1. (25) A linear filter H(f) is designed to be a matched filter for a rectangular time-limited pulse
Now, a triangular signal pulse
s(t)= √P 0
2At/T
s(t)= −(2A/T)(t−T) T/2≤t≤T
0 otherwise
is applied to the matched filter together with the usual additive white Gaussian noise n(t).
(a) Compute the signal-to-noise ratio of the matched filter output at time t = T . Express your answer only in terms of P, N0, and T, where N0 is the single-sided power spectral density of n(t). Assume that the peak, A, of the triangular pulse is chosen such that its average power equals that of the rectangular pulse (you need to determine the correct value for A).
(b) How does the SNR computed in part (a) compare with that which would be achieved if the rectangular pulse s(t) were applied to the matched filter input together with n(t)? Is there a gain or loss in SNR performance when the triangular pulse is used and by how much?
Suppose now that the problem was reversed, i.e., the filter is designed to be matched to the triangular pulse.
(c) What is the SNR of the matched filter output at time t = T when the sum of n(t) and the rectnagular pulse is applied to its input?
(d) Analogous to part (b), how does the SNR computed in part (c) compare with that which would be achieved if the triangular pulse s(t) were applied to the mathced filter input together with n(t)? Again express your answer only in terms of P, N0, and T. How does the gain or loss in this part compare with that computed for the reverse problem in part (b)?
2. (20) Consider the mathematical model of a discrete communication channel having 3 possible input messages {a, b, c} and 3 possible output symbols {1,2,3}. The channel model is completely described by the set of nine conditional probabilities
P(1|a) = 0.6 P(1|b) = 0.1 P(1|c) = 0.1
P(2|a) = 0.3 P(2|b) = 0.5 P(2|c) = 0.1
P(3|a) = 0.1 P(3|b) = 0.4 P(3|c) = 0.8
0≤t≤T otherwise
0≤t≤T/2
which specify the probability of receiving each output symbol conditioned on each input message. Assume that we know the set of three a priori probabilities with which the input messages are transmitted, namely,
P(a) = 0.3, P(b) = 0.5, and P(c) = 0.2
(a) Given that the received (channel output) symbol is 1, what is the decision of the optimum (maximum a posteriori)
receiver regarding which message is transmitted?
(b) Repeat part (a) when the received symbol is 2.
(c) Repeat part (a) when the received symbol is 3.
(d) Does the mathematical model assumed for the channel, together with the given a priori probabilities, lead to a decision rule in which each of the possible transmitted signals (a, b, and c) can at some time be selected? If your answer is “no”, then suggest a set of a priori probabilities that would make the above answer be “yes”.
3. (20)
Consider the communication system shown below. The modulated signal v(t) is linear and memoryless, with
δ(t − nT )
xn v(t) r(t) y(t) xˆn
t = nT Figure 1: Communication system with ISI.
w(t)
gT (t)
c(t)
gR (t)
detector
complex-valued symbols {xn} – one of which is transmitted every T seconds using the pulse defined by the transmit filter gT (t):
v(t) = xngT (t − nT ) n
The channel frequency response is given by
C(f)=1+be−j2πfT, |f|≤W
with b a real-valued constant, and C(f) = 0 for |f| > W. The noise w(t) is zero-mean white Gaussian with spectral density N0/2. The frequency response GT(f) of the transmit filter has bandwidth W and the signaling rate is R = 1/T < 2W .
(a) Assume that the receive filter gR(t) is matched to the transmit filter gT (t) (with gR(t) normalized to unit energy). Determine the resulting ISI pattern. That is, find yk = y(kT ) of the form
yk = desired symbol + ISI + noise
(b) Specify transmit and receive filters to achieve zero ISI communication.
(c) Suppose now that we have no knowledge about the channel impulse response, and that we implement a transmit filter according to
GT (f) = Xrc(f)
where Xrc(f) is a raised-cosine pulse. Assuming a normalized receive filter matched to GT (f), what is the resulting
ISI pattern now?
(d) Assume the possible symbols xn are determined by the complex lowpass representation of a 6-PSK constellation, with symbols spaced a distance d from each other and the origin. Maximum Likelihood detection based on the received samples yn is used. The detector ignores the presence of ISI (i.e., estimate of x, xˆn, is set equal to “6-PSK point closest to yn”). What is the maximum value bmax for the constant b for an error-free system in the absence of noise?
4. (20) Consider a QPSK receiver that uses rectangular pulse shaping as shown in Fig. 2. This system operates over an additive white Gaussian noise channel with an unknown phase offset. It uses a 10-bit training sequence (to perform phase estimation and achieve synchronization), which is then followed by 1,000 information bits. We measure a number of different signals at various points in the receiver chain; these are denoted by numbers in circles. Results of these measurements are shown in Fig. 3. The oversampling factor is 8 (i.e. we take 8 samples for each symbol). Markings on time axes denote sample numbers and not symbol numbers. The signal-to-noise ratio Eb/N0 = 10 dB for all of our measurements unless otherwise noted in the figure. In Fig. 2, BP = bandpass filter, MF = matched filter, and Dec. = Decoder.
2
(a) Match the plots A–D in Fig. 3 with corresponding measurement points in Fig. 2. Make sure to explain your answers.
(b) Study the measurements in Fig. 3. What is the main reason for the receiver’s poor performance compared to theory? Why? Which box in Fig. 2 do you suggest to be improved upon?
(c) Explain how the eye diagram E was obtained and what can be deduced from it. Which (one, or maybe several), of the measurements in Fig. 2 was used for this plot, and how?
5. (15)
signal at a rate of 8 bits/sample. Recall that to prevent aliasing, the Nyquist sampling rate must be at least twice the bandwidth of the signal, hence, fs = 16 kHz.
A baseband analog source is bandlimited to 8 kHz, and has duration of 10 seconds. We digitally sample this
(a) How many bits are collected in total?
We now consider communications of a symbol a over a passband N0/2 AWGN channel, at symbol rate 1/T symbols
per second, where T = 1 × 10−6 s (one microsecond). Assume Nyquist signaling (no ISI), with excess bandwidth 100%.
(b) If 256 QAM is used, how long will the transmission be, considering only the payload bits?
(c) What Eb/N0 will be required (roughly, within 1 dB) for an error probability of 10−6 (see BER curve below).
(d) How much signal power is required at the output of the antenna (assume a lossless system) if the the noise level is N0 = −174 dBm (referenced to 1 mW). Hint: See Eq. 8.3.48 in your text and recall how to convert raw power values to dBm.
Figure 4: BER curve for Problem 5.
3
Front End
123
Phase est.
BP
Down conv.
lacements
4
Baseband Processing
7 568
Phase corr.
MF
Dec.
Figure 2: QPSK receiver.
4
10
11
Extract training
9
Sync.
Phase est.
p
1.5
1.5
Real part
11
0.5 0.5
00
−0.5 −0.5
−1 −1
−1.5 −1.5
801 809 817 825 833 841 849 857 865 873 881 889 897 −1.5 −1 −0.5 0 0.5 1 1.5
1.5
1
0.5
1
0.5
AB
00
−0.5
−1
−1.5 −1.5
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−0.5
−1
−0.5
−1
0 0.5 1 1.5 801 809 817 825 833 841 849 857 865 873 881 889 897
CD
Real part 0 10
−1 10
−2 10
−3 10
−4 10
−5 10
−6 10
Simulated
−1
0 8 15 b0
E (Eb/N0=20 dB) BER Figure 3: QPSK receiver measurements.
5
Real part
Bit Error Rate
0 1 2 3 4 5 6 7 8 9 10
E /N [dB]
Theoretical