M-PSK Communication System

Sunil Panthi, Graduate Student, UCI spanthi@uci.edu Bhanu Pratap Reddy, Graduate Student, UCI bpreddy@uci.edu Abstract—Modeling communication system and establishing bit error rate (BER) with signal to noise ratio (SNR) is a key element in designing communication system. Based on the communication applications, communication channels vary. In cellular communication signal attenuation and fading due to the both deterministic and non-deterministic factors is common. Free space loss is the deterministic type of attenuation. Additive White Gaussian Noise (AWGN), fast fading (short term), and slow fading (long term or shadow fading) are non-deterministic type of fading. This paper focuses the non-deterministic channel modeling techniques and demonstrates the result in BER as a function of SNR. The relationship between BER and SNR are established for M-PSK system for three channel properties; AWGN, fast, and slow fading. AWGN channel was modeled by normal distribution. Rayleigh distribution were used for fast fading and lognormal distribution for slow fading is used to model. Three primary relationships created after the modeling are * BER to SNR relationship for BPSK, QPSK, 8-PSK, and 16-PSK for AWGN only channel * BER to SNR relationship for BPSK, QPSK, 8-PSK, and 16-PSK for AWGN and fast faded channel * BER to SNR relationship for BPSK for AWG, fast faded, and slow faded channel.
The use case of BER to SNR is shown in a sample link budget. Link budget helps to make key system decision such as spectral efficiency, coverage radius, link margin, and availability. Further paper proposes other techniques to improve the BER and fading for given SNR; Forward Error Correction (FEC), interleaving, and diversity just to name few. Those techniques allows improving spectral efficiency and reducing outage.
Keywords—Communication system, M-PSK, Channel simulation, AWGN, Rayleigh fading, Shadow fading, BER curves
Introduction
In communication, message has to be successfully sent from sender to the receiver. In order to successfully send the message, it should be encoded, modulated, transmitted, demodulated, and decoded. During transmission, random noise will be introduced and signal will be faded when going through the channel. It is difficult to predict the actual link attenuation and fading for non-deterministic communication channel. Statistical models help to model and predict the communication link. A model called Additive White Gaussian Noise (AWGN) is used to model the noise in the communication system. Transmitted signal is faded due to the various factors in environment. Two common fading are fast fading and slow fading. The urban environment, troposphere and ionosphere create random attenuation in a transmission signal. Rayleigh fading model is used to model fast fading in signal propagation. This distribution can be used to approximate the short-scale and very rapid signal fluctuations [1]. The second source of fading is slow fading and modeled by Shadow Fading given by lognormal distribution. Shadow fading with lognormal distribution is also used to model with diversity such as Multiple Input Multiple Output (MIMO) system [2].
This paper demonstrates information generation, transmission, and estimation using various fading models in communication system and generates the bit error rate (BER) as a function of the signal to noise ratio (SNR). Such modeling and result are valuable to efficiently design a communication link and make system level tradeoffs and design decision. Further they showed how such model can be used to select the margins for different fading situations. It is very important to successfully model the channel so that link design accounts for the necessary margin and mitigation technique to meet the link availability. In this model, we show methodology of the simulation, mathematical model behind simulation, results, and conclusion.

METHODOLOGY
Signal Generation

The first part of the modeling and simulating communication channel is the signal generation. The overview of the communication model is shown in Fig. 1. Information is generated with binary random variable with uniform distribution. Signal is gray coded based on constellation and bits per symbol. When M = 8, it is coded according to 8-PSK constellation, were bits per symbol is given by log2(8). The resulting signal is passed through pulse shaping filter and decomposed into in phase and quadrature which forms a 2-dimentional orthogonal basis. The resulting signal st for carrier frequency fc and phase ∅t is defined by:

st=cos2πfctcos∅t-sin2πfctsin⁡(∅t) (1)

Fig. 1: Communication system model for simulation
Signal Transmission

The generated signal needs to be transmitted. The signal st will have noise n introduced during transmission. The common source of the noise is the white noise. Such noise could be thermal noise, noise from communication electronics, or random process that occurs in nature. The AWGN model represents the noise described above and is given by normal distribution with zero mean and noise variance. The signal yt after AWGN introduced in signal is represented by:

yt= st+n (2) The pulse shaped signal with AWGN is shown in Fig. 2. The amplitude of the signal is distorted by the noise. The signal with AWGN yt is passed through other transmission channels.

Fig. 2: Pulse shaped signal before and AWGN introduced
During transmission, two primary channel fading is introduced; fast fading and slow fading.
The fast fading is the type of the fading that is caused due to the environment and urban areas where scattering of the signal occurs. The signal from source to the destination without line of sight passes through multiple objects. Two type of fast fading are frequency selective and frequency flat. Frequency selective fading is for the channel that has multiple paths from transmitter to receiver. The frequency response of the channel is not flat. The coherence bandwidth is same or smaller than signal bandwidth. Frequency flat fading is for the channel that has single path but not line of sight from transmitter to receiver. The frequency response of the channel is constant for all frequencies. The coherence bandwidth is significantly larger than signal bandwidth. A good example of the fast fading is signal reaching to receiver via tall buildings in urban areas. Fast fading for an amplitude response of α and phase response of β is given by:

h= αejβ (3)

Where α is given by Rayleigh distribution with standard deviation of σ as shown in equation below:

pr= rσ2 e-r22σ2 (4)

β is given by uniform distribution between 0 and 2π.

The signal that goes through AWGN and fast fading channel is given by y(t)' in equation 5.

y(t)'=sth+n (5)

The slow fading is type of the fading where receiver goes to longer fade duration. The common example is when mobile user falls behind the large building where communication is possible either due to dispersion or secondary base station. Slow fading is given by log normal distribution shown in equation below where m is mean and σ is standard deviation.

pr= 1rσ2πe-1(lnr-m)22σ2 (6)

The signal that is distorted by AWGN is send through the fast faded and slow faded channel and shown in Fig. 3. The model created selects either slow fading or fast fading channel but not both after being passed through AWGN.
Fig. 3 shows the signal using model provided byR Rayleigh and Shadow faded model. Fast faded signal faded more frequently with higher dips than slow faded.

Fig. 3: Signal after passing through fast faded (top) and slow faded (bottom) channel
Receive Signal

The constellation of the receive signal will be distorted due to AWGN and fading channels. The degree of the distortion varies based on the constellation type. Fig. 4, 5, and 6 show the constellation of the M-PSK signal received at the receiver after being passed thorough the fading channel. The constellation plot shows that phase distortion on fast faded channel is worst than slow faded channel. This is expected because of the impulse response from scattering the signal from various surface has higher phase and amplitude distortion in short period of time. The slow fading given my Lognormal distribution has higher amplitude distortion in long period of time.

Fig. 4: BPSK constellation at receiver distorted by fast (left) and slow (right) faded channel

Fig. 5:QPSK constellation at receiver distorted by fast (left) and slow (right) faded channel

Fig. 6: 8-PSK constellation at receiver distorted by fast (left) and fast (right) faded channel As the constellation of the M-PSK grows, the difference between two symbol position (phase separation) becomes smaller. Fig 6 shows the phase distortion on the 8-PSK signal for fast faded channel is visually indistinguishable for the normalized receiver power.

Equalizaiton

The receive signal is equalized with simple equalization technique to mitigate the inter symbol interference (ISI). Zero forcing equalizer is the simple equalization technique that is used in this model. It has drawbacks. It can amplify the AWGN noise. The receive signal y(t)' after the equalization is given by y(t)'' where h is the impulse channel response.

y(t)''=y(t)'h (7)
RESULT
The receive signal at the receiver is passed through the matched filter, demodulated, and estimated. The Estimation was to demonstrate the BER of M-PSK signal as a function of SNR.
Bit Error Rate for AWGN

The first estimated was demonstrated for the signal with only AWGN added. Fig. 7 shows the result of the model. BER is plotted against SNR for different M-PSK system. The result from AWGN model match with well understood relationship between BER and SNR as well as theoretical results.

Fig. 7: Bit error rate for BPSK, QPSK, 8-PSK, and 16-PSK as function of SNR for AWGN introduced channel
In this result BER of 10E-7 was achieved when SNR is about 17 dB. For same BER, QPSK, 8-PSK, and 16-PSK modulated signal require higher SNR as shown on the result above. Higher order modulation delivers higher efficiency in terms of bits per hertz. However, it requires higher SNR at the receiver. The higher SNR at the receiver can be achieved by higher transmit power, higher transmit and receive antenna gain. Transmission resources such as higher transmit power often are constrained by other factors such as regulation. This is a challenge for a communication engineer when planning the signal transmission and requires careful balance between efficiency and power resource.

Bit Error Rate for Rayleigh and Shadow Faded

The signal received from AWGN and Rayleigh faded is demodulated, equalized, and estimated for each modulation type. The BER for fast faded signal is higher than AWGN-only introduced channel for same modulation type. The bit error rate increased as the modulation order increase as shown in Fig 8. Achieving bit error rate of 10E-7 for BPSK will require significantly higher SNR than for the channel where only AWGN was introduced. In this result BER of 10E-7 require around 35 dB of SNR for BPSK modulation type.

Fig. 8: BER for BPSK, QPSK, 8-PSK, and 16-PSK as function of SNR for fast faded channel
Fig. 9 us created to highlighted the difference between AWGN, Shadow, and Rayleigh faded BPSK signal.

*
Fig. 9: BER of BPSK as function of SNR for AWGN, fast, and slow faded channel
As shown the result the bit error rate of 10E-7 is achieved around 17 dB of SNR, however achieving same error rate for Rayleigh and Shadow faded signal would require significantly higher SNR.
Link Budget

The BER to SNR relationship obtained in AWGN, fast and slow faded channel is valuable in link budgeting. It is highly desirable in communication system to have lower BER like 10E-7. From the curve, equivalent SNR can be mapped for each M-PSK constellation for desired BER. Various trade off such as carrier frequency, transmit power, outage, and coverage radius can be made. The example shown on the Table 1 shows the coverage radius for lower order modulation requiring lower SNR and higher order modulation requiring higher SNR. Coverage radius is 230 m and 73 m for BPSK and QPSK respectively using common transmission parameters in AWGN channel. For same BER, coverage radius for fast faded channel is 33 m when operating at BPSK. It is clear that impact of fast faded signal for same modulation type is large. Although, this example is focused on demonstrating coverage radius to meet the target BER, various other trade off can be made while creating a link budget. One example is that we lower the availability and offer service in larger radius for same BER. Second example is to plan for different transmit power to meet the SNR requirement of certain modulation type.

Table 1: Sample link budget based on BER, SNR, and M-PSK relation obtained from the model | AWGN, BPSK | Rayleigh BPSK | AWGN QPSK | Tx Power (dBW) | -6.00 | -6.00 | -6.00 | Freq. (Hz) | 5.20E+09 | 5.20E+09 | 5.20E+09 | Wavelength(m) | 0.0577 | 0.0577 | 0.0577 | SNR (dB) | 18.00 | 35.00 | 28.00 | BER | 1.00E-07 | 1.00E-07 | 1.00E-07 | alpha | 2.00 | 2.00 | 2.00 | Noise PSD (dBW/Hz) | -204.00 | -204.00 | -204.00 | Noise PSD 1 Mhz | 60.00 | 60.00 | 60.00 | Freq. Loss (dB) | 46.76 | 46.76 | 46.76 | Link Margin (dB) | 20.00 | 20.00 | 20.00 | Noise Figure (dB) | 6.00 | 6.00 | 6.00 | Path Loss Margin (dB) | 47.24 | 30.24 | 37.24 | Radius (m) | 230 | 33 | 73 |

discussion and conclusion

From the model, BER, SNR, and M-PSK relationship was created for three channel types; AWGN, fast faded, and slow faded. The result shows that lower order modulation, like BPSK requires lower SNR to achieve certain BER where as higher order modulation, like 16-PSK requires significantly higher SNR to achieve the same BER for same channel type. But higher order modulations have higher spectral efficiency (bits/Hz) and require higher SNR. Higher SNR can be achieved by increasing transmit power of transmitter, having directive antenna at transmitter or receiver, having diversity in the communication channel. Further, fast faded given by Rayleigh distribution and slow/shadow faded given by Lognormal distribution require significantly higher SNR to achieve same bit error rate than signal with AWGN. Comparing between slow fading and fast fading channel; depending on the duration either can preform better or worst. For the short duration slow fading has relatively worst performance. For the longer duration fast fading has relatively worst performance.
In order to improve the BER to SNR performance, channel can be diversified for the slow or fast faded case. MIMO is one of the solutions that can helps to reduce the effect of fading and decrease the outage. Further adaptive modulation in which modulation can dynamically change to adapt to the fading conditions allows change the modulation type based on the SNR and maintain the same BER. For example when receiver can receive in higher SNR it will operate at higher order modulation such as 16-PSK offering higher spectral efficiency. During fade conditions it will operate at lower order modulation such as BPSK and maintain the link reliability while offering lower spectral efficiency. Introducing Forward Error Correction (FEC) such as Low Density Parity Check (LDPC) and Turbo Product Code (TPC) further lower the bit error rate for same SNR and increases the spectral efficiency. FEC adds redundant bits before transmission and uses mathematical model to estimate the error bits without re-transmission. FEC not only improves the reliability of the communication channel but also lower the BER for same modulation type at the same SNR. Having FEC in the system coupled with adaptive modulation allows to operate at the adaptive modulation and coding technique to adapt to the changing SNR in the receive signal. The rate of the FEC vary along with the modulation coding. Interleaving is another technique to improve the error rate in the system. Interleaving is based on transforming burst packet losses into uniform packet losses, which can lower the packet loss, and effectively lowering BER. Before message are transmitted, it is read into the buffer in rows and transmitted in the column allowing receiver to recover [3]. The model met the primary objective of modeling M-PSK system for various channel conditions.

* references

[1] Jie Lai; Mandayam, Narayan B., "Fade margins for minimum duration outages in lognormal shadow fading and Rayleigh fading," Signals, Systems & Computers, 1997. Conference Record of the Thirty-First Asilomar Conference on , vol.1, no., pp.609,613 vol.1, 2-5 Nov. 1997

[2] Jie Lai; Mandayam, Narayan B., "Minimum duration outages in Rayleigh fading channels," Communications, IEEE Transactions on , vol.49, no.10, pp.1755,1761, Oct 2001

[3] Ming-Fong Tsai; Chih-Heng Ke; Hao-Ming Liang; Hsiung-Yu Huang, "Forward Error Correction with Interleaving mechanism combining Cognitive Technology for video streaming over wireless networks," Wireless and Pervasive Computing (ISWPC), 2011 6th International Symposium on , vol., no., pp.1,6, 23-25 Feb. 2011 * appendixes * * Source code for this model will be provided in a separate electronic copy.