Module 5 Carrier Modulation. Version 2 ECE IIT, Kharagpur

Transcription

1 Module 5 Carrier Modulation Version ECE II, Kharagpur

2 Lesson 5 Quaternary Phase Shift Keying (QPSK) Modulation Version ECE II, Kharagpur

3 After reading this lesson, you will learn about Quaternary Phase Shift Keying (QPSK); Generation of QPSK signal; Spetrum of QPSK signal; Offset QPSK (OQPSK); M-ary PSK; Quaternary Phase Shift Keying (QPSK) his modulation sheme is very important for developing onepts of twodimensional I-Q modulations as well as for its pratial relevane. In a sense, QPSK is an expanded version from binary PSK where in a symbol onsists of two bits and two orthonormal basis funtions are used. A group of two bits is often alled a dibit. So, four dibits are possible. Eah symbol arries same energy. Let, E: Energy per Symbol and : Symbol Duration =. b, where b : duration of 1 bit. hen, a general expression for QPSK modulated signal, without any pulse shaping, is: E π si() t = os π ft+ (i 1) ; 0 t ; i = 1,,3, where, f = n. = n. is the arrier (IF) frequeny. b On simple trigonometri expansion, the modulated signal s i (t) an also be expressed as: E π E π si( t) =.os (i 1).os π ft.sin (i 1).sinπ ft 4 4 ; 0 t 5.5. he two basis funtions are: ϕ1() t =.osπ ft ; 0 t and ϕ() t =.sinπ ft ; 0 t he four signal points, expressed as vetors, are: { os ( 1) π 4 sin ( 1) π 4 } si = E i E i si1 = s i ; i = 1,,3, Fig shows the signal onstellation for QPSK modulation. Note that all the four points are equidistant from the origin and hene lying on a irle. In this plain version of QPSK, a symbol transition an our only after at least = b se. hat is, the symbol rate R s = 0.5R b. his is an important observation beause one an guess that for a given binary data rate, the transmission bandwidth for QPSK is half of that needed by BPSK modulation sheme. We disuss about it more later. Version ECE II, Kharagpur

4 φ s 3 (0 1) * * s 4 (1 1) E * s (0 0) 0 E * s 1 (1 0) ϕ 1 os( wt + θ ) Fig Signal onstellation for QPSK. Note that in the above diagram θ has been onsidered to be zero. Any fixed non-zero initial phase of the basis funtions is permissible in general. Now, let us onsider a random binary data sequene: Let us designate the bits as odd (b o ) and even (b e ) so that one modulation symbol onsists of one odd bit and the adjaent even bit. he above sequene an be split into an odd bit sequene ( ) and an even bit sequene ( ). In pratie, it an be ahieved by a 1-to- DEMUX. Now, the modulating symbol sequene an be onstruted by taking one bit eah from the odd and even sequenes at a time as {(10), (11), (10), (11), (00), (01), (10), }. We started with the odd sequene. Now we an reognize the binary bit stream as a sequene of signal points whih are to be transmitted: { s 1, s 4, s 1, s 4, s, s 3, s 1, }. With referene to Fig.5.5.1, let us note that when the modulating symbol hanges from s 1 to s 4, it ultimately auses a phase shift of π in the pass band modulated signal [from π to + π in the diagram]. However, when the 4 4 modulating symbol hanges from s 4 to s, it auses a phase shift of π in the pass band modulated signal [from + π to 5π + in the diagram]. So, a phase hange of 0 or 4 4 π or π ours in the modulated signal every b se. It is interesting to note that as no pulse shaping has been used, the phase hanges our almost instantaneously. Sharp phase transitions give rise to signifiant side lobes in the spetrum of the modulated signal. Version ECE II, Kharagpur

5 able summarizes the features of QPSK signal onstellation. Input Dibit (b 0 ) (b e ) s Phase of Coordinates of signal points QPSK s i1 s i i π 4 + E E 1 s 0 0 3π 4 E E s π 4 E + E 3 s 1 1 7π E + E 4 able Feature summary of QPSK signal onstellation Fig.5.5. shows the QPSK modulated waveform for a data sequene For better illustration, only three arrier yles have been shown per symbol duration. Fig.5.5. QPSK modulated waveform Generation of QPSK modulated signal Let us reall that the time-limited energy signals for QPSK modulation an be expressed as, E E si() t =.os ( i 1) π 4.os wt.sin ( i 1) π 4.sin wt Version ECE II, Kharagpur

6 = E.os ( i 1) π 4.oswt Esin ( i 1) π 4. sin wt = s ϕ t + s ϕ t i = 1,, () () i1 1 i he QPSK modulated wave an be expressed in several ways suh as: s() t = E. dodd () t. os wt + E. deven () t. sinwt = E E. dodd () t os wt. deven () t sinw + t E E = dodd () t. os wt + deven () t. sinwt For narrowband transmission, we an further express s(t) as: s(t) ui () t.os wt uq( t).sin wt u t = u t + ju t is the omplex low-pass equivalent representation of s(t). where () () () I Q One an readily observe that, for retangular bipolar representation of information bits and without any further pulse shaping, u I (t) = E. d odd (t) and (t) = u Q E. d even (t) Note that while expressing Eq , we have absorbed the - sign, assoiated with the quadrature arrier sin ω t in d even (t). We have also assumed that d odd (t) = +1.0 for b o 1 while d even (t) = -1.0 when b e 1. his is not a major issue in onept building as its equivalent effet an be implemented by inverting the quadrature arrier. Fig (a) shows a shemati diagram of a QPSK modulator following Eq Note that the first blok, aepting the binary sequene, does the job of generation of odd and even sequenes as well as the job of saling (representing) eah bit appropriately so that its outputs are s i1 and s i (Eq.5.5.5). Fig (b) is largely similar to Fig (a) but is better suited for simple implementation. Close observation will reveal that both the shemes are equivalent while the seond sheme allows adjustment of power of the modulated signal by adjusting the arrier amplitudes. Inidentally, both the in-phase arrier and the quadrature phase arriers are obtained from a single ontinuous-wave osillator in pratie. Version ECE II, Kharagpur

7 Data Sequene Bit splitter / de-multi- -plexer and saling unit s i1 s i ϕ 1() t =.oswt ϕ () t =.sinwt + Σ + s(t) QPSK Modulated o/p Fig (a) Blok shemati diagram of a QPSK modulator E os wt = Eϕ1 ( t +1 b - 1 Serial to parallel onvert er (DEMUX) d odd (t) = b d even (t) Q - path I - path - Σ s(t) QPSK output ±1 E sin wt Fig (b) Another shemati diagram of a QPSK modulator, equivalent to Fig (a) but more suitable in pratie he QPSK modulators shown in Fig follow a popular and general struture known as I/Q (In-phase / Quadrature-phase) struture. One may reognize that the output of the multiplier in the I-path is similar to a BPSK modulated signal where the modulating sequene has been derived from the odd sequene. Similarly, the output of the multiplier in the Q-path is a BPSK modulated signal where the modulating sequene is derived from the even sequene and the arrier is a sine wave. If the even and odd bits are independent of eah other while ourring randomly at the input to the modulator, the Version ECE II, Kharagpur

8 QPSK modulated signal an indeed be viewed as onsisting of two independent BPSK modulated signals with orthogonal arriers. he struture of a QPSK demodulator, following the onept of orrelation reeiver, is shown in Fig he reeived signal r(t) is an IF band pass signal, onsisting of a desired modulated signal s(t) and in-band thermal noise. One an identify the I- and Q- path orrelators, followed by two sampling units. he sampling units work in tandem and sample the outputs of respetive integrator output every = b seond, where b is the duration of an information bit in seond. From our understanding of orrelation reeiver, we know that the sampler outputs, i.e. r 1 and r are independent random variables with Gaussian probability distribution. heir variane is same and deided by the noise variane while their means are ± E, following our style of representation. Note that the polarity of the sampler output indiates best estimate of the orresponding information bit. his task is aomplished by the vetor reeiver, whih onsists of two idential binary omparators as indiated in Fig he output of the omparators are interpreted and multiplexed to generate the demodulated information sequene ( d ˆ( t ) in the figure). r(t) ϕ 1 (t) I - Path = b 0 dt r 1 t I = b M O/P Q - Path = b 0 dt r U X d ˆ( t ) ϕ (t) t Q = b Fig Correlation reeiver struture of QPSK demodulator We had several ideal assumptions in the above desriptions suh as a) ideal regeneration of arrier phase and frequeny at the reeiver, b) omplete knowledge of symbol transition instants, to whih the sampling lok should be synhronized, ) linear modulation hannel between the modulator output and our demodulator input and so Version ECE II, Kharagpur

9 forth. hese issues must be addressed satisfatorily while designing an effiient QPSK modem. Spetrum of QPSK modulated signal o determine the spetrum of QPSK modulated signal, we follow an approah similar to the one we followed for BPSK modulation in the previous lesson. We assume a long sequene of random independent bits as our information sequene. Without Nyquist filtering, the shaping funtion in this ase an be written as: E gt () = ; 0 t = b After some straight forward manipulation, the single-sided spetrum of the equivalent omplex baseband signal u ~ ( t ) an be expressed as: U ( ).sin ( ) B f = E f Here E is the energy per symbol and is the symbol duration. he above expression an also be put in terms of the orresponding parameters assoiated with one information bit: ( f ) U B b = 4. E.sin ( f) Fig shows a sketh of single-sided baseband spetrum of QPSK modulated signal vs. the normalized frequeny (f b ). Note that the main lobe has a null at f b = 0.5.f. = 0.5 beause no Nyquist pulse shaping was adopted. he width of the main lobe is half of that neessary for BPSK modulation. So, for a given data rate, QPSK is more bandwidth effiient. Further, the peak of the first sidelobe is not negligibly small ompared to the main lobe peak. he side lobe peak is about 1 db below the main lobe peak. he peaks of the subsequent lobes monotonially derease. So, theoretially the spetrum strethes towards infinity. As disussed in Module #4, the spetrum is restrited in a pratial system by resorting to pulse shaping. he single-sided equivalent Nyquist bandwidth for QPSK = (1/) symbol rate (Hz) = 1 (Hz) = 1 (Hz). So, the 4 b normalized single-sided equivalent Nyquist bandwidth = ¼ = 0.5. he Nyquist transmission bandwidth of the real pass band modulated signal s(t) = x single-sided Nyquist bandwidth = 1 (Hz) = 1 (Hz) he symbol rate. b b Version ECE II, Kharagpur

10 BPSK f b Fig Normalized base band bandwidth of QPSK and BPSK modulated signals he atual transmission bandwidth that is neessary = Nyquist transmission bandwidth) x (1 + α) Hz = (1 + α). 1 Hz = (1 + α).r s Hz, where R s is the symbol rate in symbols/se. Offset QPSK (OQPSK) As we have noted earlier, forming symbols with two bits at a time leads to hange in phase of QPSK modulated signal by as muh as 180. Suh large phase transition over a small symbol interval auses momentary but large amplitude hange in the signal. his leads to relatively higher sidelobe peaks in the spetrum and it is avoidable to a onsiderable extent by adopting a simple trik. Offsetting the timing of the odd and even bits by one bit period ensures that the in-phase and quadrature omponents do not hange at the same time instant and as a result, the maximum phase transition will be limited to π at a time, though the frequeny of phase hanges over a large period of observation will be more. he resultant effet is that the sidelobe levels derease to a good extent and the demodulator performs relatively better even if the modulation hannel is slightly nonlinear in behaviour. he equivalent Nuquist bandwidth is not altered by this method. his simple and pratial variation of QPSK is known as Offset QPSK. M ary PSK his is a family of two-dimensional phase shift keying modulation shemes. Several bandwidth effiient shemes of this family are important for pratial wireless appliations. As a generalization of the onept of PSK modulation, let us deide to form a modulating symbol by grouping m onseutive binary bits together. So, the number of possible modulating symbols is, M = m and the symbol duration = m. b. Fig shows the signal onstellation for m = 3. his modulation sheme is alled as 8-PSK or Otal Phase Shift Keying. he signal points, indiated by *, are equally spaed on a irle. his implies that all modulation symbols s i (t), 0 i (M-1), are of same energy E. he dashed straight lines are used to denote the deision zones for the symbols for optimum deision-making at the reeiver. Version ECE II, Kharagpur

11 Z 5 ϕ =.sinϖ t Z 4 Z 3 Z Z 1 ϕ = 1 os wt Fig Signal spae for 8-PSK modulation he two basis funtions are similar to what we onsidered for QPSK, viz., ϕ1 () t = os π ft and ϕ () t = sin π ft ; 0 t he signal points an be distinguished by their angular loation: π θ i i = ; i = 0, 1... M M he time-limited energy signals s i (t) for modulation an be expressed in general as E π i si() t =.os π ft M Considering M-ary PSK modulation shemes are narrowband-type, the general form of the modulated signal is s t = u t os wt u t sin wt () () () I Q Fig shows a blok shemati for an M-ary PSK modulator. he baseband proessing unit reeives information bit stream serially (or in parallel), forms information symbols from groups of m onseutive bits and generates the two salars s i1 and s i appropriately. Note that these salars assume disrete values and an be realized in Version ECE II, Kharagpur

12 pratie in multiple ways. As a speifi example, the normalized disrete values that are to be generated for 8-PSK are given below in able s i1 Information Sequene Baseband Proessing Unit s i ϕ 1() t =.oswt ϕ () t =.sinwt + - Σ s(t) M-ary PSK modulated o/p Fig Blok shemati diagram of M-ary PSK modulator i s i s i able 5.5. Normalized salars for 8-PSK modulation. Without any pulse shaping, the u I (t) and u Q (t) of Eq are proportional to s i1 and s i respetively. Beside this baseband proessing unit, the M-ary PSK modulator follows the general struture of an I/Q modulator. Fig shows a sheme for demodulating M-ary PSK signal following the priniple of orrelation reeiver. he in-phase and quadrature-phase orrelator outputs are: π i ri = E os + WI, i = 0, 1 M 1 M π i rq = Esin + W Q, i = 0, 1 M M Version ECE II, Kharagpur

13 Coherent ϕ 1 (t) r(t) 0 0 dt dt r I r r Q Phase Disriminator r 1 Q θ = tan ri Look Up able (LU) O/P ϕ (t) Fig Struture of M-ary PSK demodulator W I represents the in-phase noise sample and W Q represents the Q-phase noise sample. he samples are taken one every m. b se. A notable differene with the orrelation reeiver of a QPSK demodulator is in the design of the vetor reeiver. Essentially it is a phase disriminator, followed by a map or look-up table (LU). Complexity in the design of an M-ary PSK modem inreases with m. Problems Q5.5.1) Write the expression of a QPSK modulated signal & explain all the symbols you have used. Q5.5.) What happens to a QPSK modulated signal if the two basis funtions are the ϕ t = ϕ t. same that is () ( ) 1 Q5.5.3) Suggest how a phase disriminator an be implemented for an 8-PSK signal? Version ECE II, Kharagpur