ECE 402: Communications Engineering
Homework 1 Solutions (Fall 2009)
1. The period is T = 1/2 so f0 = 2. We can evaluate the FS using Poissons Sum Formula with
central pulse
cos(2t), |t| < 1/4
0,
else
g (t) =
= (2t) cos(2t) .
By the modulation property G(
ECE 402
Communications Engineering
Quiz 3/Final Exam
2009
KEY
NAME:
STUDENT NUMBER:
INSTRUCTIONS
Write your name and student number in the space provided above.
Questions 1-4 are Quiz 3. Questions 5-8 are the Final Exam.
This quiz is closed-book and cl
ECE 402: Communications Engineering
Homework 2 (Fall 2009)
Due: Tuesday 9/15
1. Find the energy spectral density and autocorrelation of the signal
x(t) = e2t u(t)
What signal energy is located in the frequency band |f | < 1/ Hz? Hint: The following
integr
ECE 402: Communications Engineering
Homework 6 Solutions (Fall 2009)
1. (a) For polar signaling, Pb = Q( 2Eb /N0 ). From the reference sheet, we see that Q(3.719) = 104
and Q(4.264) = 105 . Hence to obtain Pb = 104 , we must have
2Eb
= 3.719 =
N0
Eb
N0
dB
ECE 402: Communications Engineering
Homework 3 (Fall 2009)
Due: Tuesday 9/22
1. Suppose that pulse code modulation with uniform quantization is used to digitize an electrocardiogram (ECG) record. The ECG consists of two channels (i.e., two signals), each
ECE 402: Communications Engineering
Homework 5 (Fall 2009)
Due: Tuesday 10/27
1. (a) Suppose that binary data are transmitted by unipolar signaling with the triangular pulse waveforms: x0 (t) = 0, x1 (t) = (t/T ). Assuming that the data consist of i.i.d.,
ECE 402: Communications Engineering
Homework 2 Solutions (Fall 2009)
1
1. From the transform table, X (f ) = 2+2jf and so Ex (f ) = |X (f )|2 =
lation is therefore
Rx ( ) = F 1 [Ex (f )] = (1/4)e2|t| .
1
.
4+4 2 f 2
The autocorre-
Using the hint, we can n
ECE 402: Communications Engineering
Homework 6 (Fall 2009)
Due: Tuesday 11/3 (due at the *beginning* of class)
1. Suppose that binary data is transmitted over an AWGN channel using polar signaling. Find the
SNR (Eb /N0 ) in dB required to achieve an error
ECE 402: Communications Engineering
Homework 7 (Fall 2009)
Due: Tuesday 11/24
1. Suppose 64QAM is used to transmit data over an AWGN channel with the Nyquist pulse g (t) =
(1/ T )sinc(t/T ).
(a) What is the spectral eciency of 64QAM?
(b) Find the Eb /N0 i
ECE 402: Communications Engineering
Homework 5 Solutions (Fall 2009)
1. (a) We can take the data amplitudes to be an = 0, 1 and the pulse waveform to be g (t) = (t/T ), so
the PSD is given by
PX (f ) =
|G(f )|2 2
+ |2 fb combfb (f )
T
where G(f ) = T sin
ECE 402: Communications Engineering
Homework 4 (Fall 2015)
Due: Thursday 10/15
1. Suppose that X(t) is a zero-mean Gaussian process with power spectral density PX (f ) =
(1/B)(f /B). Find the autocorrelation of X(t) and the joint pdf of X(1/2B) and X(1/2B
ECE 402
Communications Engineering
Quiz 1
Fall 2008
TEST
NAME:
STUDENT NUMBER:
INSTRUCTIONS
Write your name and student number in the space provided above.
This quiz is closed book and closed notes.
One handwritten 8.5 11 reference sheet may be used (b
ECE 402: Communications Engineering
Homework 3 (Fall 2016)
Due: Tuesday 9/20
1. Suppose that pulse code modulation with uniform quantization is used to digitize an electrocardiogram (ECG) record. The ECG consists of two channels (i.e., two signals), each
Sampling and Reconstruction
When can we recover a signal from samples taken every T secs?
xt xnT , n 0,1,
x t
2T T
0
T
2T
Not always possible!
Two different signals
with the same
samples:
ECE 402 Fall 16
43
Sampling and Reconstruction
xS t
An (Ideal
ECE 402
Communications Engineering
Quiz 1
Fall 2008
KEY
NAME:
STUDENT NUMBER:
INSTRUCTIONS
Write your name and student number in the space provided above.
This quiz is closed book and closed notes.
One handwritten 8.5 11 reference sheet may be used (bo
ECE 402: Communications Engineering
Homework 3 (Fall 2015)
Due: Tuesday 9/22
1. A music signal is digitally recorded using sampling and uniform quantization. Assuming the
highest frequency tone to be recorded is 15,800 Hz, what is the minimum sampling rat
ECE 402: Communications Engineering
Homework 2 Solutions (Fall 2015)
1. Any lter transforms the input (tt0 ) into h(tt0 ), where h(t) is the lter impulse response.
Thus, if the input is
k=0 (t k), the corresponding output is
y(t) =
h(t k) =
k=0
[u(t k) u
ECE 402: Communications Engineering
Homework 1 (Fall 2015)
Due: Tuesday 9/8
1. Find and sketch the Fourier Transform of
x(t) = sinc2 (t/2) cos(2t) .
2. Evaluate
cos3 (3t) [(t 1) + (t) + (t + 1)] dt
3. Classify each of the following signals as a power sign
ECE 402: Communications Engineering
Homework 6 Solutions (Fall 2015)
1. Let R be the data rate for voice. Since Pb = Q( 2Eb /N0 ) for BPSK, Q(3.090) = 103 , and
Q(4.753) = 106 (from the class reference table), we need
Eb
P
(4.753)2
=
=
= 11.3
N0
N0 R
2
fo
ECE 402: Communications Engineering
Homework 7 (Fall 2015)
Due: Tuesday 11/24
1. According to Shannons Formula, what is the minimum bandwidth needed to transmit data at 1
Mb/s over an additive white Gaussian noise channel at (Eb /N0 )dB = 20 dB? Hint: To
ECE 402: Communications Engineering
Homework 5 Solutions (Fall 2015)
1. The maximum rate of the V90 modem is R = kfb = (log2 M )fb = 8 8, 000 = 64, 000 = 64 kb/s.
(The actual rate is 56 kb/s due to robbed-bit signaling - one of the eight bits above is use
ECE 402: Communications Engineering
Homework 8 (Fall 2015)
Due: Thursday 12/3 (at the *beginning* of class)
1. A broadband router at a cable-TV company transmits digital data to a home cable modem using
several parallel bandpass channels. One channel in p
ECE 402: Communications Engineering
Homework 4 Solutions (Fall 2015)
1. The autocorrelation is RX ( ) = F 1 [PX (f )] = sinc2 (Bt). Thus for any t, X(t) is a Gaussian
with mean = 0 and variance 2 = RX (0) 2 = 1. It follows that X(1/2B) and X(1/2B)
both ha
ECE 402: Communications Engineering
Homework 6 (Fall 2011)
Due: Tuesday 11/3 (due at the *beginning* of class)
1. A satellite communication system has been designed to send binary data using BPSK at a data
rate of R = 1 Mbps and an error probability of Pb
ECE 402: Communications Engineering
Homework 7 Solutions (Fall 2009)
1. (a) As noted in class, for the sinc pulse the null bandwidth is Wnull = 1/T and the data rate is
R = k/T , so = R/Wnull = k = 6 bps/Hz.
(b) For M = 64, we want
4
= Pb
log2 M
8
10
1
1
Digital Baseband Modulation
Source coding methods like PCM convert a signal into a stream of
bits. To transmit these data over a channel, we need to represent
each bit with a waveform, which is called digital modulation (aka
line code). The simplest forms
ECE 402: Communications Engineering
Homework 6 Solutions (Fall 2009)
p
1. (a) For polar signaling, Pb = Q( 2Eb /N0 ). From the reference sheet, we see that Q(3.719) = 104
and Q(4.264) = 105 . Hence to obtain Pb = 104 , we must have
s
2Eb
= 3.719 =
N0
Eb
N
ECE 402: Communications Engineering
Homework 5 Solutions (Fall 2009)
1. (a) We can take the data amplitudes to be an = 0, 1 and the pulse waveform to be g(t) = (t/T ), so
the PSD is given by
PX (f ) =
i
|G(f )|2 h 2
+ |2 fb combfb (f )
T
where G(f ) = T
ECE 402: Communications Engineering
Homework 5 (Fall 2008)
Due: Tuesday 10/28
1. Consider a random data stream of independent binary digits, where 0s and 1s are equally
probable. Suppose that these data are encoded using polar-type signaling with amplitud
ECE 402 : Communications Engineering
Brian L. Hughes
Department of Electrical and Computer Engineering
North Carolina State University
blhughes@ncsu.edu
ECE 402 Fall 16
1
Course Overview
Communications: deals with sending messages from one point
in space
ECE 402: Communications Engineering
Homework 6 Solutions (Fall 2008)
1. (a) The baseband version of 4QAM has data amplitudes an = A jA and pulse waveform
g(t) = (t T /2)/T ). Since the data are i.i.d., we have
= Ecfw_an = (1/4)(A + jA) + (1/4)(A jA) + (
ECE 402: Communications Engineering
Homework 6 (Fall 2008)
Due: Tuesday 11/4
1. Consider a random data stream of independent binary digits, where 0s and 1s are equally
probable. Suppose that these data are encoded using 4-QAM bandpass modulation with a
re
ECE 402: Communications Engineering
Homework 5 (Fall 2009)
Due: Tuesday 10/27
1. (a) Suppose that binary data are transmitted by unipolar signaling with the triangular pulse waveforms: x0 (t) = 0, x1 (t) = (t/T ). Assuming that the data consist of i.i.d.,