CHAPTER 3 JITTER EFFECTS AND FSO MODULATION TECHNIQUES

46 CHAPTER 3 JITTER EFFECTS AND FSO MODULATION TECHNIQUES This chapter discusses the theory of jitter effects and the introduction of the various modulation schemes employed in this investigation. 3.1 JITTER EFFECT Phase Noise (PN) is the short-term instability caused by variation of frequency (phase) of a signal reference to the carrier level and a function of the carrier offset (i.e., relative noise level within a 1-Hz bandwidth). Integration of PN over a given frequency band yields phase jitter RMS. Jitter is fluctuation in the time-axis direction of a digital signal. When this fluctuation is in a long-term cycle, or more specifically less than 10 Hz, it is called "wander", and when the fluctuation is in a cycle above 10 Hz, it is called "jitter". Jitter is short-term variations of the significant instants of a digital signal from their ideal positions in time short term here means phase noise contributions are restricted to frequencies greater than or equal to 10 Hz. 3.1.1 Types of Jitter Measurements Different types of jitter measurements are available. The phase and time jitter are commonly used among researchers and the brief concept is discussed below.

47 Phase jitter or accumulated jitter:- The absolute deviation of a clock edge from its ideal position in timing. Period jitter only accounts for the variation between clock periods, phase jitter accumulates the error of each period and therefore is always larger. The wider the recording time window is, the more frequency bandwidth becomes integrated into the total phase jitter. Phase jitter can also be measured by integrating phase noise over the frequency band of interest. Phase fluctuation φ(t) is called phase noise. Phase noise is usually stated as a ratio between carrier power and noise power at an offset frequency from the carrier. All phase noise is called jitter. SSB (single side band) phase noise L(f) is usually used to express phase noise. L(f) is a function of offset frequency f, the unit of which is dbc/hz. Defined by the SSB phase noise power, it is the total power of a signal in the 1 Hz bandwidth at a frequency f m Hz away from the carrier as shown in the Figure 3.1. Figure 3.1 Schematic diagram of SSB Phase noise characteristics The integrated jitter of a phase noise is plotted over a particular jitter bandwidth. Phase noise data may be recorded as either SSB phase noise L(f) in dbc/hz or phase noise spectral density Sφ(f) in rad 2 /Hz where: L(f) = Sφ(f) 2. (3.1)

48 RMS phase jitter may be expressed in units of dbc, radians, time, or Unit Intervals (UI). By definition, period jitter compares two similar instants in time of a clock source such as two successive rising edges or two successive falling edges. Since the two instants are separated in time by approximately one period, it is reasonable to expect that higher frequency jitter components will contribute more to period jitter than lower frequency jitter components (f<<1/t.) The basic relationship between Jper (rms) and phase noise spectral density Sφ(f),it can be shown that and therefore, 2 φ RMS = 4 Sφ(f) x sin 2 (φf τ)df. (3.2) J PER (RMS) = T 0 φ 2 2π RMS. (3.3) Time jitter or Period jitter:- The short-term variation in clock period over all measured clock cycles are compared with the average clock period. This jitter measure may be specified as either an RMS or peak-to-peak quantity. Time jitter is the deviation in cycle time of a signal with respect to an ideal period over a random sample of cycles. Time jitter is important since it includes the max/min frequency and it specifics the shortest for sub-systems using clock and data signals derived by the same clock source. As shown in Figure 3.2, peak-to-peak period jitter is the difference between the largest clock period and the smallest clock period for all individual clock periods within an observation window. Figure 3.2 Period jitter

49 3.1.2 Time vs. Frequency Domain Measurements Most modern equipment that measures jitter falls into one of two broad categories namely time domain and frequency domain. Time domain equipment is typically devolved in the form of a high-speed digitizing oscilloscope with high single-shot sampling bandwidth. Frequency domain equipment is generally available in the form of a spectrum analyzer. Each of these two categories of equipment has its own advantages and disadvantages. Time domain measurements system is better for low frequency clocks and it is good with data dependent jitter. The frequency domain instruments have lower noise floor and easy detection of spurs vs. random jitter. This thesis focuses on the frequency domain analysis for jitter measurement because of its easy detection and lower noise floor. 3.1.3 Jitter Effect in Optical Propagation Through Atmospheric Turbulence When laser beam is propagating through the free space, the finite optical laser beam is vulnerable to random fluctuation in the presence of atmospheric turbulence. It causes the spreading of the optical beam due to large random inhomogeneity of the atmosphere and the resultant beam to become highly skewed from Gaussian beam profile as shown in Figure 3.3. This phenomenon is known as beam wander, which can be explained by the random displacement of the instantaneous beam centroid (i.e., point of maximum irradiance termed as hot spot) off the boresight, observed over short time periods in the receiver plane.

50 Figure 3.3 Beam wander effect is observed by the movement of hot spot within the beam By reciprocity principle (Fried & Yura 1972), the resultant beam width (observed at the receiver plane) is termed as the long-term spot size w LT, which forms the basis for the variance of beam wander instabilities, 2 w LT= w 2 L (1 + T Tot ) = w 2 L 1 + 1.33σ 2 R Ʌ ⅚ L, (3.4) where Ʌ L denotes the Fresnel ratio at the output plane and σ R 2 is Rytov 2 variance. Based on short- and long-term spot size, w LT is expressed as w 2 LT = w 2 L + w 2 2 L T ss + w L T LS, (3.5) where the term T Tot = T SS + T LS can be partitioned into a sum of small-scale (T SS ) and large-scale (T LS ) contributions. It is apparent that the long-term spot size w LT arises from the effects of pure spreading dispersion (as described by the first term w 2 LT ), and turbulent cells (or known as eddies) of all scale sizes due to random inhomogeneities of the atmosphere. Thus turbulent scale sizes larger than the beam diameter lead to refractive effects known as beam wander, which causes random movement of the instantaneous center of the incident optical beam in the receiver plane, as represented by the final term w L 2 T LS. According to the study by Fante (1980), relation between the long-term beam radius w LT, short-term beam radius w ST, and beam wander variance r c 2 of a Gaussian-beam wave is given by

51 w 2 2 LT = w ST + r 2 c. (3.6) The beam wander variance r c 2 is given by r c 2 = 2.42C n 2 L 3 w 0 1/3, (3.7) where C n 2 is refractive index structure parameter, L is link distance and w 0 is transmitter beam width. Beam jitter is defined as the whole movement of the short-term optical beam around its unperturbed position in the receiver plane, due to random inhomogeneities of the atmosphere bounded above by r 0 (r 0 L 0 ); thereby acting like an effective wavefront tilt at the transmitter, which can be significantly smaller than the wavefront tilt associated with the random fluctuations of hot spot displacement. Here r 0 = (0.16C n 2 k 2 L ) 3/5 (3.8) denotes the Fried s parameters, or the atmospheric coherence width of a reciprocal propagating point source from the receiver at distance L. The net result of total beam wander is a widening of the long-term beam profile near the boresight, which in turn leads to a slightly flattened beam as shown in Figure 3.4, in comparison to the conventional Gaussian beam profile (dashed curve) as proposed by Rytov theory.

52 Figure 3.4 Flattened beam profile (solid line) as a function of the radial distance due to the widening of the long-term beam near the boresight, in comparison to the conventional Gaussian-beam profile (dashed line) In the case of a collimated beam, the jitter-induced pointing error 2 (PE) variance σ pe is given by σ 2 pe = 0.48 λl 2 5 2w 0 3 1 C r 2 w 2 1 0 r 2 0 ) 6 2w 0 r 0 1+C 2 r w 2 0 r2, (3.9) 0 2 and for the focused beam, the jitter-induced Pointing Error(PE) variance σ pe is given by σ 2 pe = 0.54 λl 2 5 2w 0 3 1 8 C r 2 w 2 1 0 r 2 0 ) 6 2w 0 r 0 9 1+0.5C 2 r w 2 0 r2, (3.10) 0 where the parameter C r is a scaling constant typically on the order C r ~2π. 3.2 MODULATION TECHNIQUES Any communication system needs the modulation techniques to propagate the signal for the long distance communication. Various

53 modulation techniques are available for the FSO communication to analyze the jitter effects and it is discussed below. Modulation is the process by which one or more properties of the digital or analog signal are changed so that the signal can travel longer distance from its source to destination. The modulation techniques used in FSO and their performance in the presence of channel deficiencies such as noise and channel fading induced by atmospheric turbulence. There are many different types of modulation schemes which are suitable for optical wireless communication systems. The emphasis in this thesis will be on the following modulations techniques such as Pulse Amplitude Modulation (PAM), Pulse Width Modulation (PWM), Pulse Position Modulation (PPM), Amplitude Shift Keying (ASK), Binary Phase Shift Keying (BPSK) and Quadrature Phase Shift Keying (QPSK). 3.2.1 Pulse Amplitude Modulation (PAM) In the PAM system, amplitude of carrier pulse is varied in accordance with the instantaneous amplitude of the modulating signal. PAM is the simplest of all pulse modulation technique. In PAM the amplitude of the message or modulating signal is mapped to a series of pulses with two possible variant : 1) Flat top PAM:- The amplitude of each pulse is directly related to instantaneous modulating signal amplitude at the time of pulse occurrence and then keeps the amplitude of the pulse for the rest of the half cycle. 2) Natural PAM:- The amplitude of each pulse is directly proportional to the instantaneous modulating signal amplitude at the time of pulse incidence and then follows the amplitude of the

54 modulating signal for the rest of the half cycle. The carrier is in the form of train of narrow pulses. The Figure 3.5 shows the PAM output waveform. Figure 3.5 PAM output waveform Let s(t) denotes the sequence of pulses generated and PAM signal is expressed as follows (Haykin 2001) s(t) = n= m(nt s )h(t nt s ). (3.11) Where T s is the sampling period and m(nt s ) is the sample value of m(t) obtained at time t=nt s. The h(t) is a standard rectangular pulse of unit amplitude and duration T, defined as follows 1, 0 < t < T 1 h(t) = 2 t = 0, t = T 0, otherwise (3.12) The Fourier transform of the ideally sampled signal can be written as S(f) = f s n= M(f nf s ). (3.13) In natural sampling, the pulse has a finite width t. Natural sampling is also called chopper sampling because the waveform of the

55 sampled signal appears to be chopped off from the original signal waveform. Let the amplitude of the pulse is A, then the functional description of PAM is as follows S(t) = m(t) when h(t) = A 0 when h(t) = 0. (3.14) 3.2.2 Pulse Position Modulation (PPM) PPM is an orthogonal modulation technique. Here the position of the modulated pulses changes in accordance with the amplitude of modulating signal. The amplitude and frequency of the PPM wave remains constant and only position changes. A PPM receiver will require both slot and symbol synchronization in order to demodulate the information encoded on the pulse position. Nevertheless, because of its greater power efficiency, PPM is an attractive modulation technique for optical wireless communication systems particularly in deep space laser communication applications (Fettweis & Zimmermann 2008). Assuming that complete synchronization is maintained between the transmitter and receiver at all times, the optical receiver detects the transmitted signal by attempting to determine the energy in each possible time slot. The Figure 3.6 shows the PPM output waveform. Figure 3.6 PPM output waveform

56 In PPM, each block of log 2 M data bits is mapped to one of M possible symbols. Generally the notation M-PPM is used to indicate the order. Each symbol consists of a pulse of constant power, occupying one slot, along with M-1 empty slots. The position of the pulse corresponds to the decimal value of the log 2 M data bits. Hence, the information is encoded by the position of the pulse within the symbol. The slot duration Ts is associated to the bit duration by the following expression T s = Tlog 2M M. (3.15) In direct photo detection, this is equivalent to counting the number of released electrons in each Ts interval. The photo count per PPM slot can be obtained from K s = ηλp RT s hc, (3.16) where P R is the received optical power during a slot duration. In the presence of log normal atmospheric turbulence, the unconditional BER for a binary PPM modulated FSO obtained by averaging over the scintillation statistics can be approximated as (He & Schober 2009) P e = 1 n π w iq exp 2 2 σ kx i +m k i=1, (3.17) Fxp 2 σ k x i +m k +K n n n where[w] i=1 and [x i ] i=1 are the weight factors and the zeros of an n th order Hermite polynomial. m k represents the mean of ln(k s ),K n = (2σ 2 Th /(gq) 2 ) + 2FK Bg and σ k 2 ln(σ N 2 + 1). 3.2.3 Pulse Width Modulation (PWM) Pulse width modulation is a method of varying the duration of a pulse with respect to analog input. The duty cycle of a square wave is

57 modulated to encode a specific analog signal level. The pulse width changes according to the message signal (Width of the pulse is modulated) as shown in the Figure 3.7. Figure 3.7 Output wavform of pulse width modulation Pulse width modulation signals may also be used to approximate time-varying analogue signals. Pulse width modulation waveform needs a sharp rise time and fall time for pulses in order to preserve the message information. Rise time should be very less than T s. The rise time of the PWM pulse is given by t r T s. (3.18) follows The transmission bandwidth of pulse width modulation is given as B T 1 2t r. (3.19) 3.2.4 Amplitude Shift Keying (ASK) or On-Off Keying (OOK) OOK is the dominant modulation scheme employed in commercial terrestrial wireless optical communications systems. Amplitude-shift keying

58 (ASK) is a form of modulation that represents digital data as variations in the amplitude of a carrier wave. ASK uses a finite number of amplitudes, each allocated a unique pattern of binary digits. Usually, each amplitude encodes an equal number of bits. Each pattern of bits forms the symbol that is represented by the particular amplitude. For ASK, the transmitted signal can be expressed as follows S(t) = s 1(t) for symbol 1 s 2 (t) for symbol 0 for 0 t T b. (3.20) Here,s 1 (t) = 4E b T b cos(2πf c t). (3.21) s 2 (t) = 0. (3.22) The corresponding output wave form of the On-Off Keying signaling scheme is shown in the Figure 3.8. Figure 3.8 ASK output waveform If R represents the responsivity of the PIN photo detector and optical intensity is represented by I, then the received signal in an OOK system is given by, i(t) = RI j= d j g(t jt) + n(t). (3.23)

59 Where n(t) is the additive white Gaussian noise. g(t jt) is the pulse shaping function and d j =[-1,0]. At the receiver, the received signal is fed into a threshold detector which compares the received signal with a pre-assigned threshold level. A digital symbol 1 is assumed to have been received if the received signal is above the threshold level and 0 otherwise. The demodulator, which is designed precisely for the symbol-set used by the modulator, determines the amplitude of the received signal and maps it back to the symbol it represents, thus reconstructing the original data. Frequency and phase of the carrier are kept constant. The probability of error is given as follows P e = p(0) p (i 0)di + p(1) p (i 1)di, (3.24) i th i th where i th is the threshold signal level and the marginal probabilities are defined as follows and p(i 0) = 1 2πσ exp i2 2 2σ 2 (3.25) p(i 1) = 1 exp (i RI)2. (3.26) 2πσ2 2σ 2 For equiprobable symbols p(0)=p(1)=0.5, hence the optimum threshold point is at i th = 0.5RIand the conditional probability of error reduces to P ec = Q i th, (3.27) σ where Q(x) = 0.5erfc x 2. In the presence of atmospheric turbulence, the threshold level is no longer fixed midway between the signal levels representing symbols 1 and 0.

60 3.2.5 Binary Phase Shift Keying (BPSK) Phase-shift keying (PSK) is a digital modulation scheme that conveys data by changing, or modulating, the phase of a reference signal (the carrier wave). Any digital modulation scheme uses a finite number of distinct signals to represent digital data. PSK uses a finite number of phases; each assigned a unique pattern of binary digits. Transmission digits are binary in nature. So this phase shift keying is called as Binary Phase Shift Keying (BPSK). Usually, each phase encodes an equal number of bits. Each pattern of bits forms the symbol that is signified by the particular phase. In BPSK, binary symbol 1 and 0 modulate the phase of the carrier. When the symbol is altered, then the phase of the carrier is changed by 180 degrees. The BPSK signal is expressed as follows, s(t) = b(t)a cos(2πf 0 t). (3.28) Where A represents the amplitude of the carrier signal and b(t) is the binary signal which is given by Figure 3.9. +1 if binary 1 is transmitted b(t) = 1 if binary 0 is transmitted. (3.29) The output waveform of the BPSK modulation signal is given in Figure 3.9 BPSK output waveform

61 The average probability of symbol error at the receiver after propagating through the atmosphere turbulence is given by P e = 1 2 erfc E b N 0. (3.30) Where E b is transmitted signal energy per bit and N 0 is the noise spectral density. 3.2.6 Quadrature Phase Shift Keying (QPSK) Quadrature Phase Shift Keying (QPSK) is a form of Phase Shift Keying in which two bits are modulated at once, selecting one of four possible carrier phase shifts (0, 90, 180, or 270 degrees). QPSK allows the signal to carry twice as much information as ordinary PSK using the same bandwidth. The term "quadrature" indicates that there are four possible phases in (4-PSK or QPSK) which the carrier can have at a given time as shown in Figure 3.10. In PSK, information is carried through phase variations, since complete phase cannot be established. In each time period, the phase can change once while the amplitude remains constant. Figure 3.10 QPSK output waveform

62 In QPSK there are four possible phases, and therefore two bits of information transported within each time slot. The rate of change (baud) in this signal determines the signal bandwidth, but the throughput or bit rate for QPSK is twice the baud rate. For each dibit (two bits), one symbol is transmitted. One method of generating the QPSK waveform is by changing the input binary data stream into two streams: the odd- and the even bit streams consisting of the odd- and even numbered bits. Each of these binary streams can then be modulated using the BPSK, and then on adding we get the QPSK waveform. follows The modulated quadrature phase shift keying signal is expressed as S i (t) = 2E T cos 2πf c + (2i 1) π, 0 t π, (3.31) 4 4 0, elsewhere where i=1,2,3,4; E is the transmitted signal energy per symbol, f c is carrier frequency, and T is the symbol duration. The average probability of symbol error at the receiver after propagating through the atmosphere turbulence is given by P e = erfc E b N 0 (3.32) For the same bit rate, the bandwidth required by QPSK is reduced to half as compared to BPSK. Due to reduced bandwidth, the information transmission rate of QPSK is higher. The performance analysis based on the jitter effect of these modulation schemes and corresponding demodulation schemes are discussed in forthcoming chapters.