t - (2k + 1 k=0
r)(D/2r)
(12)
-D/7
where D is the data pulse width, and rect function is defined as
• / t - ton
rec'k--Xr-)--
1,1tl < T
A,
o, l t [ >
-2
(13)
Analysis of a digital PSK modulator One can see from Fig. 2, that p'(t) already has basic form of the sinewave which we want for the modulation. To extract this waveform, we apply the Fourier transform to if(t) and obtain
511 hi(t)
°°'°2
C,
L l
_
y(t) I
I
Y
p,(fo) = ( 2 sin foD - l ( - 1}ke-$2k + 1- ')(~'0/2"1. (14) --~r)k' =~o
hzlt ) ~ h l (t) Z ~ ( t - kD)
After summation of the geometric series, the above can be written as
Fig. 3.
• coD 2fOc~os-(~-ff/2r)J V e j t ~ ° / 2 ) - - ( - - 1)" e - ~('~°/2) X
I
-j_
modify the above obtained impulse response according to the new transfer function defined by
-I
(-fOe)) (15)
which contains spectra around frequencies (co- foe) and (co + fOe). Splitting the third factor in the above equation into two such parts, we have o
sin (fOD/2 - rn/2)ej~,~/2 +J sin (fOD/2 + rM2) e
--J
fO - -
toe
.~r,q2)
n(fo) H'(fo) = (2/fo) sin (foD/2r)'
where H(fo) is given in equation (19). Then, the resulting shape factor at the output of BTF would be the sinewave of equation (17).
CO + foe
(16) From a list of Fourier transform pairs, one can see that the above expression in the time domain gives rect ( D ) " sin (foal + 2 ~
(17,
which is the envelope of the PSK signal we want to produce. It is very instructive to sketch this function for, say, r = 3 and 4. The first and second factors in equation (15) are undesirable since they interfere with the desired response given by the third factor. Ignoring for the moment the first factor in equation (15), we can remove the second factor by making the transfer function of the binary transversal filter, (BTF) in Fig. 1, to be its inverse, i.e. H(fo) =
(21)
2fo, cos (foD/2r) fo2 _ o92
A SIMPLER FORM OF BINARY PSK MODULATOR
The scheme shown in Fig. 3 has two AND gates and one NOT gate, followed'by two binary transversal filters (BTFI and BTF2). The input data a(t) is sampled by a series of impulses so that the outputs of AND gates are ml(t) = ~ . a k 6 ( t -- kD) k m2(t)
= ~,, ak ¢~(t - - k O ) .
(22)
k
The output of this system can be written as y(t) = mr(t) * hi(t) + m2(t)* h2(t), where * denotes the convulution operation. The above can be further written as
(18) y(t) = htCt)* ~.(ak - dk)6Ct - kD),
(23)
k
This expression can be rewritten as H(fo) =
sin [D/2r (fo + toe)] f o + fO e
sin [D/2r(fo - fo,)], + - (19) 0.7 - - foe
since foc = rn/D. From tables of Fourier transform pairs, we obtain the time domain expression corresponding to the above H(fo), as
where we have used equations (22), and h2(t) = - h 1(0. While a~ is unipolar (0, 1), the quantity a k - - ak = Xk becomes bipolar (+ 1 , - 1). Therefore, the output of this system is given by y(t) = x(t)* hi(t),
(24)
where x(t) = ~. xk 6(t -- kD).
h(t) = rect
• cos foct,
(20)
which is the required impulse response of BTF. In the above equation rect [t/(D/r)] is a rectangular pulse of duration D/r, while the cosine waveform has a period of 2D/r. Therefore, the BTF impulse response has a shape of a half cycle of a cosine waveform. To remove also the first factor in equation (15), we would have to
k
The operation of this system is illustrated in Fig. 4, where we have assumed hi(t) =rect (t/D) sin coet with foe = 2n/D. Since the narrowest sample operating BTF is determined by the shift register speed, the filter transfer function will be affected by (sin x)/x) type distorting spectrum. This can be either equalized, or made negligible by using higher clock rates [4].
512
S.M. Bozl¢ and C. TAMVACUS REFERENCES
a(t)
, O
-D
I 2D
I 3D
I 4D
1"
x(t)
!rio V°X~,J Fig. 4.
, ~° V
'
1. L. E. Zegers, et al., Digital signal processing and LSI in modems for data transmission, Philips, J. Res. 33, 226-247 (1978). 2. P. J. Van Gerwen et al., Data modems with integrated digital filters and modulators, IEEE Trans. Commun. Technol. COM-lg, 214-222 (June 1970). 3. S. M. Bozic, Binary transversal filter, Microelectron. Reliab. 19, 219-222 (1979). 4. R. C. French, Binary transversal filters in data modems, Radio Electron, Engr 44, 357-362 (July 1974).
0026--2714/80~801-0509$02.00/0
ANALYSIS OF A DIGITAL PSK MODULATOR S. M. B o z i c a n d C. TAMVACLIS Department of Electronic and Electrical Engineering, University of Birmingham, P.O. Box 363, Birmingham B15 2TT (Received for publication 14 November 1979)
Abstract--This is a time and frequency domain study of an exclusive-OR gate used as a digital modulator. The output spectrum obtained has been partitioned into the desired PSK waveform, and the undesired part to be removed by the following filter. A simpler scheme of PSK generation has al~o been briefly discussed.
INTRODUCTION
I
The microelectronic realizations of digital modulators follow the general trend of digital processing in data communications. Papers in the technical literature deal with various theoretical and practical areas of digital filters and modulators [I, 2]. However, there are some aspects of theoretical analysis which do not appear to be sufficiently clear. This paper attempts to clarify certain points of analysis using the exclusive-OR gate as a modulator [2-1. The approach is such that it links logic function expression with the analogue system functional analysis. The result obtained for the frequency spectrum of the shape factor, at the modulator output, is split into two parts. One part produces the desired sinewave carrier modulation of data bits, and the other one, which is undesirable, defines the transfer function (and impulse response) required of the following filter. This filter is not discussed but it would be a binary transversal filter [31, designed to have a required impulse response within a given time interval. An alternative and simpler version of PSK modulator is also briefly discussed. It offers a solution particularly suitable for binary logic circuits [4].
B = ~(1
+
u),
(2)
where u is a square-wave function oscillating between + 1 and - 1 . The exclusive-OR output can then be written as
A" B
+
A" B
=
1 1~ akp(t - kD)" ~(
+
u)
~akp(t kD)'~(I+ =). -
It can be easily shown that the above reduces to
A~B=~[1-~xkp(t-kD)'u(t)~
(3)
where xk = ak -- ak is a sequence of bipolar numbers (+1, - 1 ) . The first term is d.c. on the account of ak + ak = 1 and ~.p(t - kD) = 1, because p(t) = 1 in each data interval. Ignoring d.c. term, we see that the modulation term can be written as
(4)
re(t) = x(t) " u(t), where
x(t) = ~,xkp(t
-
kD),
(5)
k EXCLUSIVE-OR AS PSK MODULATOR
The PSK modulator, Fig. 1, consists of the exclusiveOR circuit followed by the binary transversal filter (BTF). We concentrate in this paper on the analysis of the exclusive-OR, fed by the data (A) and carrier (B) signals. Both of these are in binary (0, 1) form, and one can easily obtain the output waveform, as is shown in Fig. 1. For a more detailed analysis we describe the general data signal as
C
A=o (t)
=
a(t)= ~ a k p ( t -
kD),
0
A@B
~I I
(l+u) I
_D 2
A
~ BOR =~
0 D 2
,i, ~'~T~'-
(1)
k
o
where p(t) is a square wave within each data interval D, and ak a random sequence of binary (0, 1) values carrying the information. The carrier is a periodic binary (0, 1)waveform, and as such can be described by:
I
A~B
nom m m m n n Fig. 1.
5O9
,i
I
510 and
S.M. Bozac and C. TAMVACLIS
x(t)
u(t) can be expressed either as u2.+ x cos (2n + 1)o9~t
,/ p(t)
I
(6)
n= 0,1,2,3,...
or as
2 -I
u2,+1 sin (2n + l)m~t, n=0,1,2,3 ....
depending on whether u(t) is centred at zero or shifted by T/4, (see Fig. 1 for T, in waveform B) of the centre, respectively. In either case the Fourier transform of the product in equation (4) can then be written in the following way
M(co) = Fire(t)] = ~ u2.+ , F[x(t) cos (2n +
1)coot]
or
1 M(~) = ~ u 2 . + ,
X[¢o - (2n + l)m¢]
+~u2.+,X[to+(2n+
~o t
X(Co)=P(oJ)'[~kxke-~° ~ where P(o~) is the shape factor (SF) and the factor within the brackets is the discrimination factor (DF). Therefore, equation (7) can be rewritten as
where r is an integer number [21. With reference to the waveforms in Fig. 1, one can show that r = D/T. This means there must be an integer number of carrier pulses within the data pulse, i.e. the transitions of the data signal coincide with the transitions of the carrier. Furthermore, we have e.i% ko = e - Jo~,ko
for to, given by equation (10) Therefore, if the carrier frequency is chosen to be as in equation (10), the output spectrum of the exclusiveOR gate, equation (8), can be more simply written as M(co)= t l~~ u.2 . + l P [ C o - ( 2 n + 1)oJc] +
1 M(co) = ~ u 2 . + 1P[t~ - (2n + 1)co~]
=~ u~.+,P[o~+
1
SF
(8)
THE CHOICE OF CARRIER FREQUENCY
The ideal PSK modulation we want to produce is
s(t)=[~xkp(t-kD)]costo~t, with the frequency spectrum given by
S(to) = ½P(a~-- co~)[~..xk e- ;'°'-°~,k°] Lk JDF +½P(og+
One could calculate the modulated waveform shape factor from the expression shown in equation (11). In that case we would use the input data shape factor and perform operations indicated in SF part of equation (11). However, it seems easier to derive this quantity from the modulated waveform if(t) as is illustrated in Fig. 2. Then for r carrier pulses within one datum pulse one can write for the shape factor
p'(t) =
(9)
The output of the exclusive-OR gate given by equation (8) will have the same discrimination factor as the ideal PSK of equation (9) if the carrier frequency is chosen to be R
_IDF" (11)
CALCULATION AND PARTITIONING OF THE SHAPE FACTOR
O..)c)[~Xke-J(t°+%)kD]DF.
coc = r/~,
L
The first factor in the above equation is the shape factor (SF) which at the output of the exclusive-OR gate would correspond to the modulated waveform denoted by p'(t) in Fig. 2. The frequency spectrum for p'(t) is derived and examined in the following section.
+ (2n + l)r~,]
X [~"~.Xk e -j[t°+(2n+ llc°clkD] I_ /DF.
(2n + 1)co,]
g n
x
+ ~u2.+,P[~
pl(t)
Fig. 2.
1)co,]. (7)
In the above, X(og) is the Fourier transform of equation (5) given by
i
(10)
~ ( - 1)k rect t - (2k + 1 k=0
r)(D/2r)
(12)
-D/7
where D is the data pulse width, and rect function is defined as
• / t - ton
rec'k--Xr-)--
1,1tl < T
A,
o, l t [ >
-2
(13)
Analysis of a digital PSK modulator One can see from Fig. 2, that p'(t) already has basic form of the sinewave which we want for the modulation. To extract this waveform, we apply the Fourier transform to if(t) and obtain
511 hi(t)
°°'°2
C,
L l
_
y(t) I
I
Y
p,(fo) = ( 2 sin foD - l ( - 1}ke-$2k + 1- ')(~'0/2"1. (14) --~r)k' =~o
hzlt ) ~ h l (t) Z ~ ( t - kD)
After summation of the geometric series, the above can be written as
Fig. 3.
• coD 2fOc~os-(~-ff/2r)J V e j t ~ ° / 2 ) - - ( - - 1)" e - ~('~°/2) X
I
-j_
modify the above obtained impulse response according to the new transfer function defined by
-I
(-fOe)) (15)
which contains spectra around frequencies (co- foe) and (co + fOe). Splitting the third factor in the above equation into two such parts, we have o
sin (fOD/2 - rn/2)ej~,~/2 +J sin (fOD/2 + rM2) e
--J
fO - -
toe
.~r,q2)
n(fo) H'(fo) = (2/fo) sin (foD/2r)'
where H(fo) is given in equation (19). Then, the resulting shape factor at the output of BTF would be the sinewave of equation (17).
CO + foe
(16) From a list of Fourier transform pairs, one can see that the above expression in the time domain gives rect ( D ) " sin (foal + 2 ~
(17,
which is the envelope of the PSK signal we want to produce. It is very instructive to sketch this function for, say, r = 3 and 4. The first and second factors in equation (15) are undesirable since they interfere with the desired response given by the third factor. Ignoring for the moment the first factor in equation (15), we can remove the second factor by making the transfer function of the binary transversal filter, (BTF) in Fig. 1, to be its inverse, i.e. H(fo) =
(21)
2fo, cos (foD/2r) fo2 _ o92
A SIMPLER FORM OF BINARY PSK MODULATOR
The scheme shown in Fig. 3 has two AND gates and one NOT gate, followed'by two binary transversal filters (BTFI and BTF2). The input data a(t) is sampled by a series of impulses so that the outputs of AND gates are ml(t) = ~ . a k 6 ( t -- kD) k m2(t)
= ~,, ak ¢~(t - - k O ) .
(22)
k
The output of this system can be written as y(t) = mr(t) * hi(t) + m2(t)* h2(t), where * denotes the convulution operation. The above can be further written as
(18) y(t) = htCt)* ~.(ak - dk)6Ct - kD),
(23)
k
This expression can be rewritten as H(fo) =
sin [D/2r (fo + toe)] f o + fO e
sin [D/2r(fo - fo,)], + - (19) 0.7 - - foe
since foc = rn/D. From tables of Fourier transform pairs, we obtain the time domain expression corresponding to the above H(fo), as
where we have used equations (22), and h2(t) = - h 1(0. While a~ is unipolar (0, 1), the quantity a k - - ak = Xk becomes bipolar (+ 1 , - 1). Therefore, the output of this system is given by y(t) = x(t)* hi(t),
(24)
where x(t) = ~. xk 6(t -- kD).
h(t) = rect
• cos foct,
(20)
which is the required impulse response of BTF. In the above equation rect [t/(D/r)] is a rectangular pulse of duration D/r, while the cosine waveform has a period of 2D/r. Therefore, the BTF impulse response has a shape of a half cycle of a cosine waveform. To remove also the first factor in equation (15), we would have to
k
The operation of this system is illustrated in Fig. 4, where we have assumed hi(t) =rect (t/D) sin coet with foe = 2n/D. Since the narrowest sample operating BTF is determined by the shift register speed, the filter transfer function will be affected by (sin x)/x) type distorting spectrum. This can be either equalized, or made negligible by using higher clock rates [4].
512
S.M. Bozl¢ and C. TAMVACUS REFERENCES
a(t)
, O
-D
I 2D
I 3D
I 4D
1"
x(t)
!rio V°X~,J Fig. 4.
, ~° V
'
1. L. E. Zegers, et al., Digital signal processing and LSI in modems for data transmission, Philips, J. Res. 33, 226-247 (1978). 2. P. J. Van Gerwen et al., Data modems with integrated digital filters and modulators, IEEE Trans. Commun. Technol. COM-lg, 214-222 (June 1970). 3. S. M. Bozic, Binary transversal filter, Microelectron. Reliab. 19, 219-222 (1979). 4. R. C. French, Binary transversal filters in data modems, Radio Electron, Engr 44, 357-362 (July 1974).