** Irreducible Bit Error Rate (BER) and DIGITAL MODULATION IN FREQUENCY SELECTIVE MOBILE CHANNELS**

**(i) Irreducible Bit Error Rate (BER)**

Frequency selective fading caused by multipath time delay causes interference which results in an irreducible BER floor for mobile systems. However, even if a mobile channel is not frequency selective, the time-varying Doppler spread due to motion creates an irreducible BER floor due to the random spectral spreading. These factors impose bounds on the data rate and BER that can be transmitted reliably over a frequency selective channel. Simulation is the major tool used for analyzing frequency fading effects.

It has been observed the performance of various modulation schemes in frequency selective fading channels through simulations. Both filtered and unfiltered BPSK, QPSK, and MSK modulation schemes were studied, and their BER curves were simulated as a function of the normalized rms delay spread

(d = ).

**(ii) Cause of Irreducible BER**

The irreducible error floor in a frequency selective channel is primarily caused by the errors due to the intersymbol interference, which interferes with the signal component at the receiver sampling instants. This occurs when

(a) main (undelayed) signal component is removed through multipath cancellation,

(b) a non-zero value of d causes ISI, or

(c) the sampling time of a receiver is shifted as a result of delay spread. It has been observed that errors in a frequency selective channel tend to be bursty. Based on the results of simulations, it is known that for small delay spreads (relative to the symbol duration), the resulting flat fading is the dominant cause of error bursts. For large delay spread, timing errors and ISI are the dominant error mechanisms.

Figure 11.11 shows the average irreducible BER as a function of d for different unfiltered modulation schemes using coherent detection. From the figure, it is seen that the BER performance of BPSK is the best among all the modulation schemes compared. This is because symbol offset interference does not exist in BPSK. Both OQPSK and MSK have a timing offset between two bit sequences, hence the cross-rail ISI is more severe, and their performances are similar to QPSK. Figure 11.11 shows the BER as a function of rms delay spread normalized to the bit period (d = ) rather than the symbol period as used in figure 11.11. By comparing on a bit, rather

** FIGURE 11.11 ***The irreducible BER performance for different modulations with coherent detection for a chance with a Gaussian shaped power delay profile. The parameter d is the rms delay spread normalized by the symbol period *

than symbol basis, it becomes easier to compare different modulations. This is done in figure 11.12, when it is clear that 4-level modulations (QPSK, OQPSK, and MSK) are more resistant to delay spread than BPSK for constant information throughout. Interestingly, 8-ary keying has been found to be less resistant than 4-ary keying, and this has led to the choice of 4-ary keyibng for many 2G and 3G wireless standards.

** FIGURE 11.12 ***The same set of curves as plotted in Figure 11.7 plotted as a function of rms delay spread normalized by bit period*

**SUMMARY**

■ Spread spectrum may be defined as a technique in which a transmitted signal occupies a bandwidth which is kept much larger than that which is required by a base band information signal.

■ The basic concept of spread spectrum has been illustrated in figure 17.1. Obviously, the idea is to take the energy in bandwidth *B* and spread it over the wider bandwidth B* _{RF}*. As a matter of fact, three techniques are, generally used to accomplish this as under:

(i) Direct sequence pseudonoise (DSPN)

(ii) Frequency hopping (FM)

(iii) Time hopping (TH)

■ A combination of these techniques is also available for use. Pseudorandom (PN) codes are used to achieve spreading of the spectrum. Before we discus methods of generating these codes in detail, let us first consider the techniques for widening the spectrum of a signal.

■ One of the most common method to widen the spectrum of a signal is simply to multiply (i.e., modulate) it. by wideband signal. We will observe that the spreading signal must hive properties which aid in acquiring and tracking the signal.

■ The second method of spreading the message signal is accomplished by transmitting each bit or subinteger bit (chip) on a different carrier which is selected from a wide range of frequencies.

■ Time Hop Sreading is the third technique which is used for spreading the message signal spectrum. Time hop channelizes via time slots within a given time frame.

■ The spread spectrum communication spreads the transmitted (RF) signals over a bandwidth which is as larger as is practical. This results in an RF bandwidth which is 10 to 100 times the information signal bandwidth. Frequency hopping and direct sequence modulation are two principle spread spectrum techniques.

■ Time hopping implemented by OOK may be used with PSK and FSK to achieve hybrid spread-spectrum modulation.

■ The number of such sequences has an upper bound S expressed by

S £

The equal sign applies when

*L*is a prime number. These independent sequences can he divided into two equal groups so that each number of one group has a mirror image in the other group. Mirror image sequence have the same hit-sequence when one is forward in time and the other is backward in time.

■ If the PN generator sequenced through all it’s states, then the number of states will be an even number and the sequence available at any Q or .

■ If the sequence is correlated with itself, the result is referred to as

**auto- correlation.**If a sequence is correlated with another sequence, the result is known as cross correlation. It is often desirable to use normalized correlation in our computations. We define this as under:

p =

■ The discrete modulation of the carrier frequency is usually performed as a multiple level (M-ray) PSK or frequency shift (FSK) signal.

■ A disadvantages associated with 4-PSK is a reduction in efficiency of the transmitter power amplifier.

■ In order for the receiver to detect and demodulate the received signal, certain information must be known to the receiver. The information includes the following:

- Existence of the signal mixed-up in the noise signal
- Carrier frequency of the signal
- Modulation type
- PN code rate
- The code phase (i.e., propagation time delay)

■ Our study of conventional communication techniques has acquainted us with TDM and CDMA provides a relatively new type of channel multiplexing in which many information channels can be established in the same frequency channel simultaneously without interference between channels.

**SHORT QUESTIONS WITH ANSWERS**

**Q.1 Explain the concept of spread spectrum. **

**Ans. **Pseudorandom (PN) codes are used to achieve spreading of the spectrum. Before we discuss methods of generating these codes in detail, let us first consider the techniques for widening the spectrum of a signal.

**diagram**

**FIGURE 11.13*** Illustration of the concept of Spread Spectrum*

**Q.2. What do you mean by frequency HOP spreading? Explain. **

**Ans. **The second method of spreading the message signal is accomplished by transmitting each bit or subinteger hit (chip) on a different carrier which is selected from a wide range of frequencies.

The simplified arrangement has been shown in figure 11.14.

**diagram**

**FIGURE 11.14*** Frequency hopping modulation*

**Q.3. What are the three basic spread-spectrum techniques. **

**Ans. **As a matter of fact, three techniques are, generally used to accomplish this as under:

(i) Direct sequence pseudonoise (DSPN)

(ii) Frequency hopping (FH)

(iii) Time hopping (TH)

**Q.4. Explain Time Hop spreading? **

**Ans. **Time Hop Sreading is the third technique which is used for spreading the message signal spectrum. Time hop channelizes via time slots within a given time frame. This contrast with frequency hoping, where channels are established via frequency slots. The basic tune hop modulator has been shown in figure 11.15.

**diagram**

**FIGURE 11.15*** Time hop modulation*

**QUESTIONS**

- A true random waveform has no DC term. Why is there a DC term in the power density of the PN code ?
- (a) What are the sequences generated by the polynomials
*x*^{5}+*x*^{4}+*x*^{3}+*x*^{2}+ 1 and*x*^{5}+*x*^{4}+*x*^{2}+*x*+ 1 ? Assume an initial conditions of (11111) in both cases.

(b) Compute the plot the autocorrelation function for each sequence in part (a).

(c) Compute and plot the cross-correlation function for the two sequences.

(d) Draw the circuit diagram for generating each of the sequences.

- A PN sequences is 12
^{15}– 1 in length. How many runs of four is would be expected ? - Explain the principles of spread spectrum.
- With the help of block diagram, explain frequency Hopping modulation.
- Explain about time Hop spreading.
- What do you mean by Burst transmission? Explain.
- Explain the generation of pseudorandom sequences.

**APPENDICES**

**APPENDIX – A**

**DEFINITIONS **

X(w) = F[x(t)] = (t) e^{-jwt} dt

x(t) = F-1 [X(w)] = (w) e^{-jwt} dw

Parseval’s theorems :

**Table A.1 : Properties of the Fourier Transform**

Property |
x(t) | X(w) |

Linearity Time shifting Scaling Time reversal Duality Frequency shifting Modulation Time differentiation Frequency differentiation Integration Convolution Multiplication |
a_{1}x_{1}(t) + a_{2}x_{2}(t) x(t – t)_{0} x(at) x(-t)X(t) x(t)e^{jw0t}x(t)cos w_{0}tx‘(t)– jtx(t)(t) dt x_{1}(t) Ä x_{2}(t) x_{1}(t) x_{2}(t) |
a_{1}X_{1}(w) +a_{2}X_{2}(w)X(w)e ^{-jwt0}X X(-w) 2πx(-w) X(w-w _{0})[X(w-w _{0})+X(w+w_{0})]jwX(w) X'(w) X(w) + πX(0) (w) X _{1}(w)X_{2}(w)X _{1}(w) Ä X_{2}(w) |

**Table A.2 : Properties of the Fourier Transform**

x(t) | X(w) |

(t) (t-t _{0})1 u(t) sgn(t) e ^{jw0t}cos w _{0}tsin w _{0}te ^{-at} u(t) a > 0te ^{-at} u(t) a > 0e ^{-at} u(t) a > 0e ^{-t2j (2} ^{2}^{)}equationequationequation |
1 e ^{-jwt0}2π (w) π (w) + – j sin(w)2π (w-w _{0})π[ (w-w _{0}) + (w +w_{0})]-jπ[ (w-w_{0}) – (w +w_{0})] equation equation equation-jsgn(w) X(w) |

**APPENDIX B **

Bessel functions of the first kind of order n and argument β:

**Generating Function and Definition **

**equation**

**equation**

**equation**

Properties of J_{n}(m_{f})

- J
_{-n}(β) = (-J)^{n}J_{n}(β) - J
_{n-1}(β) + J_{n+1}(β) = J_{n}(β)

**Table B : 1 Selected Values of J _{n}(**

**β**

**)**

n/β | 0.1 | 0.2 | 0.5 | 1 | 2 | 5 | 8 | 10 |

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 |
0.997 0.050 0.001 |
0.990 0.100 0.005 |
0.938 0.242 0.031 0.003 |
0.765 0.440 0.115 0.020 0.002 |
0.224 0.577 0.353 0.129 0.034 0.007 0.001 |
-0.178 -0.328 0.047 0.365 0.391 0.261 0.131 0.053 0.018 0.006 0.001 |
0.172 0.235 -0.113 -0.291 -0.105 0.286 0.338 0.321 0.224 0.126 0.061 0.026 0.010 0.003 0.001 |
-0.246 0.043 0.255 0.058 -0.220 -0.234 -0.014 0.217 0.318 0.208 0.208 0.123 0.063 0.029 0.012 0.005 0.002 |

**APPENDIX C **

The Complementary Error Function Q(z)

**equation**

Q(z) = Q(-z) = 1 -Q(z) z __>__ 0

Q(z) = -erf (z)

erf(z) =

Q(z) = z > > (z > 4)

** Table C.1 : Q(z)**

z | Q(z) | z | Q(z) | z | Q(z) | z | Q(z) |

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 |
0.5000 0.4801 0.4602 0.4404 0.4207 0.4013 0.3821 0.3632 0.3446 0.3264 0.3085 0.2912 0.2743 0.2578 0.2420 0.2266 0.2169 0.1977 0.1841 0.1711 |
1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 |
0.1587 0.1469 0.1357 0.1251 0.1151 0.1056 0.0968 0.0885 0.0808 0.0735 0.0668 0.0606 0.0548 0.0495 0.0446 0.0401 0.0359 0.0322 0.0287 0.0256 |
2.00 2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 |
0.0228 0.0202 0.0179 0.0158 0.0139 0.0122 0.0107 0.0094 0.0082 0.0071 0.0062 0.0054 0.0047 0.0040 0.0035 0.0030 0.0026 0.0022 0.0019 0.0016 |
3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70 3.75 3.80 3.85 3.90 3.95 4.00 4.25 4.75 5.20 5.60 |
0.00135 0.00114 0.00097 0.00082 0.00069 0.00058 0.00048 0.00040 0.00034 0.00028 0.00023 0.00019 0.00016 0.00013 0.00011 0.00009 0.00007 0.00006 0.00005 0.00004 0.0003 10 ^{-5}10 ^{-6}10 ^{-7}10 ^{-8} |

**APPENDIX D**

D.1. **Trigonometric Identities**

e^{±}^{j}^{q} = cosq ± j sin q

cos q = (e^{j}^{q} + e^{-j}^{q})

sin q = (e^{j}^{q} – e^{–j}^{q})

sin^{2}q + cos^{2} q = 1

cos 2q = cos^{2}q – sin^{2} q = 2 cos^{2} q – 1 = 1-2 sin^{2}q

sin 2q = 2cosq sinq

con2 q = (1 + cos 2q)

sin^{2} q = (1 – cos 2q)

cos (a ± b) = cos α cos b sin α sin b

sin (a ± b) = sin α cos b cos α sin b

tan (a __+__ b) =

cos α cos b = cos (α – b) + cos (α + b)

sin α sin b = cos (α – b) – cos (α + b)

sin α cis b = sin (α – b) + sin (α + b)

a cos x + b sin x = Ccos (x + q) where C = and q = – tan-^{1 }

**D.2. Series Expansions and Approximations**

**equation**

**equation**

**equation**

e* ^{x}* = e

^{x}^{Ina}= 1 +

*x*In

*a*+ (

*x*In

*a*)

^{2}+ …

cos

*x*= 1 –

*x*

^{2}+

*x*

^{4}– …

sin

*x*= x –

*x*

^{3}+

*x*

^{5}– …

In (1 +

*x*) =

*x*–

*x*

^{2}+

*x*

^{3}– …

When |

*x*| < < 1

(1 +

*x*)

*= 1 + n*

^{n}*x*

e

*1 +*

^{x}*x*

cos

*x*1

sin

*x*

*x*

a

*1 +*

^{x}*x*In

*a*

In (1 +

*x*)

*x*

**D.3. Integrals**

**Indefinite integral**

a

*x*d

*x*= sin a

*x*

a

*x*d

*x*= – cos a

*x*

a

*x*cos b

*x*d

*x*= a

^{2}b

^{2}

a

*x*sin b

*x*d

*x*= a

^{2}b

^{2}

a

*x*cos b

*x*d

*x*= – a

^{2}b

^{2}

a

*x*d

*x*=

a

*x*d

*x*=

d

*x*=

cos b

*x*d

*x*= (a cos

*bx*+

*b*sin b

*x*)

sin b

*x*d

*x*= (a sin

*bx*+

*b*cos b

*x*)

= tan

^{-1}

= = tan

^{-1}

a > 0

a > 0

a > 0

**equation**

a > 0

a > 0

a > 0, b > 0

a > 0, b > 0

**Integration by Parts**

a > 0, b > 0

**ABBREVIATIONS**

ac Alternating current

ADPCM adaptive differential pulse-code modulation

AM amplitude modulation

ARQ automatic-repeat-request

ASCII American Standard Code for Information Interchange

ASK amplitude-shift keying

ATM asynchronous transfer mode

BPF bandpass filter

B-ISDN broadband ISDN

BSC binary symmetric channel

CCITT Consultative Committee for International Telephone and Telegraph

CDM code-division multiplexing

CDMA code-division multiple access

CPFSK continuous-phase frequency-shift keying

CW continuous wave db decibel dc direct current

DFT discrete Fourier transform

DM delta modulation

DPCM differential phase-code modulation

DPSK differential phase-shift keying

DSB-SC double-sideband suppressed carrier

Ds/BPSK direct sequence/binary phase-shift keying exp exponential

FDM frequency-division multiplexing

FDMA frequency-division multiple access

FFT fast Fourier transform

FH frequency-hop/M-ary frequency-shift keying

FSK frequency-shift keying

Hz Hertz

IDFT Inverse discrete Fourier transform IF intermediate frequency

ISDN integrated services digital network

ISI inter-symbol interference

LDM linear delta modulation

In natural logarithm (log

_{e}*x*)

log logarithm (log

_{10}

*x*)

LPF lowpass filter

ms milisecond

ms microsecond

modem modulator-demodulator

MSK minimum-shift keying

NRZ non-return to zero

OOK on-off keying

OQPSK offset QPSK

OSI open-system interconnection

PAM pulse amplitude modulation

PCM pulse-code modulation]

PDM pulse-duration modulation

PPM pulse-position modulation

PG processing gain

PLL phase-locked loop

PN pseudo-noise

PSK phase-shift keying

QAM quadrature amplitude modulation

QPSK quadrature phase-shift keying

RF radio frequency

rms root-mean-square

RZ return-to-zero

SDMA space-division multiple access

SNR signal-to-noise ratio

STM synchronous transfer mode

TDM time-division multiple access

TDMA time-division multiple access

TSI time-slot interchanging

T-S-T time-space-time switching

TV Televiosion

UHF Ultra-high frequency

VCO Voltage controlled oscillator

VHF very high frequency

VLSI very-large-scale integration

WDM wavelength-division multiplexing