ECE5884 Wireless Communications – Quiz 3 8 August 2022
1. Write only ONE or TWO sentences for each of the following topics. Just write as you understood .
(a) Coherence time
(b) Coherence bandwidth
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(c) Fast fading channel (d) Flat fading channel
(e) Narrowband communications
2. (MATLAB) A multipath fast-fading channel of wireless communications sys-
tem is given as
h = hr + jhi
which is a complex number. We assume that h follows a circularly symmetric complex Gaussian distribution, i.e., h ∼ CN (0, Ωp), where Ωp is the average envelop power. This implies that hr, hi ∼ N (0, Ωp/2) which are Gaussian distributions with mean zero and variance Ωp/2. Then, the envelop of h, i.e., ∣h∣, follows a Rayleigh distribution.
PDFs and CDFs of ∣h∣ and ∣h∣2 can be given as follows (also see Slide#12 of the Lecture note or Slide#10 of the Workshop note):
Using MATLAB simulations, please verify above PDFs and CDFs when
Ωp = 5 dBm!
(Hint: For an example, you may generate a random variable h in MATLAB
h = sqrt(omega/2) ∗ randn(1, 1) + 1i ∗ sqrt(omega/2) ∗ randn(1, 1).
Try to get your plots as in Figure 1 ) 1
PDFof∣h∣: f (z)= e−z ;z≥0 (1)
CDFof∣h∣2: F (z)=1−e− t (4)
(z)=1−e−z (2)
CDF of ∣h∣: F
PDFof∣h∣2: f (z)=1e−t (3)
0.05 0.1 0.15 0.2
300 200 100
0.1 0.15 0.2
Simulation Analytical
Simulation
Analytical
00 0 0.01 0.02 0.03 0.04
0 0.01 0.02 0.03 0.04
Simulation Analytical
Simulation Analytical
Figure 1: Example.
3. We have a simple wireless system with a transmitter and receiver pair which are implemented with a single antenna. Then, the received signal at the receiver at time t may be written as
r(t) = h s(t) + n(t). (5)
Here, h is the wireless channel as explained in Problem#2; s(t) is the trans- mitted signal which has the average transmit power Ps (W); and n(t) is the additive white Gaussian noise (AWGN) which follows a circularly symmetric complex Gaussian distribution, i.e., n(t) ∼ CN (0, N0), where N0 (W/Hz) is the average noise power. The channel bandwidth of the wireless system is B (Hz). We neglect path-loss and shadowing effects.
(a) Write an expression for the instantaneous received signal-to-noise ratio (SNR) γ in terms of h, Ps, N0 and B.
(b) Derive an expression for the SNR outage probability, when the received SNR falls below a threshold γth. Please provide details of your derivation!
(c) The wireless system requires the SNR outage probability less than 10−5. What is the minimum signal power Ps (W) which satisfies this require- ment when N0 = 10−9(W/Hz), B = 1(Hz), γth = 5(dB) and Ωp = 5 (dBm)?
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