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Master of Computer Science 1 - MOB Mobile Internet and Surrounding

MOB sujet 4

Transmission techniques - DS CDMA

1. CDMA Exercises

The principle of the spread spectrum modulation consists in coding the information by the way

of a bandwidth signal larger than the band strictly necessary, with a goal of fighting the interfering

signals and the distortions tied with the propagation.

The signal is coded at the beginning: a code is assigned with each user in order to allow

decoding in the arrival. The spread is ensured by a pseudo-random signal called spread code. At

the reception, the signal is seen as the noise if the receiver does not have the code.

The spread spectrum modulation is optimized to fight against the noise, by which it limit better

the effects.

1.0.1 Mathematical reminds

- Correlation measure the manner whose two signals vary simultaneously. A strong

correlation implies a great resemblance between the two compared signals.

- Auto-correction quantifies the correlation of a signal with a version of itself shifted in

time, according to this temporal shift.

- Cross-correlation quantifies the correlation of two shifted signals the one in comparing

with the other in time, according to this temporal shift.

- Orthogonality: two periodic signals from period T are known as orthogonal if their

cross-correlation is null for a null temporal shift.

CDMA modulation (Code Division Multiple Access) is a spread spectrum modulation. It

allows transmitting two signals simultaneously in the same band, their separation being realized at

the reception in getting profit of their orthogonality. Several mobile telephony systems (IS95,

UMTS…) have chooses this technique. The CDMA is then also used in order to manage the

multi-users aspect. Among the spread spectrum methods are:

- Frequency Hopping - CDMA (FH-CDMA), seemed to be the most interesting technique

at the beginning, but its difficult implementation took the interest back on spread spectrum

by directed sequence;

- Direct Sequence - CDMA (DS-CDMA), is used in particular in the 3G European UMTS

system (Mobile Universal Telephone System).

The principle of the DSCDMA is as follows. On each transmitting station, is allocated a

single code of m chips called "chip sequence" constituted from "-1" and from "+1". To

transmit a bit to "1", the station sends its sequence chip. To transmit a bit to “0”, the station

sends the opposite of its sequence chip. The duration Tc of transmission of a chip is m time

lower than the duration Tb of a bit.

Example: let us suppose that the chip sequence of A is "-1-11-1111-1 " (m = 8). To transmit

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a binary "1", A sends "-1-11-1111-1"; to transmit a binary "0", A sends "11-11-1-1-11"

Power spectral density (PSD) of a signal corresponds to the power distribution of the signal

according to the frequency. For a code of forming, the PSD is proportional to the square of

the Fourier transform of this forming. The forming is a form of wave given starting from

which one announces a "0" or a "1". If this is about one rectangular function of width T, the

Fourier transform of such a function is a cardinal sine whose first cancellation is of f = 1/T.

One thus finds a PSD which has the form of the square of a cardinal sine:

A signal DS-CDMA corresponds to a succession of rectangles of width Tc, the duration of

a chip. Its PSD thus has the form of the square of a cardinal sine in the first cancellation by

f = 1/Tc.

By opposition, a digital signal is constituted by a succession of rectangles of width Tb, the

bit duration, has a PSD of the form of the square of a cardinal sine with the first

cancellation of f = 1/Tb.

The chip sequence consists of m chips. Each bit of the digital signal in starting is replaced

1 m

by the chip code so that Tb = mTc. And thus,

TC T b

The principal lobe of DS-CDMA signal is thus m time broader than the principal lobe of

the starting signal. It is said that the signal was spread with a spread factor of m. Two

signals have the same total power, but power of the starting signal is more concentrated in

the band, where the appellation of “signal with narrow-band ".

1.0.2 Exercises

1. In your opinion, what are the advantages and disadvantages in the use of the CDMA for the

cellular networks?

Advantages: The spread spectrum modulation is optimized to fight against the noise, by

which it limit better the effects.

Disadvantages: In CDMA, any speaker can talk at any time; however each uses a different

language. Each listener can only understand the language of their partner. As more and

more couples talk, the background noise (representing the noise floor) gets louder, but

because of the difference in languages, conversations do not mix.

2. If one wishes to send information of b bits/s with a key of m chips, what will be the flow in

chips per second?

The chip sequence is constituted of m chips. Each bit digital signal of departure is replaced

1 m

by the code chip so that Tb = mTc. So, f c mfb .

Tc Tb

According to Shannon’s theorem, we deduce: Cc = mCb = mb chips/s.

3. Which factor will the frequency band occupied by the signal be increased with?

The data bit rate and noise power ratio.

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The main lobe of DS-CDMA signal is m times larger than the main lobe of the signal of

departure. The frequency band occupied is increased by a factor m. They say that the signal

was unfurled with a spreading factor of m. Both signals have the same overall power, but

the signal of departure is more concentrated in the band, hence the name "narrow band

signal”.

4. Two keys are orthogonal if the standardized internal product of these both keys is null.

Remind: internal product of [a, b] [a’, b’] = a a’ + b b’.

__

Show that if two keys S and T are orthogonal, S and T are too.

Suppose S = [s1, s2,..,sm], T = [t1, t2,…,tm]

__

Thus T = [t1, t2,…, tm]

We have S T = (s1 t1+ s2 t2 + … + sm tm)/m = 0

__

So S T = (s1 t1 + s2 t2 + … + sm tm)/m = (s1 t1+ s2 t2 + … + sm tm)/m = 0

__

e.g S and T are orthogonal.

5. One considers 4 transmitters having each one a key:

A: (-1 -1 -1 +1 +1 -1 +1 +1)

B: (-1 -1 +1 -1 +1 +1 +1 -1)

C: (-1 +1 -1 +1 +1 +1 -1 -1)

D: (-1 +1 -1 -1 -1 -1 +1 -1)

a) Show that B, C and D are orthogonal at A.

A B = [(-1) (-1) + (-1) (-1) + (-1) (+1) + (+1) (-1) + (+1) (+1) + (-1) (+1) + (+1)

(+1) + (-1) (-1)]/8 = (1 + 1 – 1 – 1 + 1 – 1 + 1 – 1)/8 = 0

b) A code has a good autocorrelation if its product intern with itself is high and its internal

product with itself is shifted of 1 or several chips are weak. That means that the receiver

will be able to recognize easily the emitted signal. Show that the key of Barker

(+1,-1,+1,+1, -1, +1,+1,+1, -1,-1,-1) is a good autocorrelation.

We have: Bk Bk = 1

The value is even more important that the key is long, but it takes to get complete the

concept of autocorrelation view the product's internal key with it even when it is offset by 1

chip, 2 chips, ...

Suppose Bk’ is Bk shifted of 1 chip to the left, thus:

Bk’ = (-1,+1,+1, -1, +1,+1,+1, -1,-1,-1,+1).

We have: Bk Bk’ = [1 x (-1) + (-1) x 1 + 1 x 1 + 1 x (-1) + (-1) x 1 + 1 x 1 + 1 x 1 + 1 x

(-1) + (-1) x (-1) + (-1) x (-1) + (-1) x 1]/11 = [(-1) + (-1) + 1 + (-1) + (-1) + 1 + 1 + (-1) + 1

+ 1 + 1]/11 = –1/11

The product of the key Barker (+1 -1 +1 +1 -1 +1 +1 +1 -1 -1 -1) with the same key but

shifted by two chips to the left, Bk” (+1 +1 -1 +1 +1 +1 -1 -1 -1 +1 -1) is:

Bk Bk” = (1 – 1 – 1 + 1 – 1 + 1 – 1 – 1 + 1 – 1 + 1)/11 = –1/11

We deduce that this key and the same key shifted by one chip or two chips are nearly

orthogonal. The higher the value of domestic product, the lower the receiver can easily

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interpret the result (signal codes with keys "pseudo-orthogonal").

c) When a station wants to transmit a binary "1", it sends its key and when it wants to

transmit a binary "0", it sends the opposite of this key. If only C wishes to send a bit of

"1", what the receiver will receive and will make to interpret the answer?

If only C wishes to send a bit of "1", C send chip sequence S = (-1 +1 -1 +1 +1 +1 -1 -1).

The receiver will calculate, it had: S A = 0 and S B = 0 and S D = 0 and S C = 1, so

receiver know C was sent bit "1".

d) If only C wishes to send a bit of "0", what the receiver will receive and will make to

interpret the answer?

If C send a binary "0", C send chip sequence S = (+1 -1 +1 -1 -1 -1 +1 +1).

The receiver will calculate, it had: S A = 0 and S B = 0 and S D = 0 and S C = –1, so

receiver know C was sent bit "0".

e) The signals are added linearly, the transmitters are synchronous and the signals are

received with the same power by the receiver. If one supposes that the stations A, B

and C decide to send at the same time the bit "0", what will be the sequence read by

receiver?

The stations A, B and C decide to send at the same time the bit "0", chip sequence S at the

receiver:

S = (+3 +1 +1 -1 -3 -1 -1 +1)

(Remember we have: A = (+1 +1 +1 -1 -1 +1 -1 -1) and B = (+1 +1 -1 +1 -1 -1 -1 +1) and

C = (+1 -1 +1 -1 -1 -1 +1 +1) )

f) If the receiver receives the sequence “-1 +1 -3 +1 -1 -3 +1 +1”, what does it deduce

from?

Suppose S = (-1 +1 -3 +1 -1 -3 +1 +1)

We have:

S A = (-1 +1 -3 +1 -1 -3 +1 +1) (-1 -1 -1 +1 +1 -1 +1 +1) = (1 - 1 + 3 + 1 – 1 + 3 + 1 +

1)/8 = 1, so A sent bit “1”.

S B = (-1 +1 -3 +1 -1 -3 +1 +1) (-1 -1 +1 -1 +1 +1 +1 -1) = (1 – 1 – 3 – 1 – 1 – 3 + 1 –

1)/8 = -1, so B sent bit “0”.

S C = (-1 +1 -3 +1 -1 -3 +1 +1) (-1 +1 -1 +1 +1 +1 -1 -1) = (1 + 1 + 3 +1 – 1 – 3 – 1 – 1)/8

= 0, so C has not sent anything.

S D = (-1 +1 -3 +1 -1 -3 +1 +1) (-1 +1 -1 -1 -1 -1 +1 -1) = (1 + 1 + 3 – 1 + 1 + 3 + 1 – 1)/8

= 1, so D sent bit “1”.

6. Suppose that one wants to transmit a data flow of 56 kbit/s by using a spread spectrum

technique.

a) Find the band-width of the necessary channel when the signal-with-noise report is 0,1;

0,01 and 0,001.

Case 1: SNR = 0,1

C = B log2(1 + SNR) B = C / log2(1 + SNR) = 56 103 / log2(1 + 0,1) = 407.262Hz

407,2KHz

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Case 2: SNR = 0,01

C = B log2(1 + SNR) B = C / log2(1 + SNR) = 56 103 / log2(1 + 0,01) = 3.901.000Hz

3,9 MHz

Case 3: SNR = 0,001

C = B log2(1 + SNR) B = C / log2(1 + SNR) = 56 103 / log2(1 + 0,001) = 38.835.647Hz

38,835 MHz

b) In a ordinary system (without spread spectrum)

In a regular system (without spread-spectrum), so that R = B

Shannon's formula C = Blog2 (1 + S / N) applied with C = B = R gives

log2 (1 + S / N) = 1. This implies that S / N = 1. Thus, spread-spectrum without a S / N

much greater is required.

7. Prove that a receiving station can get the data sent by a specific sender if it multiplies the

entire data on the channel by the sender’s chip code and then divides it by the number of

stations.

Let us prove this for the first station, using our previous four-station example. We can say that

the data on the channel:

D = (d1 c1 + d2 c2 + d3 c3 + d4 c4)

The receiver which wants to get the data sent by station 1 multiplies these data by c1.

D c1 = (d1 c1 + d2 c2 + d3 c3 + d4 c4) c1

= d1 c1 c1 + d2 c2 c1 + d3 c3 c1 + d4 c4 c1

= d1 N + d2 0 + d3 0 + d4 0

d1 = (D c1 ) / N

2. Practical - Spread spectrum techniques

The goal of this practical is to understand the impact of spread spectrum techniques on the

emitted signal and on the received signal by the recipient. You will be able with this practical:

– to visualize and study impact of the code used and of the modulation type on the spectrum

of a signal;

– to observe the properties of two signals according to the used keys (DS-CDMA);

– to visualize quality of reception of a signal 1, while varying the signal with noise report

(SNR) as well as the ratio (C/I) of the power of the signal has to interpret (Carrier: signal 1)

by compared with a second signal (Interferor: signal 2) which will come to disturb the

reception of the first;

– to trace the curves of Bit Error Rate (BER), according to coding for a SNR or a fixed C/I.

For that you will use the simulation tool developed for this effect. Copy the script tme_cdma in

your folder:

Carry out the script with the help of Octave, the digital calculation, visualization and programming

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software similar to Matlab. Appear the following menu:

Choose an option

[1] signal key 1 (useful)

[2] signal key 2 (interfering)

[3] display the spectrum of the useful signal (red) and the interfering signal

(green)

[4] choose the modulation type

[5] trace the autocorrelation of the key 1 and the cross-correlation between

two keys

[6] trace the constellation diagram in reception of signal 1 (useful) for a SNR

and a fixed C/I

[7] calculate the bit error rate in reception for a fixed SNR

[8] calculate the bit error rate in reception for a fixed C/I

[9] quit

Menu description

[1]: allows visualizing or modifying the spread spectrum key of the "signal 1". The key by defect is

the key of Barker used in Wifi?

[2]: allows visualizing or modifying the spread spectrum key of the "signal 2". The key by defect

is a sequence of "1".

[3]: trace the curves of the spectral density of signals 1 and 2 after modulation with and without

filtering. The two traced spectra, in red for signal 1 and in green for signal 2 have both the

same power. This spectrum is obtained for each signal by getting the same continuation

random of 10 000 bits. Each signal is coded with its own key which can be visualized and

modified while typing [1] or [2] in the menu.

[4]: allows choosing the modulation type which will be applied to signal 1. You have the choice

between: PSK-2 (BPSK), QAM-4 (QPSK), QAM-16 and QAM-64.

[5]: allows visualizing the curve of the autocorrelation of the signal 1 key and cross-correlation

between the keys of signals 1 and 2.

[6]: allows visualizing in the form of points in a constellation diagram, the value of the different

symbols received by the receiver. One can thus see the amplitude and the phase of these

symbols. The configuration of the receiver is adapted according to the modulation type of

signal 1, of the CDMA keys of the two signals, and the values of ratios SNR and C/I.

[7]: allows tracing curves of BER according to report C/I for a fixed SNR?

[8]: allows tracing curves of BER according to report SNR for a fixed C/I?

3. Properties and spectral of the emitted signal

The principle of the DS-CDMA is as follows. At each transmitting station is allocated a

single code of m chips called "sequence chip " constituted from "1 " and from "+1". From a

code constituted from "0" and from "1" like the code of Barker, it is advisable to replace the

"0" by "1" to obtain the sequence chip. To transmit a bit to "1", the station sends the

opposite of its sequence chip. To transmit a bit to "0", the station sends the opposite of its

sequence chip. The duration Tc of transmission of a chip is m time lower than the duration

Tb of a bit.

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1. What is the key value by defect of the signal 2? What does it mean?

For the first part of the tests, we keep the key value of the signal 1 on

"10110111000" and the key of the signal 2 on "1". There is thus nothing to modify.

2. (a) Choose a modulation type by using the menu [4] and display then the spectral

density of power of signal 1 and signal 2 by using the menu [3]. What do you

observe by comparing the form of the primary and secondary lobes of signals 1 and

2 without filtered for the selected modulation? These observations are they still

valid for the other modulations?

(b) What is the power reduction between the primary and secondary lobe for the

two curves?

(c) What is the effect of the filtering on the frequential spectrum?

(d) What is the interest of the filtering?

4. Properties and correlation

The autocorrelation is studied in considering the profile of the autocorrelation function

obtained by shifting the digital sequence x[k] (of length N completed beforehand by zeros) of a

gradually increasing value n and by comparing the initial sequence and the shifted sequence.

The formula used here to estimate the autocorrelation function is written:

The correlation coefficient obtained will be larger as much as the digital sequence and its

shifted version are similar. Conversely, if the sequences are opposed, the correlation will be

negative. A null autocorrelation indicates that there is not correlation between the initial

sequence and the shifted sequence for the considered shift.

The study of the autocorrelation allows us to evaluate the capacity of the receiver has to

synchronize with the flow of chips of the transmitter. In effect, the receiver makes a correlation

between the received signal and the used key in the emission. A peak of correlation must be

observed when on the duration of a bit, the key used for encoder a bit and the key used by the

receiver are synchronous. In the other hand, the value of γx[n] must be near to zero for a shift not

null representing a desynchronization of the receiver. Thus, a good key must have such

properties. It will be said that it has good properties of autocorrelation.

1. A good autocorrelation must present a maximum for a null shift and must be near to zero

elsewhere. Which interest is presented in the case where the channel is the subject of

multi-traffic?

2. Observe the autocorrelation of the Barker key of the IEEE standard 802.11

"10110111000" by using the menu [5]. It appears good to you?

3. Which property appears important to you to choose the keys of two distinct transmitters?

By basing you on that which was known as the autocorrelation properties, what appears

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important to you for the cross-correlation?

4. Give the mathematic expression of an estimator of the cross-correlation of two

sequences x[k] and y[k].

5. Are the keys of signals 1 and 2 orthogonal?

6. Modify the keys of two transmitters and observe their autocorrelation and their

cross-correlation.

5. Reception and interpretation of the signal 1

The menu [6] makes it possible to visualize the value of the amplitude and of the phase of

different symbols received by the receiver. The simulator allows observing the impact of the

modulation; of interferences linked to the additive noise (SNR) and to the disturbance caused by

another signal (signal 2) synchronous with the first.

1. What does it mean SNR or ratio C/I positive, null and negative?

2. Reposition the key value of the signal 1 to “10110111000" (Barker key) and that of

signal 2 to “1”, i.e. to the default values. Using the menus [4] and [6], vary the SNR, the

C/I and the modulation type and fill out table 1. What do you observe? what does it

occur when ratio C/I tends towards 0?

Tab. 1 – Synthesis of observations on the constellation diagrams.

Modulation SNR C/I Quality of the reception of signal

(Good/ Medium /Bad)

BPSK 100 15

QPSK 100 15

16QAM 100 15

64QAM 100 15

BPSK 15 100

QPSK 15 100

16QAM 15 100

64QAM 15 100

BPSK 15 0

QPSK 15 0

16QAM 15 0

64QAM 15 0

BPSK 15 3

QPSK 15 3

16QAM 15 3

64QAM 15 3

3. Position the key value of the signal 1 to "10110111000" (Barker key) and that of signal 2

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MOB sujet 4

Transmission techniques - DS CDMA

1. CDMA Exercises

The principle of the spread spectrum modulation consists in coding the information by the way

of a bandwidth signal larger than the band strictly necessary, with a goal of fighting the interfering

signals and the distortions tied with the propagation.

The signal is coded at the beginning: a code is assigned with each user in order to allow

decoding in the arrival. The spread is ensured by a pseudo-random signal called spread code. At

the reception, the signal is seen as the noise if the receiver does not have the code.

The spread spectrum modulation is optimized to fight against the noise, by which it limit better

the effects.

1.0.1 Mathematical reminds

- Correlation measure the manner whose two signals vary simultaneously. A strong

correlation implies a great resemblance between the two compared signals.

- Auto-correction quantifies the correlation of a signal with a version of itself shifted in

time, according to this temporal shift.

- Cross-correlation quantifies the correlation of two shifted signals the one in comparing

with the other in time, according to this temporal shift.

- Orthogonality: two periodic signals from period T are known as orthogonal if their

cross-correlation is null for a null temporal shift.

CDMA modulation (Code Division Multiple Access) is a spread spectrum modulation. It

allows transmitting two signals simultaneously in the same band, their separation being realized at

the reception in getting profit of their orthogonality. Several mobile telephony systems (IS95,

UMTS…) have chooses this technique. The CDMA is then also used in order to manage the

multi-users aspect. Among the spread spectrum methods are:

- Frequency Hopping - CDMA (FH-CDMA), seemed to be the most interesting technique

at the beginning, but its difficult implementation took the interest back on spread spectrum

by directed sequence;

- Direct Sequence - CDMA (DS-CDMA), is used in particular in the 3G European UMTS

system (Mobile Universal Telephone System).

The principle of the DSCDMA is as follows. On each transmitting station, is allocated a

single code of m chips called "chip sequence" constituted from "-1" and from "+1". To

transmit a bit to "1", the station sends its sequence chip. To transmit a bit to “0”, the station

sends the opposite of its sequence chip. The duration Tc of transmission of a chip is m time

lower than the duration Tb of a bit.

Example: let us suppose that the chip sequence of A is "-1-11-1111-1 " (m = 8). To transmit

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a binary "1", A sends "-1-11-1111-1"; to transmit a binary "0", A sends "11-11-1-1-11"

Power spectral density (PSD) of a signal corresponds to the power distribution of the signal

according to the frequency. For a code of forming, the PSD is proportional to the square of

the Fourier transform of this forming. The forming is a form of wave given starting from

which one announces a "0" or a "1". If this is about one rectangular function of width T, the

Fourier transform of such a function is a cardinal sine whose first cancellation is of f = 1/T.

One thus finds a PSD which has the form of the square of a cardinal sine:

A signal DS-CDMA corresponds to a succession of rectangles of width Tc, the duration of

a chip. Its PSD thus has the form of the square of a cardinal sine in the first cancellation by

f = 1/Tc.

By opposition, a digital signal is constituted by a succession of rectangles of width Tb, the

bit duration, has a PSD of the form of the square of a cardinal sine with the first

cancellation of f = 1/Tb.

The chip sequence consists of m chips. Each bit of the digital signal in starting is replaced

1 m

by the chip code so that Tb = mTc. And thus,

TC T b

The principal lobe of DS-CDMA signal is thus m time broader than the principal lobe of

the starting signal. It is said that the signal was spread with a spread factor of m. Two

signals have the same total power, but power of the starting signal is more concentrated in

the band, where the appellation of “signal with narrow-band ".

1.0.2 Exercises

1. In your opinion, what are the advantages and disadvantages in the use of the CDMA for the

cellular networks?

Advantages: The spread spectrum modulation is optimized to fight against the noise, by

which it limit better the effects.

Disadvantages: In CDMA, any speaker can talk at any time; however each uses a different

language. Each listener can only understand the language of their partner. As more and

more couples talk, the background noise (representing the noise floor) gets louder, but

because of the difference in languages, conversations do not mix.

2. If one wishes to send information of b bits/s with a key of m chips, what will be the flow in

chips per second?

The chip sequence is constituted of m chips. Each bit digital signal of departure is replaced

1 m

by the code chip so that Tb = mTc. So, f c mfb .

Tc Tb

According to Shannon’s theorem, we deduce: Cc = mCb = mb chips/s.

3. Which factor will the frequency band occupied by the signal be increased with?

The data bit rate and noise power ratio.

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The main lobe of DS-CDMA signal is m times larger than the main lobe of the signal of

departure. The frequency band occupied is increased by a factor m. They say that the signal

was unfurled with a spreading factor of m. Both signals have the same overall power, but

the signal of departure is more concentrated in the band, hence the name "narrow band

signal”.

4. Two keys are orthogonal if the standardized internal product of these both keys is null.

Remind: internal product of [a, b] [a’, b’] = a a’ + b b’.

__

Show that if two keys S and T are orthogonal, S and T are too.

Suppose S = [s1, s2,..,sm], T = [t1, t2,…,tm]

__

Thus T = [t1, t2,…, tm]

We have S T = (s1 t1+ s2 t2 + … + sm tm)/m = 0

__

So S T = (s1 t1 + s2 t2 + … + sm tm)/m = (s1 t1+ s2 t2 + … + sm tm)/m = 0

__

e.g S and T are orthogonal.

5. One considers 4 transmitters having each one a key:

A: (-1 -1 -1 +1 +1 -1 +1 +1)

B: (-1 -1 +1 -1 +1 +1 +1 -1)

C: (-1 +1 -1 +1 +1 +1 -1 -1)

D: (-1 +1 -1 -1 -1 -1 +1 -1)

a) Show that B, C and D are orthogonal at A.

A B = [(-1) (-1) + (-1) (-1) + (-1) (+1) + (+1) (-1) + (+1) (+1) + (-1) (+1) + (+1)

(+1) + (-1) (-1)]/8 = (1 + 1 – 1 – 1 + 1 – 1 + 1 – 1)/8 = 0

b) A code has a good autocorrelation if its product intern with itself is high and its internal

product with itself is shifted of 1 or several chips are weak. That means that the receiver

will be able to recognize easily the emitted signal. Show that the key of Barker

(+1,-1,+1,+1, -1, +1,+1,+1, -1,-1,-1) is a good autocorrelation.

We have: Bk Bk = 1

The value is even more important that the key is long, but it takes to get complete the

concept of autocorrelation view the product's internal key with it even when it is offset by 1

chip, 2 chips, ...

Suppose Bk’ is Bk shifted of 1 chip to the left, thus:

Bk’ = (-1,+1,+1, -1, +1,+1,+1, -1,-1,-1,+1).

We have: Bk Bk’ = [1 x (-1) + (-1) x 1 + 1 x 1 + 1 x (-1) + (-1) x 1 + 1 x 1 + 1 x 1 + 1 x

(-1) + (-1) x (-1) + (-1) x (-1) + (-1) x 1]/11 = [(-1) + (-1) + 1 + (-1) + (-1) + 1 + 1 + (-1) + 1

+ 1 + 1]/11 = –1/11

The product of the key Barker (+1 -1 +1 +1 -1 +1 +1 +1 -1 -1 -1) with the same key but

shifted by two chips to the left, Bk” (+1 +1 -1 +1 +1 +1 -1 -1 -1 +1 -1) is:

Bk Bk” = (1 – 1 – 1 + 1 – 1 + 1 – 1 – 1 + 1 – 1 + 1)/11 = –1/11

We deduce that this key and the same key shifted by one chip or two chips are nearly

orthogonal. The higher the value of domestic product, the lower the receiver can easily

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interpret the result (signal codes with keys "pseudo-orthogonal").

c) When a station wants to transmit a binary "1", it sends its key and when it wants to

transmit a binary "0", it sends the opposite of this key. If only C wishes to send a bit of

"1", what the receiver will receive and will make to interpret the answer?

If only C wishes to send a bit of "1", C send chip sequence S = (-1 +1 -1 +1 +1 +1 -1 -1).

The receiver will calculate, it had: S A = 0 and S B = 0 and S D = 0 and S C = 1, so

receiver know C was sent bit "1".

d) If only C wishes to send a bit of "0", what the receiver will receive and will make to

interpret the answer?

If C send a binary "0", C send chip sequence S = (+1 -1 +1 -1 -1 -1 +1 +1).

The receiver will calculate, it had: S A = 0 and S B = 0 and S D = 0 and S C = –1, so

receiver know C was sent bit "0".

e) The signals are added linearly, the transmitters are synchronous and the signals are

received with the same power by the receiver. If one supposes that the stations A, B

and C decide to send at the same time the bit "0", what will be the sequence read by

receiver?

The stations A, B and C decide to send at the same time the bit "0", chip sequence S at the

receiver:

S = (+3 +1 +1 -1 -3 -1 -1 +1)

(Remember we have: A = (+1 +1 +1 -1 -1 +1 -1 -1) and B = (+1 +1 -1 +1 -1 -1 -1 +1) and

C = (+1 -1 +1 -1 -1 -1 +1 +1) )

f) If the receiver receives the sequence “-1 +1 -3 +1 -1 -3 +1 +1”, what does it deduce

from?

Suppose S = (-1 +1 -3 +1 -1 -3 +1 +1)

We have:

S A = (-1 +1 -3 +1 -1 -3 +1 +1) (-1 -1 -1 +1 +1 -1 +1 +1) = (1 - 1 + 3 + 1 – 1 + 3 + 1 +

1)/8 = 1, so A sent bit “1”.

S B = (-1 +1 -3 +1 -1 -3 +1 +1) (-1 -1 +1 -1 +1 +1 +1 -1) = (1 – 1 – 3 – 1 – 1 – 3 + 1 –

1)/8 = -1, so B sent bit “0”.

S C = (-1 +1 -3 +1 -1 -3 +1 +1) (-1 +1 -1 +1 +1 +1 -1 -1) = (1 + 1 + 3 +1 – 1 – 3 – 1 – 1)/8

= 0, so C has not sent anything.

S D = (-1 +1 -3 +1 -1 -3 +1 +1) (-1 +1 -1 -1 -1 -1 +1 -1) = (1 + 1 + 3 – 1 + 1 + 3 + 1 – 1)/8

= 1, so D sent bit “1”.

6. Suppose that one wants to transmit a data flow of 56 kbit/s by using a spread spectrum

technique.

a) Find the band-width of the necessary channel when the signal-with-noise report is 0,1;

0,01 and 0,001.

Case 1: SNR = 0,1

C = B log2(1 + SNR) B = C / log2(1 + SNR) = 56 103 / log2(1 + 0,1) = 407.262Hz

407,2KHz

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Case 2: SNR = 0,01

C = B log2(1 + SNR) B = C / log2(1 + SNR) = 56 103 / log2(1 + 0,01) = 3.901.000Hz

3,9 MHz

Case 3: SNR = 0,001

C = B log2(1 + SNR) B = C / log2(1 + SNR) = 56 103 / log2(1 + 0,001) = 38.835.647Hz

38,835 MHz

b) In a ordinary system (without spread spectrum)

In a regular system (without spread-spectrum), so that R = B

Shannon's formula C = Blog2 (1 + S / N) applied with C = B = R gives

log2 (1 + S / N) = 1. This implies that S / N = 1. Thus, spread-spectrum without a S / N

much greater is required.

7. Prove that a receiving station can get the data sent by a specific sender if it multiplies the

entire data on the channel by the sender’s chip code and then divides it by the number of

stations.

Let us prove this for the first station, using our previous four-station example. We can say that

the data on the channel:

D = (d1 c1 + d2 c2 + d3 c3 + d4 c4)

The receiver which wants to get the data sent by station 1 multiplies these data by c1.

D c1 = (d1 c1 + d2 c2 + d3 c3 + d4 c4) c1

= d1 c1 c1 + d2 c2 c1 + d3 c3 c1 + d4 c4 c1

= d1 N + d2 0 + d3 0 + d4 0

d1 = (D c1 ) / N

2. Practical - Spread spectrum techniques

The goal of this practical is to understand the impact of spread spectrum techniques on the

emitted signal and on the received signal by the recipient. You will be able with this practical:

– to visualize and study impact of the code used and of the modulation type on the spectrum

of a signal;

– to observe the properties of two signals according to the used keys (DS-CDMA);

– to visualize quality of reception of a signal 1, while varying the signal with noise report

(SNR) as well as the ratio (C/I) of the power of the signal has to interpret (Carrier: signal 1)

by compared with a second signal (Interferor: signal 2) which will come to disturb the

reception of the first;

– to trace the curves of Bit Error Rate (BER), according to coding for a SNR or a fixed C/I.

For that you will use the simulation tool developed for this effect. Copy the script tme_cdma in

your folder:

Carry out the script with the help of Octave, the digital calculation, visualization and programming

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software similar to Matlab. Appear the following menu:

Choose an option

[1] signal key 1 (useful)

[2] signal key 2 (interfering)

[3] display the spectrum of the useful signal (red) and the interfering signal

(green)

[4] choose the modulation type

[5] trace the autocorrelation of the key 1 and the cross-correlation between

two keys

[6] trace the constellation diagram in reception of signal 1 (useful) for a SNR

and a fixed C/I

[7] calculate the bit error rate in reception for a fixed SNR

[8] calculate the bit error rate in reception for a fixed C/I

[9] quit

Menu description

[1]: allows visualizing or modifying the spread spectrum key of the "signal 1". The key by defect is

the key of Barker used in Wifi?

[2]: allows visualizing or modifying the spread spectrum key of the "signal 2". The key by defect

is a sequence of "1".

[3]: trace the curves of the spectral density of signals 1 and 2 after modulation with and without

filtering. The two traced spectra, in red for signal 1 and in green for signal 2 have both the

same power. This spectrum is obtained for each signal by getting the same continuation

random of 10 000 bits. Each signal is coded with its own key which can be visualized and

modified while typing [1] or [2] in the menu.

[4]: allows choosing the modulation type which will be applied to signal 1. You have the choice

between: PSK-2 (BPSK), QAM-4 (QPSK), QAM-16 and QAM-64.

[5]: allows visualizing the curve of the autocorrelation of the signal 1 key and cross-correlation

between the keys of signals 1 and 2.

[6]: allows visualizing in the form of points in a constellation diagram, the value of the different

symbols received by the receiver. One can thus see the amplitude and the phase of these

symbols. The configuration of the receiver is adapted according to the modulation type of

signal 1, of the CDMA keys of the two signals, and the values of ratios SNR and C/I.

[7]: allows tracing curves of BER according to report C/I for a fixed SNR?

[8]: allows tracing curves of BER according to report SNR for a fixed C/I?

3. Properties and spectral of the emitted signal

The principle of the DS-CDMA is as follows. At each transmitting station is allocated a

single code of m chips called "sequence chip " constituted from "1 " and from "+1". From a

code constituted from "0" and from "1" like the code of Barker, it is advisable to replace the

"0" by "1" to obtain the sequence chip. To transmit a bit to "1", the station sends the

opposite of its sequence chip. To transmit a bit to "0", the station sends the opposite of its

sequence chip. The duration Tc of transmission of a chip is m time lower than the duration

Tb of a bit.

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1. What is the key value by defect of the signal 2? What does it mean?

For the first part of the tests, we keep the key value of the signal 1 on

"10110111000" and the key of the signal 2 on "1". There is thus nothing to modify.

2. (a) Choose a modulation type by using the menu [4] and display then the spectral

density of power of signal 1 and signal 2 by using the menu [3]. What do you

observe by comparing the form of the primary and secondary lobes of signals 1 and

2 without filtered for the selected modulation? These observations are they still

valid for the other modulations?

(b) What is the power reduction between the primary and secondary lobe for the

two curves?

(c) What is the effect of the filtering on the frequential spectrum?

(d) What is the interest of the filtering?

4. Properties and correlation

The autocorrelation is studied in considering the profile of the autocorrelation function

obtained by shifting the digital sequence x[k] (of length N completed beforehand by zeros) of a

gradually increasing value n and by comparing the initial sequence and the shifted sequence.

The formula used here to estimate the autocorrelation function is written:

The correlation coefficient obtained will be larger as much as the digital sequence and its

shifted version are similar. Conversely, if the sequences are opposed, the correlation will be

negative. A null autocorrelation indicates that there is not correlation between the initial

sequence and the shifted sequence for the considered shift.

The study of the autocorrelation allows us to evaluate the capacity of the receiver has to

synchronize with the flow of chips of the transmitter. In effect, the receiver makes a correlation

between the received signal and the used key in the emission. A peak of correlation must be

observed when on the duration of a bit, the key used for encoder a bit and the key used by the

receiver are synchronous. In the other hand, the value of γx[n] must be near to zero for a shift not

null representing a desynchronization of the receiver. Thus, a good key must have such

properties. It will be said that it has good properties of autocorrelation.

1. A good autocorrelation must present a maximum for a null shift and must be near to zero

elsewhere. Which interest is presented in the case where the channel is the subject of

multi-traffic?

2. Observe the autocorrelation of the Barker key of the IEEE standard 802.11

"10110111000" by using the menu [5]. It appears good to you?

3. Which property appears important to you to choose the keys of two distinct transmitters?

By basing you on that which was known as the autocorrelation properties, what appears

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important to you for the cross-correlation?

4. Give the mathematic expression of an estimator of the cross-correlation of two

sequences x[k] and y[k].

5. Are the keys of signals 1 and 2 orthogonal?

6. Modify the keys of two transmitters and observe their autocorrelation and their

cross-correlation.

5. Reception and interpretation of the signal 1

The menu [6] makes it possible to visualize the value of the amplitude and of the phase of

different symbols received by the receiver. The simulator allows observing the impact of the

modulation; of interferences linked to the additive noise (SNR) and to the disturbance caused by

another signal (signal 2) synchronous with the first.

1. What does it mean SNR or ratio C/I positive, null and negative?

2. Reposition the key value of the signal 1 to “10110111000" (Barker key) and that of

signal 2 to “1”, i.e. to the default values. Using the menus [4] and [6], vary the SNR, the

C/I and the modulation type and fill out table 1. What do you observe? what does it

occur when ratio C/I tends towards 0?

Tab. 1 – Synthesis of observations on the constellation diagrams.

Modulation SNR C/I Quality of the reception of signal

(Good/ Medium /Bad)

BPSK 100 15

QPSK 100 15

16QAM 100 15

64QAM 100 15

BPSK 15 100

QPSK 15 100

16QAM 15 100

64QAM 15 100

BPSK 15 0

QPSK 15 0

16QAM 15 0

64QAM 15 0

BPSK 15 3

QPSK 15 3

16QAM 15 3

64QAM 15 3

3. Position the key value of the signal 1 to "10110111000" (Barker key) and that of signal 2

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