Matlab question

Fall 2017/2018 EE 635

Homework #3 Instructions:

1) Submit the homework online on Blackboard. No late submission please. If you have to, submit an incomplete answer on time.

2) Submit all typed notes, program results and plots, and handwritten notes in one pdf file. The pdf file name should have the following format:

EE635_HW_nn_lastname.pdf where

• nn is the homework number in two digits, i.e. 01, 11 • lastname is your last name in small case without any spaces or special characters, e.g.

alghamdi. 3) If you need to attach other files (other than the notes, e.g. m files, etc.), combine all files in one

zip file (not rar file). The zip file name should have the following format: EE635_HW_nn_lastname.zip

4) After each problem list all the references that you have used to answer the problem, including books, journal articles, solution manuals, web pages, persons, etc.

5) You must work individually on the homework. Oral consultations and discussion among students is ok. You must list the names of students you have consulted in the references.

1) Consider the transmitter and receiver show in the figure below. The transmitter and receiver are at the same height and the receiver moves to the right at a speed of 𝑣𝑣 m/s. Use the two-ray channel model. a) Find the time-varying impulse response 𝑐𝑐(𝜏𝜏, 𝑡𝑡) of the channel for 𝑑𝑑 ≫ ℎ. b) Find 𝐾𝐾1(𝑡𝑡, 𝜏𝜏)

h

d vt� v

2) Using MATLAB, generate two independent zero-mean Gaussian random variables 𝑋𝑋 and 𝑌𝑌 with variance 5. Compute 𝑍𝑍 = √𝑋𝑋2 + 𝑌𝑌2 and 𝜙𝜙 = arctan(𝑌𝑌/𝑋𝑋). Use MATLAB to demonstrate that 𝑍𝑍 and 𝜙𝜙 are independent and that 𝑍𝑍 is Rayleigh distributed and 𝜙𝜙 is uniformly distributed.

3) Consider a random, time-varying channel with impulse response 𝑐𝑐(𝜏𝜏, 𝑡𝑡) = 𝛼𝛼1(𝑡𝑡)𝛿𝛿(𝜏𝜏 − 1) + 𝛼𝛼2(𝑡𝑡)𝛿𝛿(𝜏𝜏 − 2)

where 𝛼𝛼1(𝑡𝑡) and 𝛼𝛼2(𝑡𝑡) are jointly-WSS complex random processes with 𝐸𝐸[𝛼𝛼1(𝑡𝑡 + Δ𝑡𝑡)𝛼𝛼1∗(𝑡𝑡)] = 𝑅𝑅1(Δ𝑡𝑡) = 𝑎𝑎1 sinc(𝑓𝑓1Δ𝑡𝑡) 𝐸𝐸[𝛼𝛼2(𝑡𝑡 + Δ𝑡𝑡)𝛼𝛼2∗(𝑡𝑡)] = 𝑅𝑅2(Δ𝑡𝑡) = 𝑎𝑎2 sinc(𝑓𝑓2Δ𝑡𝑡) 𝐸𝐸[𝛼𝛼1(𝑡𝑡 + Δ𝑡𝑡)𝛼𝛼2∗(𝑡𝑡)] = 𝑅𝑅12(Δ𝑡𝑡) = 0

and 𝑎𝑎1, 𝑎𝑎2 > 0, 𝑓𝑓2 > 𝑓𝑓1 > 0. a) Show that this channel satisfies the WSSUS conditions. b) Find the scattering function of the channel. c) Find the power-delay profile of the channel, the mean delay spread and the rms delay

spread. d) Find the coherence bandwidth, coherence time and Doppler spread. e) Find all the variants of 𝐴𝐴𝑐𝑐, 𝑆𝑆𝑐𝑐 and 𝑆𝑆𝐶𝐶 defined in the lectures.

4) For a narrowband fading environment, define the outage probability as 𝑃𝑃out = 𝑃𝑃(𝑧𝑧2 < 2𝑃𝑃min) = 𝑃𝑃(𝑧𝑧 < �2𝑃𝑃min

where 𝑧𝑧(𝑡𝑡) is the fading envelope and 𝑃𝑃min is the minimum allowable received power for proper performance. a) Derive a formula for 𝑃𝑃out for the case of Rayleigh fading. You result should be in terms of

𝜌𝜌 = �𝑃𝑃min/𝑃𝑃𝑟𝑟 b) Compute the outage probability for a Rayleigh fading channel with 𝑃𝑃𝑟𝑟 = −70 dBm and

𝑃𝑃min = −70 dBm.

c) Suppose that we require an outage probability of 0.01 with 𝑃𝑃min = −70 dBm. What is the required average received power?

d) Use MATLAB to numerically compute the outage probability for a Rician fading channel with 𝑃𝑃𝑟𝑟 = −70 dBm and 𝑠𝑠2/2 = −80 dBm.

Note: When computing outage probabilities, you must use powers in absolute scale and not dBm scale!

5) Consider a wideband channel characterized by the autocorrelation function

𝐴𝐴𝑐𝑐(𝜏𝜏, Δ𝑡𝑡) = � sinc(𝑊𝑊Δ𝑡𝑡), 0 ≤ 𝜏𝜏 ≤ 10 µs 0, otherwise

where W= 100 Hz and sinc(𝑥𝑥) = sin(𝜋𝜋𝑥𝑥) /(𝜋𝜋𝑥𝑥). a) Does this channel correspond to an indoor channel or an outdoor channel, and why? b) Sketch the scattering function of this channel. c) Compute the channel’s average delay spread, rms delay spread, and Doppler spread. d) Over approximately what range of data rates will a signal transmitted via this channel

exhibit frequency-selective fading? e) Would you expect this channel to exhibit Rayleigh or rather Rician fading statistics? Why? f) Assuming that the channel exhibits Rayleigh fading, what is the average length of time that

the signal power is continuously below its average value? g) Assume a system with narrowband binary modulation sent over this channel. Your system

has error correction coding that can correct two simultaneous bit errors. Assume also that you always make an error if the received signal power is below its average value and that you never make an error if this power is at or above its average value. If the channel is Rayleigh fading, then what is the maximum data rate that can be sent over this channel with error-free transmission? Make the approximation that the fade duration never exceeds twice its average value.

6) Let a scattering function 𝑆𝑆𝑐𝑐(𝜏𝜏, 𝜌𝜌) be nonzero over 0 ≤ 𝜏𝜏 ≤ 0.1 ms and −0.1 ≤ 𝜌𝜌 ≤ 0.1 Hz. Assume that the power of the scattering function is approximately uniform over the range where it is nonzero. a) What are the multipath spread and the Doppler spread of the channel? b) Suppose you input to this channel two identical sinusoids separated in time by Δ𝑡𝑡. What is

the minimum value of Δ𝑓𝑓 for which the channel response to the first sinusoid is approximately independent of the channel response to the second sinusoid?

c) For two sinusoidal inputs to the channel 𝑢𝑢1(𝑡𝑡) = sin 2𝜋𝜋𝑓𝑓𝑡𝑡 and 𝑢𝑢2(𝑡𝑡) = sin 2𝜋𝜋𝑓𝑓(𝑡𝑡 + Δ𝑡𝑡), find the minimum value of Δ𝑡𝑡 for which the channel response to 𝑢𝑢1(𝑡𝑡) is approximately independent of the channel response to 𝑢𝑢2(𝑡𝑡).

d) Will this channel exhibit flat fading or frequency-selective fading for a typical voice channel with a 3-kHz bandwidth? For a cellular channel with a 30-kHz bandwidth?