frequency hopping
TRANSCRIPT
-1-
Frequency Hopping: Mathematical Approach,
Applications and Simulation
Mathematics 9 – Probability and Stochastic Processes
Second Year – Electronics and Communication Department
أيمه أسامة السيد عمرو عصام حسان كريم سعيد محمد حافظSection: 7 Number:210 Section: 7 Number:191 Section:3 Number:83
مصطفى محمود محمد
خيرهللا
مروان محمد عادل عبد مروان مصطفى مصطفى
العظيمSection: 10 Number: 295 Section: 10 Number:281 Section: 10 Number:280
Group N42
شريف ربيع. د.أ
-2-
Table of contents
1. Introduction 3
2. Spread spectrum 3
3. Direct sequence vs. frequency hopping 3
4. Pseudorandom process 4
5. Basic concept 4
6. Basic frequency hopping spread spectrum algorithm 5
7. Frequency hopping sequences and hit probability 5 8. Frequency Hopping Code Division Multiple Access (FH-CDMA) 7 9. Plotting the distribution of hitting 8
Appendix A. MatLab code for the plot 9 Appendix B. MatLab simulation for sequence generation 9 Appendix C. Adaptive frequency hopping for unlicensed bands 11 Appendix D. References 12
-3-
1. Introduction
In 1895, Guglielmo Marconi opened the way for modern wireless
communications by transmitting the three-dot Morse code for the letter ‘S’ over a
distance of three kilometers using electromagnetic waves. From this beginning,
wireless communications has developed into a key element of modern society. From
satellite transmission, radio and television broadcasting to the now ubiquitous mobile
telephone, wireless communications has revolutionized the way societies function.
To understand how the signal is transmitted in air, we make an analogy between
electromagnetic waves and sound. At the ear of the listener, the waves impinge upon
the eardrum of the listener and are translated into familiar words, phrases, and tones.
When information is transmitted through wireless means such as in radio
transmission, this information must first be converted to electrical signals. In order to
understand the benefits of radio transmission, it is helpful to discuss the nature of the
term "spectrum."
2. Spread Spectrum
Spread-spectrum techniques are methods by which a signal (e.g. an electrical,
electromagnetic) generated in a particular bandwidth is deliberately spread in
the frequency domain, resulting in a signal with a wider bandwidth. These techniques
are used for a variety of reasons, including the establishment of secure
communications, increasing resistance to natural interference and jamming, to prevent
detection … etc. Most commercial spread spectrum systems transmit an RF signal
bandwidth in the neighborhood of one to two orders of magnitude greater than the
bandwidth of the information that is being sent.
There are at least two problems with conventional wireless communications that
can occur under certain circumstances. First, a signal whose frequency is constant is
subject to catastrophic interference. This occurs when another signal is transmitted on,
or very near, the frequency of the desired signal. Catastrophic interference can be
accidental it can be deliberate. Second, a constant-frequency signal is easy to
intercept, and is therefore not well suited to applications in which information must be
kept confidential between the source (transmitting party) and destination (receiving
party).
That's why there are two popular forms of spread spectrum modulation, Direct
Sequence and Frequency Hopping.
3. Direct sequence vs. Frequency Hopping
Direct sequence spread spectrum, the stream of information to be transmitted is
divided into small pieces, each of which is allocated across to a frequency channel
across the spectrum. A data signal at the point of transmission is combined with a
higher data-rate bit sequence (also known as a chipping code) that divides the data
according to a spreading ratio. The redundant chipping code helps the signal resist
interference and also enables the original data to be recovered if data bits are damaged
during transmission.
On the other hand, Frequency-hopping spread spectrum (F.H.S.S) is a method of
transmitting radio signals by rapidly switching a carrier among many
frequency channels, using a pseudorandom sequence known to
both transmitter and receiver. An FHSS signal simply appears as an increase in the
background noise to a narrowband receiver. An eavesdropper would only be able to
intercept the transmission if the pseudorandom sequence was known.
Spread-spectrum transmissions can share a frequency band with many types of
conventional transmissions with minimal interference. The spread-spectrum signals
-4-
add minimal noise to the narrow-frequency communications, and vice versa. As a
result, bandwidth can be utilized more efficiently.
4. Pseudorandom process
It is a process that appears to be random but it is not. Pseudorandom sequences
typically exhibit statistical randomness while being generated by an entirely
deterministic causal process. Such a process is easier to produce than a genuine
random one, and has the benefit that it can be used again and again to produce exactly
the same numbers, useful for testing and fixing software. Besides, it have to be known
by transmitter and receiver only so when any other one try to enter the sequence he
just hear noises and no information can be heard.
5. Basic Concept
Any communication system has a band of transmission. This band is divided into
channels, and each user occupies a channel to perform his call.
Channel 1
User 1
User 1
User 1
User 1
Channel 2
User 2
User 2
User 2
User 2
Channel 3
User 3
User 3
User 3
User 3
Channel 4
User 4
User 4
User 4
User 4
Time slot 1
Time slot 2
Time slot 3
Time slot 4
However, if an external noise acts on a certain channel, the user occupying this
channel will get a poor service, while he is paying the same fare as the other users,
which is unfair. Consequently, the service provider wants to provide an equal service
for all users, but with a minimum percentage of poor service, i.e. minimum noise
interference.
To do so, each user occupies all channels during different time slots. Given that a
time slot is in the order of micro or milliseconds, the time a certain user is subjected to
a noisy channel will be small. So the signal by the users contains two types of data;
the normal data and the frequency hopping sequence, which can be considered as the
path of the data sent.
Channel 1
User 1
User 4
User 2
User 3
Channel 2
User 2
User 3
User 1
User 4
Channel 3
User 3
User 1
User 4
User 2
Channel 4
User 4
User 2
User 3
User 1
Table 5.1 : Normal Frequency Distribution
Table 5.2 : Frequency Hopping
-5-
It is clear that if a fifth user entered the system, then probability that any user is
interfered will be 1/m.
But this path is randomly generated according to a certain algorithm called the
frequency hopping sequence. So there is a probability that two users interfere with
each other on the same channel at the same time slot, which is called hit probability.
This hit probability is system dependent:
If the systems is self contained, like , the algorithm adopted can assure that as
long as the number of active users is less than or equal to the number of
carriers, the hit probability equals zero, and increases as the number of active
users exceeds the number of carriers.
If the system works on an unlicensed band, such as Blue-tooth, the hit
probability depends not only on the algorithm, but also on different systems
using the same frequency.
6. Basic Frequency Hopping Spread Spectrum Algorithm
The basic algorithm for successfully transmitting and receiving an FHSS signal
is:
Step 1 – The calling/initiating party sends a request via a control channel or
other pre-defined frequency.
Step 2 – The receiving party then sends a seed number back to the initiating
party.
Step 3 – The seed number is then used as the key variable in the pre-defined
algorithm for the FHSS communications device that then calculates the series
of frequencies to use during the communication session. Many times, this
period of frequency change is pre-defined so that a single base station can
service a number of communication connections.
Step 4 – The calling/initiating party then sends a synchronization signal on the
first frequency in the calculated sequence.
Step 5 – The communication session between the two parties commences and
each party shifts frequencies in sync with the other.
7. Frequency Hopping Sequences and Hitting Probability
Frequency hopping sequence is a sequence of randomly chosen numbers from a
given set & indicates the order of the channel the user will use at each time slot. The
choice of each sequence to be used by the user is a random choice of a given family of
sequences while the generation is a Pseudo Random process which is implemented
mostly using hardware digital circuit consists of several components; mainly linear
feedback shift registers. For two sequences of length n we need to minimize that the
two users are using the same channel at the same time slot which is called the Hit
Probability.
There are several ways to generate families of sequences with minimal hit
probability, we will discuss the approach delivered by this paper; "Families of
sequences with optimal Hamming correlation properties" published 1974, then we
will state other approaches.
We will discuss briefly the principle concept used in generating the sequence:
a. Hamming Correlation: A method to compare sequences with each other. It is
divided into two ways :
-6-
Auto Correlation: The sequence is compared to itself but with
time shift.
Cross Correlation: Two different sequences are compared
together.
b. M - Sequence :
It was found mathematically that its range is a Prime numbers of channels, i.e.
{0, 1... P-1}. Also can be implemented using linear feedback registers.
For every prime number and positive integer n, there exist an M-sequence of
length q = Pn-1; n is a parameter.
Now we are going to talk about the proposed approach to develop optimal
families of sequences:
Consider P = { 0, 1, ......., P-1 }
PK {0, 1... Pk-1} where P
K: Set of all word of length K (Transformation
parameter) over P.
It can be proved that if X is an M-sequence then for each K, the σK Transform
of X is an Optimal sequence,
H(Y) = Pn-k
-1, where H(Y) is the Auto Correlation.
Using one of M-Sequence properties we can obtain an optimal family F where
M ( Yr , Ys ) = Pn-K
( Cross Correlation ) for every pair of distinct members of
F considering the case of P = 2 , n=K+1
H(Y) = 2 - 1 = 1
M (Yr, Ys) = 2
So the frequency hopping sequence has a maximum of two hits with any other
sequence.
In 2004, Ryoh Fuji Hara, Ying Miao and Miwako Mishima proposed a paper
in which they stated 13 new approaches.
We can conclude that for any frequency hopping sequence family there are
three parameters:
v: Length of the sequence.
m: Number of channels.
λ: Maximum number of hits of two users having same sequence
H(Y).
It is obvious that the probability that two users having the same sequence hit at
given time slot is (λ/v).
And the probability that two users having the same sequence is (1/m).
If the two users have the same sequence Phit = λ/vm, then the probability that
two users not having the same sequence hit equals (λ+1)/v.
The probability that they don't have the same sequence equals (m-1)/m.
Therefore,
Total Phit = λ/vm + [λ+1/v]*[(m-1)/m]
-7-
And the number of users interfering a given user: x, can be modeled as a
binomial random variable, as each user independently interfere the given user
or not. Then,
Phit = 𝑈−1𝑥
phitx
(1- phit )U-x-1
, phit is the PDF of x and U is the number of active
users.
8. Frequency Hopping Code Division Multiple Access (FH-CDMA)
Code Division Multiple Access (CDMA) is a channel transmission standard that
allows several transmitters to send information simultaneously over a single
communication channel. This ensures that the bandwidth is used in a perfect way and
in an optimized manner.
CDMA is popular because it provides privacy, protection against multi-path
interference, anti-jamming capabilities, and a low probability of interception (LPI). In
CDMA each sender is assigned a unique code sequence that encodes the information-
bearing signal. The receiver also knows the code sequence of the sender and decodes
the received signal upon reception, recovering the original data. FH-CDMA technique
is a relatively less widely used CDMA scheme in real applications. The reason for its
less wide acceptance is owing to several factors. First, the FH technique requires a
very accurate reference clock in the whole wireless system which uses the FH-CDMA
technique for user separation. This accurate network-wide reference clock is very
costly to implement using currently available digital technology.
In an FH-CDMA system, a transmitter "hops" between available frequencies
according to a specified algorithm, which can be either random or preplanned. The
transmitter operates in synchronization with a receiver, which remains tuned to the
same center frequency as the transmitter. A short burst of data is transmitted on a
narrowband. Then, the transmitter tunes to another frequency and transmits again. The
receiver thus is capable of hopping its frequency over a given bandwidth several times
a second, transmitting on one frequency for a certain period of time, then hopping to
another frequency and transmitting again. Frequency hopping requires a much wider
bandwidth than is needed to transmit the same information using only one carrier
frequency.
The length of time the transmitted carrier is unchanged is called the dwell time
(TD). After this time has elapsed the transmitted carrier may change, i.e. hop, to
another carrier. There is a sequence of frequency hops that is given to a user. This
sequence is a frequency hopping code, and in general this code sequence continuously
repeats while a user is transmitting data. Figure 8.1 shows a code sequence for the
situation of eight carrier frequencies. Each shaded block represents a data modulated
signal positioned at a specific frequency. Suppose the modulated data are binary
frequency shift keying (BFSK) where a logical 1 is represented by frequency fA and a
logical 0 by a frequency fB. If the duration of a data bit is Tb, then the BFSK output is
either fA or fB and lasts for Tb seconds. Should TD < Tb, then it follows that for one bit
there will be a number of frequency hops as shown in figure 8.2. This is the case of
FFH-CDMA On the other hand, if TD ≥ Tb, each hop may last for a packet of data
and this is referred to as (SFH-CDMA) shown in figure 8.3 . The military often use
FFH-CDMA, where the hopping rate may be very fast making it difficult to
effectively jam the transmitted signal. However, the technology is complex and
expensive for the commercial market where SFH-CDMA is preferred.
-8-
Fig. 8.1 Fig. 8.2 Fast FHSS
Fig. 8.3 Slow FHSS
9. Plotting the distribution of hitting
Fig 9.1 v=31,
lambda=2,m=16,number of active
users=100
Fig 9.2 v=63,
lambda=2,m=32,number of active
users=100
-9-
It is clear from the above figure that as the number of channels gets greater and
the length of sequence gets longer, the expected value for the number of users
interfering a given user decreases, which is good.
Appendix A. MatLab code for the plot
n=2; while (n<10) v=(2^n)-1;%length of sequence l=2;%auto correlation m=2^(n-1);%number of channels p=(l/(v*m))+(((1+1)/v)*((m-1)/m));%hit probability x=1:1:99;%number of intrfering users y=binopdf(x,99,p);%binomial distribution of x plot(x,y,'+') title(['v=' num2str(v) ',lambda=' num2str(l) ',m=' num2str(m)
',number of active users=100'])
n=n+1;%next iteration end
Appendix B. MatLab simulation for sequence generation
We decided to make a simulation by generating two frequency hopping
sequences of the same family, and calculate the number of hits between a sequence
and itself, and between two different sequences, to verify the relation for maximum
hit probability derived in a previous section.
Our problem was that we had not deal with generating m-sequences before.
Fortunately there is an open source function to generate an m-sequence within the
ranges in the following table.
p n
2 2,3,4,5,6,7,8,9,10
3 2,3,4,5,6,7
5 2,3,4
Table B.1
We took that m-sequence and made the transformation twice on it, and then we
calculated the number of hits between a sequence and itself, and between the two
sequences.
Here is the code for that:
function [h M]=fh_sequence(p,n,k)
ms=transpose(mseq(p,n));
for j=p^n:(p^n)+k-1 ms(j)=ms(j-p^n+1); end
-10-
for i=1:(p^n)-1 fhs(i) = 0; for j=1:k o=i+j-1; fhs(i) = fhs(i)+ms(o)*(p^(j-1)); end end
for i=1:(p^n)-1 if i<(p^n)-1 fhs2(i+1)=fhs(i); else fhs2(1)=fhs(i); end end
h=0; for i=1:(p^n)-1 if fhs2(i)==fhs(i) h=h+1; end end
for j=1:(p^n)-1 ms2(j)=ms(j); end
ms2=circshift(ms2,[0 60]);
for j=p^n:(p^n)+k-1 ms2(j)=ms2(j-p^n+1); end
for i=1:(p^n)-1 fhs3(i) = 0; for j=1:k o=i+j-1; fhs3(i) = fhs2(i)+ms2(o)*(p^(j-1)); end end
M=0; for i=1:(p^n)-1 if fhs3(i)==fhs(i) M=M+1; end end
And here are some sample runs in the next page.
-11-
Fig B.1 p=2, n=3, k=2, U=100
Fig B.2 p=5, n=4, k=2, U=150
It appears that in both runs the output values are within the limits derived before.
Appendix C. Adaptive frequency hopping for unlicensed bands
Some technologies operate in unlicensed bands, just as Bluetooth, and wireless
LANS, both work at the band 2.4≈2.485 GHz, with a channel increment of 1 MHz
Therefore, a concern arises. How can we deal with the interference between them?
Nevertheless, how can we deal with interference among several Bluetooth networks
for example?
To solve this problem, a concept arisen called Adaptive Frequency Hopping.
-12-
Adaptive frequency hopping has two main types:
a) Adaptive frequency rolling:
The band is divided by the network in interest into sub-bands, the
transmissions of the whole network rolls between these sub-bands, while the
users hop inside the band. If the network detects an error while in a certain
sub-band, it randomly jumps to another sub-band.
b) Dynamic adaptive frequency hopping:
The network initially uses the whole band. If it detects and error, the band is
divided into two sub-band and the network rolls between them. If an error is
detected again, the band is divided into four sub-bands… Etc.
Appendix D. References
1. Introduction and definitions of spread spectrum and its types:
Tutorials from.
www.searchnetworking.com
www.sss-mag.com
2. FH-sequences and hit probability:
Families of sequences with optimal hamming correlation properties,
Abraham Lempel and Haim Greenberger, IEEE transactions on
information theory, January 1974.
Optimal frequency hopping sequences:
a combinatorial approach, Ryoh Fuji-Hara, Ying Miao and Miwako
Mishina, IEEE transactions on information theory, June 2004.
3. CDMA:
GSM, cdmaOne and 3G Systems, Raymond Steele , Chin-Chun Lee and
Peter Gould, publisher: john wiley and sons, 2001
An Overview of CDMA Techniques for Mobile Communications, M.F.L.
Abdullah and Mayada Faris Ghanim, Journal of Mobile
Communicat ion, 2011 .
The next generation CDMA technologies, Hsiao-Hwa Chen, publisher:
Wiley.
Multi-Rate FH-CDMA wireless systems, a lecture by Guu Chang Yung,
Department of Electrical Engineering, National Chung-Hsing University,
Taiwan.
4. MatLab:
Tutorials by Eng. Wael El Sharkasy, IEEE Alexandria Student Branch,
Summer 2009.
http://www.mathworks.com/matlabcentral