# cmpen 462 hw 2

deadline is a bit longer than 2 days i will extend the deadline

CMPEN 462
Homework #2: Wireless Fundamentals
Due: March 4, 2021 (10:35am)
(worth 3 pts total)
Wireless PHY layer characteristics
1. (0.3 pts) If a 14 Mbps LTE data rate on the downlink is desired, what would the
minimum bandwidth be and why, if QPSK modulation is being used and the entire
bandwidth is able to be allocated to a single user if needed? Recall that LTE has 1.4, 3, 5,
10, 15, and 20 MHz BW capabilities.
2. (0.3 pts) 5G systems are based upon a ‘Numerology’ that permits varying slot times and
varying subcarrier bandwidths (15, 30, 60, 120, and 240 kHz) to provide flexibility,
robustness, higher data rates, and lower latencies. What are the associated symbol
periods/durations/times (Ts) for each of these physical layer options?
Modulation
3. (0.3 pts) Given the QPSK constellation diagram shown below, using MATLAB plot the
QPSK signal given a bit rate of 10Mbps and a carrier frequency of 10MHz for the bit
sequence [1 0 1 1 0 1 0 0]. The amplitude and phase must be correct.
Beamforming
4. (0.4 pts) Consider two antennas A and B separated by a distance of
2� and transmitting inphase signals of wavelength . A receiver R is placed on a line passing through the mid-point of
the two antennas and making an angle relative to them as shown in Figure 1. Determine the
phase difference between signals of A and B that arrive at R. Plot the phase difference as a
function when varies from 0 to 100 (provide three plots for three different values of : 0,
90, and 45 degrees).
5. (0.4 pts) Consider a uniform linear array of N antennas as shown in Figure 2. The antenna
separation is d as shown, and all antennas transmit signals in phase, at a wavelength of .
Consider a receiver at an angle to a receiver very far away from the array. Since the receiver is
far, we can assume that lines joining the antennas to the receiver are parallel to each other,
making an angle _ above the horizontal line. Find all angles where the sum of arriving
signals from all antennas add up to 0, creating a perfect null.
Wireless Channel
6. (0.3 pts) If a signal is transmitted at a power of 275 mWatts (mW) and the noise in the
channel is 15 uWatts (uW), if the signal BW is 50MHz, what is the maximum capacity of
the channel? (this is a CMEPN 362 flashback question).
7. (0.3 pts) The cyclic prefix is intended to be long enough to mitigate the majority of
expected multipath (e.g., in LTE CP durations are 4.7 usec). Since multipath comes from
signals traveling longer distances than that of the shortest path signal, if the shortest
path between Tx and Rx is 2.5km, what is the distance of the longest traveled multipath
signal if the multipath delay should be significantly less than the CP duration (use factor
of 3)? Signal propagation speed, c = 3.0 * 10^8 (m/s).
8. (0.3 pts) Suppose you transmit a packet consisting of 5 data symbols and 3 redundant symbols
for the purposes of error correction. Because of the redundancy, the receiver can decode all the
symbols correctly as long as it receives any 6 or more of the 8 transmitted symbols correctly. We
now introduce a new metric called dispersion. Suppose a symbol ̅= 3 − was transmitted
based on a QAM modulation. Because of noise, and distortions in the channel, the received
symbol at the receiver was ̅= 3.2 − 1.1 . The difference ̅= ̅− ̅= 0.2 − 0.1 is called the
dispersion of the channel. In other words, the received symbol ̅= ̅+ ̅. If d is low, the symbol
would be mapped correctly to its bits, otherwise, there may be errors. For example, the
dispersion ̅
1is low in Figure 3(a), so the transmitted symbol ̅
1 = 1 − is decoded correctly as
1101. However, in Figure 3(b), the dispersion ̅
2 is high enough to cause incorrect decoding of
the symbol ̅
1 as 1001.
Suppose a transmitter has predicted the following dispersions for 8 symbols it would transmit.
[0.4970 + 0.5760i, -0.3471 – 0.1771i, 0.4611 – 0.3750i, 0.2999 + 1.1440i, 0.0026 +
0.0061i, 0.1322 + 0:1071i, 0.1777 + 0.9221i,1.1458 – 0.1548i]
You have a choice among BPSK, QPSK, and 16QAM for modulation at the transmitter (shown in
Figure 4). Pick the best modulation that allows the highest data rate where at least 6 of the
transmitted symbols can be received correctly (For sake of simplicity, assume that a dispersion
of outer symbols as shown in Figure 5 will cause an error).
Similarly, find the best modulation for the following dispersions.
• [-1.1060 – 0.6606i, -0.6343 + 0.4213i, 0.8692 + 1.0015i, -0.3148 – 0.0896i,
-1.9127 – 0.2194i, 0.0733 – 0.6894i, 1.8146 – 0.8009i; 0.3974 + 0.2512i]
• [0.1636 – 2:4676i, -0.4199 + 1.0603i, 0.7282 + 1.0495i, -1.1825 + 1.5902i, 1.1519
+ 1.4941i, 4.9096 – 0.8516i, 1.7176 – 1.1025i, 3.5888 – 1.4198i]
OFDM Decoding
9. (0.4 pts) Below given is a set of 16 downconverted, and downsampled I&Q samples in the form
of I+jQ.
• [-1.1250 + 0.6250j, -0:6856 – 1.0786j, -0.2286 + 0.0518j, -0.1426 – 0.1738j, 0.1250 –
0.1250j, 0.6064 + 0.1341j, 0.3018 + 0.4786j, -0.1841 + 0.5210j, -0.3750 + 0.3750j, 0.5089
+ 1.1518j, 0.4786 – 0.3018j, 0,3194 – 0.2530j; -0.1250 + 0:1250j, -0.4296 + 0.2926j,
-0.0518 – 0.2286j, 0.0073 – 0.5942j]
• [-1.3750 + 0.8750j, 0.3276 – 0.2039j, -0.0884 – 0.7437j; 0.6804 – 0.4068j, -0.2500 –
0.7500j, 0.4890 + 0.1362j, 0.6187 + 0.3902j, -0.1344 – 0.4494j, -1.3750 + 0.1250j,
-0.1508 + 0.0271j, 0.0884 + 0.4937j, -0.8572 + 0.2300j, 0.0000 + 0.7500j, -0.6658 +
0.0406j, -0.6187 – 0.1402j, 0.3112 + 0.6262j]
• [-1:3750 + 0.1250j, 0.1560 – 0.7033j, -0.1250 + 0.7589j, -0.3474 + 0:4705j, 0.1250 +
0.1250j, -0.2517 – 0.2396j, -0.6553 + 0.8321j, 0.3870 + 0.7990j, -1.1250 + 0.8750j,
0.0207 + 0.2766j, -0:1250 – 1.0089j, 0.1706 – 0.3973j, -0.1250 – 0.1250j, 0.0749 +
0.1663j, 0.4053 – 0.5821j, -0.2102 – 0.3722j]
• [-1.2500 + 0.1250j, 0.4724 + 0.3162j, 0.1768 + 0.0518j, 0.1603 + 1.0842j, 0.0000 +
0.1250j, 0.3352 – 0.3913j, 0.3536 + 0.1250j, -0.4119 – 0.0292j, -1.0000 + 0.8750j, –
0.0456 – 0.7430j, -0.1768 – 0.3018j, -0.0871 – 1.0110j, 0.2500 – 0.1250j, -0.2620 +
0.3181j, -0.3536 + 0.1250j, 0.8386 + 0.4560j]
If the transmitter used a 16 channel OFDM modulation with 16 QAM, can you decode the
transmitted message (use standard 16QAM constellation shown in Figure 3(b))?
The transmitted message is an 8-character word. Each character is transmitted in the form of a
binary ASCII string (‘01100001’ is the ASCII binary code for character ‘a’). The first 4 bits of the
first character were modulated as a 16QAM symbol and loaded on the negative most frequency.
The last 4 bits of the first character were modulated as a 16QAM symbol and loaded on the
second most negative frequency. The first 4 bits of the second character were modulated as a
16QAM symbol and loaded on the third most negative frequency. Similarly, the last 4 bits of the
last character were modulated as a 16QAM symbol and loaded on the positive most frequency.
The other characters are similarly loaded on frequencies in between.

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