# Analyze BER Performance Of Wireless FSK Systems

Modern communication systems employ digital modulation for a variety of reasons, including improved immunity to noise and channel impairments as well as enhanced security compared to analog modulation. In addition, advances in very large-scale integration (VLSI) and digital signal processing (DSP) technology have made digital modulation more cost effective than analog transmission systems. Digital transmissions accommodate digital error-control codes that detect and/or correct transmission errors, and support complex signal conditioning and processing techniques such as source coding, encryption, and equalization to improve the performance of the overall communication link. By using the MATLAB simulation software from The MathWorks, various digitally modulated systems, including 2-, 4-, and 8-level FSK systems in an additive white Gaussian noise (AWGN) channel, will be analyzed to understand bit-error-rate (BER) performance under different operating conditions.

In digital wireless communication systems, the modulating signal may be represented as a time sequence of symbols or pulses, where each symbol has m finite states. Each symbol represent n bits of information, where n = log2m bits/symbol. Some digital modulation techniques have subtle differences between them, and each technique belongs to a family of related modulation methods. For example, frequency-shift keying (FSK) may be coherently or noncoherently detected, and may have 2, 4, 8, or more levels per symbol.^{1}

Several factors influence the choice of a digital modulation scheme. A desirable modulation scheme provides low bit error rates at low received signal- to-noise ratios (SNRs), performs well under multipath and fading conditions, occupies a minimum bandwidth, and is easy and cost effective to implement. In reality, depending on the demands of a particular application, tradeoffs must be made when selecting a digital modulation scheme.

The performance of a modulation scheme is often measured in terms of its power efficiency and bandwidth efficiency. Power efficiency describes the ability of a modulation technique to preserve the fidelity of the digital message at low power levels. In a digital communication system, in order to increase noise immunity, it is necessary to increase signal power. However, the amount by which the signal power should be increased to obtain a certain level of fidelity (i.e., an acceptable bit error probability) depends on the particular type of modulation employed. The power efficiency (sometimes called energy efficiency) of a digital modulation scheme is a measure of how favorable this tradeoff between fidelity and signal power, and is often expressed as the ratio of the signal energy per bit to noise power spectral density (E_{b}/N_{0}) required at the input of the receiver for a certain probability of error (say 10^{-3}).

Bandwidth efficiency describes the ability of a modulation scheme to accommodate data within a limited bandwidth. In general, increasing the data rate implies decreasing the pulse width of a digital symbol, which increases the bandwidth of the signal. Thus, there is an unavoidable relationship between data rate and bandwidth occupancy. However, some modulation schemes perform better than others in making this tradeoff. Bandwidth efficiency reflects how efficiently the allocated bandwidth is utilized and is defined as the ratio of the throughput data rate per Hertz in a given bandwidth. If R is the data rate in bits per second, and B is the bandwidth occupied by the modulated radio frequency signal, then bandwidth efficiency, η_{B}, is

In terms of bits per second, the number of bits are conveyed or processed per unit of time.

The system capacity of a digital communication system is directly related to the bandwidth efficiency of the modulation scheme, since a modulation with a greater value of ?_{B} will transmit more data in a given spectrum allocation. There is a fundamental upper bound on achievable bandwidth efficiency. Shannon's channel coding theorem states that for an arbitrary small probability or error, the maximum possible bandwidth efficiency is limited by the noise in the channel, and is given by channel capacity formula. The Shannon's bound for AWGN non-fading channel is given by Eq. 2.

In designing a digital communication system, very often a tradeoff exists between bandwidth efficiency and power efficiency. For example, adding error control coding to a message increases bandwidth occupancy (and, in turn, reduces the bandwidth efficiency), but at the same time reduces the required power for a particular bit error rate, and hence trades bandwidth efficiency for power efficiency. On the other hand, higher level modulation schemes (M-ary keying), except M-ary FSK, decrease bandwidth occupancy but increase the required received power, trading power efficiency for bandwidth efficiency.

While power and bandwidth considerations are very important, other factors also affect the choice of a digital modulation scheme. For example, for all personal communication systems that serve a large user community, the cost and complexity of the subscriber receiver must be minimized, and a modulation that is simple to detect is most attractive. The performance of a modulation scheme under various types of channel impairments such as Rayleigh and Ricean fading and multipath time dispersion, given a particular demodulator implementation, is another key factor in selecting a modulation. In wireless systems where interference is a major issue, the performance of a modulation scheme in an interference environment is extremely important. Sensitivity to detection of time jitter, caused by timevarying channels, is also an important consideration in choosing a particular modulation scheme. In general, the modulation, interference, and implementation of the time-varying effects on a channel as well as the performance of the specific demodulator are simulated as a complete system.^{1-8}

As its name suggests, an FSK transmitter has its frequency shifted by the message.^{1-15} Although there could be more than two frequencies in FSK, this experiment will use a binary bit stream, with only two frequencies. The word keyed' suggests that the message is of the on-off' variety, such as one (historically) generated by a binary sequence (*Fig. 1*).^{2}

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The transmitter may consist of two oscillators (frequencies f_{1} and f_{2}), with only one being connected to the output at any one time (*Fig 2*).^{3} Unless there are special relationships between the two oscillator frequencies and the bit clock there will be abrupt phase discontinuities of the output waveform during transitions of the message.

In a FSK system, the binary symbols are represented by Eqs. 3 and 4,

S_{0}=Acos(2? f_{0}t) 0

S_{1}=Acos(2?f_{1}t ) elsewhere (4)

where

A = a constant,

f_{0} and f_{1} = the transmitted frequencies, and

T = the bit duration.

The signal has a power P = A2/2, so that A = 2P.

Equations 3 and 4 can be written as:

where E = PT = the energy contained in a bit duration.

For orthogonally, f_{0} = m/T and f_{1} = n/T for integer n > integer m and f_{1} - f_{0} must be an integer multiple of 1/2T. It is possible to take f_{1}(t) = (2T)^{0.5}cos2f_{0}t and f_{2}(t) = (2T)^{0.5}sin2 f_{1}t as the orthogonally basis functions.^{3} The signal constellation diagram of an orthogonal BFSK signal is shown in Fig. 3.^{3} Phase continuity is maintained at the transitions. Furthermore, the BFSK signal is the sum of two BASK signals generated by two modulating signals m_{0}(t) and m_{1}(t). Therefore, the Fourier transform of the BFSK signal s(t) is given by Eq. 6 (on p. 85).

**Figure 4** shows the amplitude spectrum of the BFSK signal when m0(t) and m_{1}(t) are periodic pulse trains. An alternative representation consists of letting f_{0} = f_{c} - f and _{f}1 = f_{c} + f. Then, as Eqs. 7 and 8 show

f_{1} - f_{0} = 2?f (7)

s (t) = Acos2? ( f_{c} + ?f )t (8)

where f_{c} = the carrier frequency,

f =

B = the frequency deviation,

= the modulation index, and

B = 1/T = the bandwidth of the modulating signal.

When f >> 1/T, the condition defines a wideband BFSK signal. The bandwidth is approximately equal to 2 f. When f

From probability theory, it is known that a Raleigh distributed random variable R, with probability distribution function

is related to Gaussian random variables X and X by Eqs. 9 and 10:

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Generating a Rayleigh distributed random variable with the computer yields Eq. 11:

where M = a uniformly distributed random variable in the interval (0, 2?). Solving Eq. 11 results in Eq. 12:

By generating a second uniformly distributed random variable N, and defining N ?^{2}, then from Eqs. 9, 10, and 12 two independent Gaussian distributed random variables X and X are obtained as

A model for the simulation of coherent binary FSK system in AWGN channel is shown in Fig. 5.

Since the signals are orthogonal, when a 0 1(t)> is transmitted, the correlation outputs are r_{0} = + n_{1} and r_{1} = n_{1}. When a 1 2(t)> is transmitted, the correlation outputs are r1 = n0. **Figure 6** shows the results of the simulation for the transmission of 40 kb at different values of E_{b}/N_{0} and how it compares with theory.

A model for the simulation of a noncoherent (square-law detection) binary FSK system in an AWGN channel is shown in **Fig. 7**. Since the signals are orthogonal, when s_{1}(t) is transmitted, the first demodulator output is

where

n_{1I}, n_{1Q} n_{2I} n_{1Q} = mutually statistically independent zero-mean Gaussian random variables with variance s^{2} and f = the channel-phase shift.

The square-law detector computes r1 = r1I^{2} + r1Q ^{2}r^{2} = r2I ^{2} + r2Q ^{2} and selects the information bit corresponding to the larger of these two decision variables. **Figure 8** shows the results of the simulation for the transmission of 40 kb at several different values of Eb/ N0 and how it compares with theory. The result is acceptable and theoretical is similar to simulation.

**Figure 9** shows a model for the simulation of a coherent 4-level FSK system in an AWGN channel.2 The block diagram of the simulation of a noncoherent 4-level FSK system in an AWGN channel is similar to that in **Fig. 7**; the only difference is the number of correlate (demodulator) outputs. The block diagram of the simulation of coherent 8-level FSK is similar to that depicted in **Fig. 9**, while the model for simulation of a noncoherent 8-level FSK system is similar to that shown in **Fig. 7**.

**Figure 10**, **Figure 11**, **Figure 12**, **Figure 13** illustrate the results of the simulations for the transmission of 40 kSymbols at several different values of E_{b}/N_{0} and how it compares with theory. With an increase in M state, the BER decreases. Also, the value of E_{b}/N_{0} is smaller in an M-8 configuration than for an M-4 configuration. **Figure 14** and **Figure 15** compare simulated bit and symbol error probabilities for coherent and noncoherent 2, 4, and 8-level FSK systems. At a particular error probability, the required energy efficiency (E_{b}/N_{0}) is lowest for an 8-level FSK system and largest for a binary FSK system. At a constant E_{b}/N_{0}, an 8-level FSK system has the lowest error probability, while a binary FSK system has the largest.

REFERENCES

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