Chapter 3 Introduction of Equalizer
3.4 Algorithm
Figure 3-5 Sorts of equalizers
In the design of equalizers there exist different types of design criteria [25].
The most frequently encountered two criteria with their efficiency are told in the sequel. Some equalizers are designed to minimize mean square error (MSE) at the slicer input with the constraint of zero ISI. These are called Zero-Forcing (ZF) equalizers. Some equalizers are designed to minimize the MSE at the input of the slicer by reducing the signal slightly at the slicer input. This reduction of signal results in reduction in MSE, so overall MSE is smaller than that of the ZF equalizer. These equalizers are called MSE equalizers. The MSE
enhancement.
The linear equalizer is cheap in implementation but its noise performance is not very good. So, in the literatures, some equalizer types which introduce nonlinearity are searched. The most popular of these nonlinear equalizers is the decision feedback equalizer (DFE). The DFE is first proposed by Austin in [27].
This equalizer results in less MSE against linear equalizer, but it has the disadvantage of error propagation in its feedback loop.
As it is told before, most of the time, the channel’s and, consequently, the transmission system’s transfer functions are not known. Also, the channel’s impulse response may vary with time and fade. The result of this is that the equalizer can't be designed a priori, frequently. So, mostly preferred scheme is to exploit adaptive equalizers. Adaptive equalizers use adaptive algorithms to converge to the true coefficients and have the benefit of tracking the changes in the channel impulse response. But, to achieve this, it adds additional complexity to the receiver structure.
Also, the adaptation algorithm plays a significant role for the performance of the equalizer. The most popular algorithm from the aspect of performance and complexity is the Least Mean Squares (LMS) algorithm. It has a good performance and low complexity. It is globally convergent if the desired values are given correctly. The handicap of LMS algorithm for equalizer if the desired symbols are not correct, it does not converge. So, the equalizer using LMS algorithm requires a priori known symbols in case the decisions of the equalizer are wrong.
The LMS algorithm is a linear adaptive filtering algorithm that belongs to the family of the stochastic gradient algorithms [26]. The stochastic gradient
algorithms differ from the steepest descent algorithms in that the gradient is not calculated deterministically. The LMS algorithm has two parts. In the first part, the output of a transversal filter is computed according to the tap inputs and the error term is generated according to the difference between the filter output and the desired response. In the second part, the adjustment of the tap weights is done according to the error term.
The algorithm forms a feedback loop by the error term fed back. The filter produces an output and the difference between the output and the desired term is obtained. This difference is the estimation error term. The estimation error is given to the Adaptation Control Block. Adaptation Control Block multiplies the estimation error with the input taps’ complex conjugate and a step size α.
The results of the corresponding taps are added to the corresponding filter taps.
So, the new filter is obtained [24]:
( ) ( ) ( ) ( ) ( ) ( )
T ˆ where f(k) is the filter vector at time k, and rˆ* (k) is the complex conjugate of the input vector at time k, α is the step size parameter, e(k) is the estimation error, I(k) is the desired response at time k. In equation (2.3), fn(k) is the nth tap of the filter at time k, and r*(k-n) is the complex conjugate of the input at time k-n, and other parameters are the same as first equation. Equations (2.2) and (2.3) are equivalent. The small step size will result in less excess error but in slow convergence rate. The large step size will result in high excess error but high convergence rate.Chapter 4
The Concept of New Optical Modulation System
4.1 Preface
There are three parts in optical communication systems : optical transmitter, communication channel and optical receiver. Optical transmitter converts an electrical input signal into the corresponding optical signal and then launches it into the optical fiber serving as a communication channel. The role of an optical receiver is to convert the optical signal back into electrical form and recover the data transmitted through the lightwave system. In this chapter, we will do an introduction about the external Mach-Zehnder Modulator (MZM), constructing a model of new ROF system.
4.2 Mach-Zehnder Modulator (MZM)
Direct modulation and external modulation are two modulations of generated optical signal. When the bit rate of direct modulation signal is above 10 Gb/s, the frequency chirp imposed on signal becomes large enough. Hence, it is difficult to apply direct modulation to generate microwave/mm-wave.
However, the bandwidth of signal generated by external modulator can exceed 10 Gb/s. Presently, most RoF systems are using external modulation with Mach-Zehnder modulator (MZM) or Electro-Absorption Modulator (EAM).
The most commonly used MZM are based on LiNbO3 (lithium niobate) technology. According to the applied electric field, there are two types of LiNbO3 device : x-cut and z-cut. According to number of electrode, there are two types of LiNbO3 device: dual-drive Mach-Zehnder modulator (DD-MZM)
and single-drive Mach-Zehnder modulator (SD-MZM) [6].
4.3 Single-drive Mach-Zehnder modulator
The SD-MZM has two arms and an electrode. The optical phase in each arm can be controlled by changing the voltage applied on the electrode. When the lightwaves are in phase, the modulator is in “on” state. On the other hand, when the lightwaves are in opposite phase, the modulator is in “off ” state, and the lightwave cannot propagate by waveguide for output.
4.4 The architecture of ROF system
4.4.1 Optical transmitter
Optical transmitter concludes optical source, optical modulator, RF signal, electrical mixer, electrical amplifier, etc.. Presently, most RoF systems are using laser as light source. The advantages of laser are compact size, high efficiency, good reliability small emissive area compatible with fiber core dimensions, and possibility of direct modulation at relatively high frequency.
The modulator is used for converting electrical signal into optical form.
Because the external integrated modulator was composed of MZMs, we select MZM as modulator to build the architecture of optical transmitter.
There are two schemes of optical transmitter generated optical signal. One scheme is used two MZM. First MZM generates optical carrier which carried the data. The output optical signal is BB signal. The other MZM generates optical subcarrier which carried the BB signal and then output the RF signal, as shown in Fig. 4-1 (a). The other scheme is used a mixer to get up-converted
shown in Fig. 4-1 (b). Fig. 4-1 (c) shows the duty cycle of subcarrier biased at different points in the transfer function.
Figure 4-1 (a) and (b) are two schemes of transmitter and (c) is duty cycle of subcarrier biased at different points in the transfer function. (LO: local oscillator)
4.4.2 Optical signal generations based on LiNbO3 MZM
Figure 4-2 Optical microwave/mm-wave modulation scheme by using MZM.
The microwave and mm-wave generations are key techniques in RoF systems. The optical mm-waves using external MZM based on double-sideband (DSB), single-sideband (SSB), and double-sideband with optical carrier suppression (DSBCS) modulation schemes have been demonstrated, as shown in Fig. 4-2. Generated optical signal by setting the bias voltage of MZM at quadrature point, the DSB modulation experiences performance fading problems due to fiber dispersion, resulting in degradation of the receiver sensitivity. When an optical signal is modulated by an electrical RF signal, fiber chromatic dispersion causes the detected RF signal power to
DSB
SSB
Dual-driveMZM
DSBCS
Single-drive MZM Single-drive
MZM
have a periodic fading characteristic. The DSB signals can be transmitted over several kilo-meters. Therefore, the SSB modulation scheme is proposed to overcome fiber dispersion effect. The SSB signal is generated when a phase difference of π/2 is applied between the two RF electrodes of the DD-MZM biased at quadrature point. Although the SSB modulation can reduce the impairment of fiber dispersion, it suffers worse receiver sensitivity due to limited optical modulation index (OMI). The DSBCS modulation is demonstrated optical mm-wave generation using DSBCS modulation. It has no performance fading problem and it also provides the best receiver sensitivity because the OMI is always equal to one. The other advantage is that the bandwidth requirement of the transmitter components is less than DSB and SSB modulation. However, the drawback of the DSBCS modulation is that it can’t support vector signals, such as phase shift keying (PSK), quadrature amplitude modulation (QAM), or OFDM signals, which are of utmost importance in wireless applications.
4.4.3 Communication channel
Communication channel concludes fiber, optical amplifier, etc.. Presently, most RoF systems are using single-mode fiber (SMF) or dispersion compensated fiber (DCF) as the transmission medium. When the optical signal transmits in optical fiber, dispersion will be happened. DCF is use to compensate dispersion. The transmission distance of any fiber-optic communication system is eventually limited by fiber losses. For long-haul systems, the loss limitation has traditionally been overcome using regenerator witch the optical signal is first converted into an electric current and then
rege
Fig
4.5 The new proposed model of optical modulation system
Figure 4-6 The proposed RoF system based on a single-electrode MZM.
Figure 4-6 schematically depicts the concept of the proposed RoF system.
The MZM driving signal at a subcarrier frequency of f1 and a sinusoidal LO signal with a frequency of f2, as indicated in insets (a)-(c) of Fig. 4-6 The frequency, f2 of the LO signal is half the desired mm-wave frequency of operation. To achieve the double sideband with carrier suppression (DSBCS) modulation scheme, the MZM is biased at the null point to suppress the optical carrier. Inset (d) of Fig. 4-6 shows the generated optical signal and LO spectrum that has two upper-wavelength sidebands (USB1, USB2) and two lower-wavelength sidebands (LSB1, LSB2) with carrier suppression at the output of the MZM. After square-law photo detection, the generated photocurrent can be written as
(USB1+USB2+LSB1+LSB2) .2
I =
(1)
Expanding the above equation produces the following terms:
DC=USB12+USB22+LSB12+LSB2 .2 (2) Signal at the sum frequency=
USB1 LSB2 USB2 LSB1.× + × (3) Signal at the frequency difference=
USB1 USB2 LSB1 LSB2.× + × (4) Beat noise=USB1 LSB1 USB2 LSB2× + × . (5) The beating terms of USB1 × LSB2 and USB2 × LSB1 generate the desired electrical signals at the sum frequency (f2+f1). The beating terms of USB1 × USB2 and LSB1 × LSB2 generate electrical signals at the frequency difference (f2-f1), which are well below the desired mm-wave frequency band and are filtered off by low-pass filter at receiver part. Notably, a frequency multiplication factor of two (2) can be achieved by properly choosing frequencies f1 and f2. This reduces the bandwidth requirements of the RoF transmitter allowing for the use of low-frequency electrical and optical components, including the MZM (< 40 GHz), which are readily available and have very good performance (e.g. flat frequency response).
In this system, the target sum frequency is 60 GHz. Two main issues will be crucial to the system performance and will be explained in details in following.
First, the RF fading issue, as shown in the equation (12), comes from the interaction between the two copies of the desired signals, which are generated at the photodiode, namely USB1 × LSB2 and USB2 × LSB1, respectively.
After fiber transmission, the relative phase between the two generated RF signals will change with transmitted distance owing to the slight difference in the propagation speeds of the two sideband pairs induced by fiber chromatic dispersion. As the relative phase reaches 180°, the electrical RF signal will
vanish. This is the RF fading problem. Here is also the reason why we use an interleaver to filter out USB1 and LSB2. The second issue comes from beat noise of two signals USB1 × LSB1. If the center frequency of single carrier signals is not properly chosen, the beat noise will fall into the signal band and severely degrade the system performance.
Chapter 5
The Theoretical Calculations of Proposed System
5.1 Introduce MZM
For MZM with configuration as Fig. 5-1, the output E-filed for upper arm is EU E · a · e∆φ (1)
∆φ π· π (2)
∆φ is the optical carrier phase difference that is induced by v , where a is the power splitting ratio.
The output E-filed for upper arm is
EL E · √1 a · e∆φ (3)
∆φ is the optical carrier phase difference that is induced by v
∆φ π· π (4) The output E-filed for MZM is
ET E · a · b · e∆φ √1 a · √1 b · e∆φ (5) where a and b are the power splitting ratios of the first and second Y-splitters in MZM, respectively. The power splitting ratio of two arms of a balanced MZM is 0.5. The electrical field at the output of the MZM is given by
ET · E · e∆φ e∆φ (6) ET E · cos ∆φ ∆φ · exp j∆φ ∆φ (7) For single electro x-cut MZM. The electrical field at the output is given by EOUT E · cos ∆φ ∆φ · exp j∆φ ∆φ (8) Add time component, the electrical field is
EOUT E · cos ∆φ · cos t (9) where E0andωcdenote the amplitude and angular frequency of the input optical carrier, respectively; V t is the applied driving voltage, and ∆φ is the optical carrier phase difference that is induced by between the two arms of the MZM. The loss of MZM is neglected. consisting of an electrical sinusoidal signal and a dc biased voltage can be written as,
cos t (10) where is the dc biased voltage, and are the amplitude and the angular frequency of the electrical driving signal, respectively. The optical carrier phase difference induced by is given by
∆φ Vπ Vπ ·π (11) Eq. (10) can be written as:
EOUT E · cos V V cos ωt
Vπ ·π
2 · cos ω t E · cos b m · cos ωRFt · cos ω t E · cos ω t · cos b · cos m · cos ωRFt
sin b · sin m · cos ωRFt (12) where V π is a constant phase shift that is induced by the dc biased voltage, and V π is the phase modulation index.
cos x sin θ J x 2 J x cos 2nθ
∞
sin x sin θ 2 J x sin 2n 1 θ
∞
cos x cos θ J x 2 1 J x cos 2nθ
∞
sin x cos θ 2 1 J x cos 2n 1 θ
∞
(13) Expanding Eq. (12) using Bessel functions, as detailed in Eq. (13). The electrical field at the output of the MZM can be written as:
EOUT E · cos ·
cos · 2 · 1 · m · cos 2
∞
sin · 2 · 1 · · cos 2 1
∞
(14) where is the Bessel function of the first kind of order n. the electrical field of the mm-wave signal can be written as
EOUT E · cos · J m · cos ω t
E · cos · · cos
∞
2 π
E · cos · · cos 2
∞
E · sin · · cos
∞
2 1
E · sin · · cos
∞
2 1
(15)
Figure 5-1 The principle diagram of the optical mm-wave generation using balanced MZM.
5.2 Theoretical calculation of single drive MZM
5.2.1 Bias at maximum transmission point
When the MZM is biased at the maximum transmission point, the bias voltage is set at 0, and cos 1 and sin 0. Consequently, the electrical field of the mm-wave signal can be written as
· · cos
· · cos
∞
2
· · cos 2
∞
(16) The amplitudes of the generated optical sidebands are proportional to those of the corresponding Bessel functions associated with the phase modulation index . With the amplitude of the electrical driving signal equal to , the
decreases and increases with the order of Bessel function and m, respectively, as shown in Figure 5-2. , , , and are 0.5668, 0.2497, 0.069, and 0.014, respectively. Therefore, the optical sidebands with the Bessel function higher than can be ignored, and Eq. (14) can be further simplified to
· · cos
· · cos 2 · · cos 2 · · cos 4
· · cos 4 (17)
0 2 4 6 8 10
-0.5 0.0 0.5 1.0
m
J0 J1 J2 J3
Figure 5-2 The different order of Bessel functions vs. m.
5.2.2 Bias at quadrature point
When the MZM is biased at the quadrature point, the bias voltage is set at
V , and cos √ and sin √ . Consequently, the electrical field of the mm-wave signal can be written as
EOUT 1 and cos 0 and sin 1. Consequently, the electrical field of the mm-wave signal using DSBCS modulation can be written as
EOUT E · J m · cos ω ωRF t
5.3 Theoretical calculations and simulation results
5.3.1 mm-Wave Signal Generation Based on The Proposed System
Here, we present a theoretical basis of the proposed mm-wave generation and transmission system. The concept behind the generation of the 60-GHz wireless signal is shown in Fig. 4-6, where only one single-electrode MZM is utilized. The optical field at the input of the single-electrode MZM is given by
( ) cos( )
in o c
E t =E ωt , where Eo and ωc are the amplitude and angular frequency of the optical field, respectively. The driving RF signal V(t) consisting of two sinusoidal signals at different frequencies MZM is V t( )=V1⋅cosω1t V+ 2⋅cosω2t, where V1 and V2 are the signal amplitudes at frequency ω1 and ω2, respectively.
To simplify the analysis, the power splitting ratio of the MZM is set as 0.5. In order to suppress the undesired optical carrier, the single-electrode MZM is biased at the null point. The optical field at the output of the MZM is then given by
1 1 2 2
( ) cos cos[( 2 ) ( cos cos )].
out o c
E t =E ⋅ ωt⋅ π Vπ ⋅Vπ + ⋅V ωt V+ ⋅ ω t (6) Using Bessel function expansion, the output optical field at the output of the MZM can be rewritten as
where m1 and m2 are the modulation indices defined as V1π 2Vπ and V2π 2Vπ , respectively. Jn() is the nth order Bessel function of the first kind. For a small modulation index the magnitude of Bessel function of the first kind is proportional to the order of the function. As shown in the Fig. 5-3, when the modulation index is small, the output optical field can be further simplified to
0 2 1 1 1 0 1 1 2 2
( ) { ( ) ( ) cos[( ) ] ( ) ( ) cos[( ) ]}.
out o c c
E t =E ⋅ J m J m ⋅ ω ±ω t +J m J m ⋅ ω ±ω t (8) After square-law photo detection the photocurrent of the mm-wave at frequency of ω ω1+ 2 can be expressed as
1 2
2 0( 1) (0 2) (1 1) (1 2),
iω ω+ = ⋅R E J m J m J m J mo (9) where R is the respresivity of photodiode.
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Figure 5-3 The magnitude of Bessel functions versus different RF
modulation index.
5.3.2 Dispersion Induce RF Fading Analysis and Beat Noise
When optical RF signals are transmitted over a standard single-mode fiber with dispersion, a phase shift to each optical sideband relative to optical carrier is induced by fiber dispersion. The propagation constant of fiber can be expressed as [29]
= = is the derivative of the propagation constant evaluated at ω ω= c. To simplify the analysis, the effect of high order fiber dispersion (i.e. 3rd order and higher) at 1550-nm band is neglected. For carrier tones with central frequency at ω ω= c±nωRF , we have c is light speed in free space, D is the chromatic dispersion parameter, and fc
is the frequency of the optical carrier. For a standard single-mode fiber, D is 17-ps/(nm.km). Therefore, after transmission over a standard single-mode fiber of length z, the electrical field can be written as
0 2 1 1 1 0 1 1 2 12
After square-law photo detection, the photocurrent at the frequency of can be expressed as
1 2 2 0 1 0 2 1 1 1 2 2 22 12
( ) ( ) ( ) ( ) ( ) cos[1 ( )].
o 2
iω ω+ t = ⋅R E J m J m J m J m ⋅ β ωz −ω (12)
Due to fiber dispersion effect, the RF fading issue would be observed. The RF signal power is related to cos[1 2 ( 22 12)]
2β ωz −ω . Therefore, the RF fading issue would become serious when the magnitude of sum frequency ( f2+ f1) and
frequency difference ( f2−f1) become large. Figure 5-4 Simulated RF power of the generated mm-wave signal versus standard single-mode fiber length for various input frequency differences (i.e.
f2 - f1)
For 60-GHz applications, the sum frequency ( f2+ f1) is fixed at 60.5 GHz, and the frequency difference (f2− f1) will dictate the performance of RF fading.
As shown in the Fig. 5-4, when the frequency difference increases, the RF power will drop off rapidly. For frequency differences of 10 GHz and 40 GHz, the first deep appears following 6 and 1.6-km fiber transmission, respectively.
Not only does the smaller frequency difference result in a longer fiber transmission distance, but it also reduces the bandwidth requirements of the transmitter. However, the drawback of a small frequency difference is the risk of having beat noise interference. For example, if we choose 5.5 GHz as the frequency difference, and set the input frequencies at ω =33 GHz and ω =27.5
GHz, then with 7-GHz signal bandwidth, the generated signal will occupy frequencies from 24 to 31 GHz. As a result the beat noise (i.e. USB1 × LSB1) will fill the band from 48 to 62 GHz. Since the generated signal will fill the band from 57 to 64 GHz, the beat noise will fall in-band as shown in Fig. 5-5
GHz, then with 7-GHz signal bandwidth, the generated signal will occupy frequencies from 24 to 31 GHz. As a result the beat noise (i.e. USB1 × LSB1) will fill the band from 48 to 62 GHz. Since the generated signal will fill the band from 57 to 64 GHz, the beat noise will fall in-band as shown in Fig. 5-5