CHAPTER 12 Applications of Discrete-Time Signals and Systems
12.4.3 Spread Spectrum and Orthogonal Frequency-Division Multiplexing
The objective of TDM is to put several users or different types of data together sharing the same bandwidth at different times. Likewise, FDM users share part of the available bandwidth all the time.
TDM and FDM are examples of how to use bandwidth in an efficient way. In other situations, like in quadrature-amplitude modulation (QAM), the objective is to send two messages over the same bandwidth using the orthogonality of the carriers to recover them. In spread spectrum, the objective is to use the orthogonality of the carriers associated with different users to share the available spectrum, while spreading the message in frequency so that it occupies a bandwidth much larger than that of the message. On the other hand, orthogonal frequency-division multiplexing (OFDM) is a multicarrier system where the carriers are orthogonal.
Sharing the radio spectrum among users, or multiple access, is a basic strategy of wireless commu- nication systems. Basic modalities are derived from FDM, TDM, and spread spectrum. In FDMA the spectrum is shared by assigning specific channels to users, permanently or temporarily. TDMA allows access to all of the available spectrum, but each user is assigned a time interval in which to access it. CDMA uses spread spectrum, where a user’s message is spread or encrypted over the available
spectrum using a code to differentiate the different users. The objective of these three techniques is to maximize the radio spectrum utilization.
Spread Spectrum—A Famous Actress Idea
Not surprisingly, the first mention of the use of frequency hopping, a form of spread spectrum, for secure communications came from a patent by Nikolas Tesla in 1903. As you recall, Tesla is the world-renowned Serbian-American inventor, and physicist, and mechanical and electrical engineer who pioneered amplitude modulation.
The most celebrated invention of frequency-hopping was, however, that of Hedy Lamarr and George Antheil, who in 1942 received a U.S. patent for their “secret communications system,” in which they used a piano-roll for frequency-hopping.
This was during World War II, and their idea was to stop the enemy from detecting or jamming radio-guided torpedoes.
To avoid the jammer, in frequency-hopping spread spectrum the transmitter changes in a quasi-random way the center frequency of the transmitted signal. Hedy Lamarr (1913–2000) was an Austrian-American actress and communications technology innovator, while George Antheil (1900–1959) was an American composer and pianist. Their patent was never applied, and it would be many years before the technology was actually deployed. Ms. Lamarr conceived the idea of hopping from frequency to frequency just as a piano player plays the same notes, but in different octaves. Their concept eventually provided the basis for the CDMA airlink, which Qualcomm commercialized in 1995. Today, CDMA and its core principles provide the backbone for wireless communications, thanks to the creative vision of an extraordinary woman [70, 74, 62].
Spread Spectrum
A spread-spectrum system is one in which the transmitted signal is spread over a wide frequency band, much wider than the bandwidth required to transmit the message. Such a system would take a baseband voice signal with a bandwidth of a few kilohertz and spread it to a band of many megahertz.
Two types of spread-spectrum systems are:
n Direct-sequence system:A digital code sequence with a bit rate higher than the message is used to obtain the modulated signal.
n Frequency-hopping system: The carrier frequency is shifted in discrete increments in a pattern dictated by a code sequence. We will not consider this here.
Direct-Sequence Spread-Spectrum
Suppose the messagem(t)we wish to transmit is a polar binary signal, and that a spreading codec(t), also in polar binary form, is modulated by the message to obtain the modulated baseband signal
x(t)=m(t)c(t) (12.21)
The sequencec(t)is pseudo-random, unpredictable to an outsider, but that can be generated deter- ministically. Each user is assigned uniquely one of these sequences—that is, the spreading codes assigned to two users are not related at all. Moreover, the bit rate ofc(t)is much higher than that of the message. As in many other modulation systems, the modulated baseband signalx(t)has a much higher rate than the message, and as such its spectrum is much wider than that of the message that is already wide given that it is a sequence of pulses. This can also be seen by considering thatx(t)as the product ofm(t), andc(t), its spectrum is the convolution of the spectrum ofm(t)with the spectrum ofc(t)with a bandwidth equal to the sum of the bandwidths of these spectra.
12.4 Application to Digital Communications 737
When transmitting over a radio link the baseband signalx(t)modulates an analog carrier to obtain the transmitting signals(t). At the receiver, if no interference occurred in the transmission, the received signalr(t)=s(t), and after demodulation using the analog carrier frequency, the spread signalx(t)is obtained. If we multiply it byc(t)we get
x(t)c(t)=c2(t)m(t)=m(t) (12.22)
sincec2(t)=1 for allt. See Figure 12.12.
Two significant advantages of direct-sequence spread spectrum are:
n Robustness to noise and jammers:The above detection or despreading is idealized. The received signal will have interferences due to channel noise, interference from other users, and even, in military applications, intentional jamming. Jamming attempts to corrupt the sent message by adding to it either a narrowband or a wideband signal. If at the receiver, the spread signal con- tains additive noiseη(t)and a jammerj(t), it is demodulated by the BPSK system. The received baseband signal is
ˆr(t)=x(t)+ ˆη(t)+ ˆj(t) (12.23) where the noise and the jammer have been affected by the demodulator.
Multiplying it byc(t)gives
ˆr(t)c(t)=m(t)+ ˆη(t)c(t)+ ˆj(t)c(t) (12.24) or the desired message with a spread noise and jammer. Thus, the transmitted signal is resistant to interferences by spreading them over all frequencies.
FIGURE 12.12 Direct-sequence
spread-spectrum system.
Spreader BPSK Modulator
× LPF ×
×
×
BPSK Demodulator
Despreader m(t)
A cos(Ωct)
A cos(Ωct)
c(t)
r(t)=s(t)+η(t)+j(t)
c(t)
s(t)
m(t∧ )
n Robustness to interference from other users:Assuming no noise or jammer, if the received baseband signal comes from two users—that is,
ˆr(t)=m1(t)c1(t)+m2(t)c2(t) (12.25) where the codesc1(t)andc2(t)are the corresponding codes for the two users, andm1(t)andm2(t) their messages. At the receiver of user 1, despreading using codec1(t)we get
ˆr(t)c1(t)=m1(t)c21(t)+m2(t)c2(t)c1(t)≈m1(t) (12.26) since the codes are generated so thatc21(t)=1 and c1(t)andc2(t)are not correlated. Thus, we detect the message corresponding to user 1. The same happens when there is interference from more than one user.
Simulation of direct sequence spread spectrum. In this simulation we consider that the mes- sage is randomly generated and that the spreading code is also randomly generated (our code does not have the same characteristics as the one used to generate the code for spread-spectrum systems).
To generate the train of pulses for the message and the code we use filters of different length (recall the spreading code changes more frequently than the message). The spreading makes the transmitting signal have a wider spectrum than that of the message (see Figure 12.13).
The binary transmitting signal modulates a sinusoidal carrier of frequency 100 Hz. Assuming the communication channel does not change the transmitted signal and perfect synchronization at the analog receiver is possible, the despread signal coincides with the sent message. In practice, the effects of multipath in the channel, noise, and possible jamming would not make this possible.
%%%%%%%%%%%%%%%%
% Simulation of
% spread spectrum
%%%%%%%%%%%%%%%%
clear all; clf
% message
m1 = rand(1,12)>0.9;m1 = (m1-0.5)∗2;
m = zeros(1,00);
m(1:9:100) = m1 h = ones(1,9);
m = filter(h,1,m);
% spreading code
c1 = rand(1,25)>0.5;c1 = (c1-0.5)∗2;
c = zeros(1,100);
c(1:4:100) = c1;
h1 = ones(1,4);
c = filter(h1,1,c);
Ts = 0.0001; t = [0:99]∗Ts;
s = m.∗c;
figure(1)
12.4 Application to Digital Communications 739
0 1 2 3 4 5 6 7 8 9
−1 0 1
m(t)
Message
0 1 2 3 4 5 6 7 8 9
−1 0 1
c(t)
Code
0 1 2 3 4 5 6 7 8 9
−1 0 1
s(t)
t (sec) Spread Message
−50000 −4000−3000−2000−1000 0 1000 2000 3000 4000 20
40 60
|M(f)|
Message Spectrum
−50000 −4000−3000−2000−1000 0 1000 2000 3000 4000 10
20 30
|S(f)|
f (Hz) Spread Signal Spectrum
0 1 2 3 4 5 6 7 8 9
−1 0 1
0 1 2 3 4 5 6 7 8 9
−1 0 1
r(t)
0 1 2 3 4 5 6 7 8 9
−1 0 1
0 1 2 3 4 5 6 7 8 9
−1 0 1
t (sec)
×103
×103
×103
×103
×103
×103
×103 sa(t)ma(t)m1(t)
(a) (b)
(c)
FIGURE 12.13
Simulation of direct-sequence spread-spectrum communication. (a) Displays from top to bottom the message, the code, and the spread signal. (b) Displays the spectrum of the message and of the spread signal (notice it is wider than that of the message). (c) Displays the band-pass signals sent and received (assumed equal), the despread analog, and the binary message.
subplot(311)
bar(t,m); axis([0 max(t) -1.2 1.2]);grid; ylabel(‘m(t)’) subplot(312)
bar(t,c); axis([0 max(t) -1.2 1.2]);grid; ylabel(‘c(t)’) subplot(313)
bar(t,s); axis([0 max(t) -1.2 1.2]);grid; ylabel(‘s(t)’); xlabel(‘t (sec)’)
% spectrum of message and spread signal M = fftshift(abs(fft(m)));
S = fftshift(abs(fft(s)));
N = length(M);K = [0:N-1];w = 2∗K∗pi/N-pi; f = w/(2∗pi∗Ts);
figure(2) subplot(211)
plot(f,M);grid; axis([min(f) max(f) 0 1.1∗max(M)]); ylabel(‘—M(f)—’) subplot(212)
plot(f,S); grid; axis([min(f) max(f) 0 1.1 ∗max(S)]);ylabel(‘—S(f)—’); xlabel(‘f (Hz)’)
% analog modulation and demodulation s = s.∗cos(200∗pi∗t);
r = s.∗cos(200∗pi∗t);
% despreading mm = r.∗c;
for k = 1:length(mm);
if mm(k)>0 m2(k) = 1;
else
m2(k) = -1;
end end figure(3) subplot(411)
plot(t,s); axis([0 max(t) 1.1∗min(s) 1.1∗max(s)]);grid; ylabel(‘s a(t)’) subplot(412)
plot(t,r); axis([0 max(t) 1.1∗min(r) 1.1∗max(r)]);grid; ylabel(‘r(t)’) subplot(413)
plot(t,mm); axis([0 max(t) 1.1∗min(mm) 1.1∗max(mm)]);grid; ylabel(‘m a(t)’) subplot(414)
bar(t,m2); axis([0 max(t) -1.2 1.2]);axis([0 max(t) 1.1∗min(mm) 1.1∗max(mm)]) grid;ylabel(‘\m(t)’); xlabel(‘t (sec)’)
Orthogonal Frequency-Division Multiplexing
OFDM is a multicarrier modulation technique where the available bandwidth is divided into nar- rowband subchannels. It is used for high data-rate transmission over mobile wireless channels [27, 60, 4].
If{dk, k=0,. . .,N−1}are symbols to be transmitted, the OFDM-modulated signal is s(t)=
∞
X
m=−∞
N−1
X
k=0
dkej2πfktp(t−mT) (12.27)
whereT is the symbol duration, fk=f0+k1f for a subchannel bandwidth1f =1/T with initial frequency f0, and p(t)=u(t)−u(t−T). Thus, the carriers are conventional complex exponentials.
Considering a baseband transmission, at the receiver the orthogonality of these exponentials in [0,T] allows us to recover the symbols. Indeed, assuming that no interference is introduced by the transmission channel (i.e., the received signalr(t)=s(t)), multiplying r(t)by the conjugate of the exponential carrier and smoothing the result we obtain fork=0,. . .,N−1, andm≤t≤(m+1)T
12.4 Application to Digital Communications 741
(wherep(t−mT)=1), 1 T
(m+1)T
Z
mT
r(t)e−j2πfktdt= 1 T
(m+1)T
Z
mT N−1
X
`=0
d`ej2πf`te−j2πfktdt
=
N−1
X
`=0
d`1 T
(m+1)T
Z
mT
e−j2π(fk−f`)tdt
=
N−1
X
`=0
d`δ[k−`]=dk
for any−∞<m<∞, and where we letfk−f` =(k−`)1f =(k−`)/T.
OFDM Implementation with FFT
If the modulated signals(t), 0≤t≤T, in Equation (12.27) is sampled att=nT/N, we obtain for a frame the inverse DFT
s[n]=
N−1
X
k=0
dkej2πfknT/N=
N−1
X
k=0
dkej2πkn/N 0≤n≤N−1 (12.28) where 2πfkT/N=2πk/Nare the discrete frequencies in radians. At the receiver, with no interferences present, the symbols{dk}are obtained by computing the DFT of the baseband received signal. Given that the inverse and the direct DFT can be efficiently implemented by the FFT, the OFDM is a very effi- cient technique that is used in wireless local area networks (WLANs) and digital audio broadcasting (DAB).
Figure 12.14 gives a general description of the transmitter and receiver in an OFDM system: The high- speed data in binary form coming into the system are transformed from serial to parallel and fed into an IFFT block giving as output the transmitting signal that is sent to the channel. The received signal is then fed into an FFT block providing estimates of the sent symbols that are finally put in serial form.
FIGURE 12.14
Discrete model of baseband OFDM. The blocks S/P and P/S convert a serial into a parallel stream and a parallel to serial, respectively.
S/P IFFT FFT P/S
d0
d1 Channe
{dk}
dN−1
s(n) r(n)
d∧0
{d∧k} dN−1
∧
d∧1
. . .
. . .