Exercise 2: Path loss and log normal shadowing

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Exercises for Drahtlose Kommunikation

Prof. Dr. Hannes Frey Daniel Schneider Winter term 2019/2020

Assignment 4

General remarks

• The exercises will be discussed on Thursday, January 23, 12:15.

Exercise 1: Doppler shift

If a sender moving at velocity vsends a signal at carrier frequency f,an observer at a fixed position will receive a shifted frequencyf⁰.The difference between original frequencyf and received frequency f⁰ is called Doppler shift and is given by

f⁰=f· c

c±v, (1)

where the sign of ±v is positive if the sender moves away from the observer and negative if it approaches the observer.

Assume that a sender and a receiver, both operating at 900 MHz, are mounted on a vehicle driving at 100 km/h towards a wall.

(a) At which frequency is the wave propagating in direction of travel (i.e. what is the frequency observed at the wall)?

(b) At which frequency does the receiver mounted on the vehicle receive the signal after reflection at the wall?

Hint: Consider suitable inertial frames of reference and take into account the effect on the propagation speed (in particularcmay be treated as classical entity). Alternatively you can also use the combined formula for transmitter and receiver velocities

https://en.wikipedia.org/wiki/Doppler effect#General.

Remark: The classical physics formula (1) is a very good approximation of the accurate relativistic one. The error is of the order ^v_c²2.

Exercise 2: Path loss and log normal shadowing

Consider a scenario with three nodes A, B and C having a pairwise distance of d = 100m to one another (triangle). The channel behaves according to log-normal shadowing. The transmit power is Ptx =−10 dBm, the path loss (PL) coefficient is n= 2.5 and the standard deviation is σ = 5.

To be able to correctly receive data, the received powerPrx has to be at least -80 dBm. The PL at reference distance 1m isP L(d₀ = 1m) = 10 dB.

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Exercises for Drahtlose Kommunikation WS 2019/2020

(a) What is the PL at distancedon average?

(b) Assume a receiver is located at distance d from a sender. Calculate the probability that the receiver is able to correctly receive data.

(c) When using multi-hop communication not all pairwise links in the network need to be working.

It is possible to route data multi-hop from node to node until they reach the destination. Thus, for multi-hop it is sufficient that there exists a working path between each pair of nodes.

What is the probability that the nodes A, B andC are connected in terms of multi-hop communication?

(d) Each (single-hop) link should be working with a probability of 99%. How large must the distance dbe chosen?

Exercise 3: Two-ray ground model

(a) Assume a two-ray ground model and let both nodes have a constant distance to the reflecting ground. As the distance d between sender and receiver tends to infinity, does the phase shift converge to a fixed value or does it continue periodically? Give a short explanation.

(b) Which channel models from the lecture are suitable for modeling the channel behavior in the following scenarios? Give a short explanation.

(i) The receiver moves away from sender at constant velocity.

(ii) Sender and receiver move at same speed into the same direction.

(iii) Sender and receiver are not moving.

Exercise 4: Hamming distance

What minimum Hamming distance is needed to be able to:

(a) Detect 10 bit errors?

(b) Correct 10 bit errors?

Show your calculation.

Exercise 5: LNS Regression

In this excercise we would like to determine the path loss coefficient nof the log-normal-shadowing model fromMempirical path loss measurements (di, yi), i= 1,· · · , M,wheredidenotes the distances in meter and y_i denotes the pathlossP L(d_i) in dB. Thus we consider the linear model

yi =P L(d0)[dB] +n·xi+εi, i= 1,· · · , M,

where x_i = 10 log₁₀(d_i/d₀), and where ε_i are normally distributed fluctuations with expectation 0 and varianceσ², and d0 is a reference distance.

(a) Assuming that free space propagation is valid within the first meters, use the Friis equation to calculate P L(d0) for d0 = 1m for a transmitter/receiver pair with gainGt=Gr= 1 and carrier frequency f = 2.4 GHz.

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(b) Download the files “demo.py”(Python 2.7 code also below) and “data.csv” and use linear regression to estimate nand σ as follows:

(i) Center they-data:y_i⁰ :=y_i−P L(d₀)[dB].

(ii) Run a linear regression on (x_i, y⁰_i) (use fit intercept = False, i.e. the desired function vanishes at the origin due to the centering) to find the steepness n.

(iii) Use the estimates

d7→P L(d0)[dB] + 10·n·log₁₀(d/d0) to expressεi and find the standard deviation σ.

[The true nequals 2.2 and the true σ equals 5.0 dB.]

f r o m _ _ f u t u r e _ _ i m p o r t d i v i s i o n i m p o r t n u m p y as np

f r o m m a t p l o t l i b i m p o r t p y p l o t as plt

f r o m s k l e a r n.l i n e a r _ m o d e l i m p o r t L i n e a r R e g r e s s i o n i m p o r t p a n d a s as pd

def l i n _ r e g (X, y):

reg = L i n e a r R e g r e s s i o n(f i t _ i n t e r c e p t = F a l s e).fit(X, y) r e t u r n reg.coef_, reg.i n t e r c e p t _

def t e s t _ r e g ():

X = np.a r r a y([1 ,2 ,3]).r e s h a p e( -1 ,1)

y = np.a r r a y([2.1 , 3.9 , 6 . 2 ] ) .r e s h a p e( -1 ,1) m,b = map(float, l i n _ r e g(X,y))

p r i n t " f o u n d f u n c t i o n : "

p r i n t " y = {} x + ( { } ) ".f o r m a t(m,b)

for x in X:

p r i n t m*f l o a t(x)+b

def r e a d _ d a t a (f i l e n a m e):

df = pd.D a t a F r a m e.f r o m _ c s v(f i l e n a m e) r e t u r n np.a r r a y(df)

def t e s t _ s t d _ d e v (s i g m a):

d a t a = np.r a n d o m.n o r m a l(2 , sigma, 1 0 0 0 ) p r i n t " e s t i m a t e = ", np.std(data, d d o f = 1)

if _ _ n a m e _ _ == ’ _ _ m a i n _ _ ’:

p r i n t " L i n e a r r e g r e s s i o n d e m o "

t e s t _ r e g()

# r e a d in d a t a

d a t a = r e a d _ d a t a(" d a t a . csv ")

# f l a t t e n d a t a

X = d a t a[: ,0].r e s h a p e( -1 ,1) y = d a t a[: ,1].r e s h a p e( -1 ,1)

# d e m o of e s t i m a t i o n of std d e v i a t i o n s i g m a = .5

p r i n t " e s t i m a t i n g std d e v i a t i o n , t r u e v a l u e = ", s i g m a

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t e s t _ s t d _ d e v(s i g m a)

p r i n t " P l o t t i n g d a t a "

plt.p l o t(X,y) plt.s h o w()

Exercise 6: Binary linear codes

Show that for binary linear codes (a) S1, S2∈ C implies 0∈ C.

(b) The minimum Hamming distance dmin ofC satisfies d_min = min

c∈C\{0}d(c,0).

We wish you a Merry Christmas. . . . . . and a successful, Happy New Year

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