179 EE - Introduction to Communications - Spring 2009
Homework 2: Due Thursday 4/23 at 4pm.
1. (15 points) Bandwidth Ecient Modulation. This problem illustrates a form of modulation that uses less spectrum than standard modulation by exploiting the symmetry of real signals. Consider the communication system shown in the gure below. The signal x(t), with spectrum X(f ) as shown in the gure, is multiplied by a cosine of frequency fc = 10 KHz to obtain the signal z(t). Then z(t) is passed through a low-pass lter H(f ) with cuto frequency fc (H(f ) = 1 for |f | fc and zero otherwise) to produce the transmitted signal y(t). Assuming the channel is a wire, the received signal is also y(t), which in the receiver is multiplied by a cosine of frequency f0 . The resulting signal w(t) is passed through a low-pass lter G(f ) with cuto frequency B and gain A (G(f ) = A for |f | B and zero otherwise). The signal at the output of the lter G(f ) is denoted by x(t).
cos(2 f ct) x(t) x X(f) 1 1KHz 0 1KHz z(t) H(f) 1 f c 0 f c y(t) Channel cos(2 f 0t) w(t) x B 0 B G(f) A ^ x(t)
y(t)
(t)
(a) Sketch Z(f ). (b) Sketch Y (f ) (this is the modulated signal). (c) Find the values of f0 , B, and A such that x(t) = x(t) (so we recover the original signal from the modulated signal). Also, sketch the spectrum W (f ) of the signal w(t) that results from this choice of f0 . (d) Repeat part (c) assuming we remove the lter H(f ) (so z(t) = y(t)). Compare the passband bandwidth of the transmitted signal y(t) without H(f ) versus the passband bandwidth of y(t) with H(f ) (i.e. the bandwidth of Y (f ) found in part (b)). 2. (15 points) Voice Scrambling. One of the big advantages of digital modulation is that signals can be easily encrypted using standard encryption techniques. Most analog communication signals can be easily listened to using an antenna and simple receiver. AM This problem explores a simple encryption strategy for analog voice signals that distorts the transmitted signal to make eavesdropping more dicult. Consider the voice scrambling system shown below, with input voice signal X(f ) as shown in the gure. The highpass lter H1 (f ) passes all frequency components above f1 and removes all frequency components below f1 . The low pass lter passes all frequency components in [B, B] and removes all other frequency components. (a) Sketch Y (f ) assuming f1 >> B.
X(f)
B 0
1
Z(f)
1
B
f
Highpass Filter
B
Lowpass Filter
0
B
f
x(t)
H 1 (f)
y(t)
H 2 (f)
z(t)
cos(2 f1t) H 1(f)
1
cos(2 f2 t) H 2(f)
4
f 1
0
f1
f
B
0
B
f
(b) Find f2 as a function of f1 and B such that Z(f ) is as shown in the gure. (c) Explain how you would descramble the scrambled signal, i.e. how you would recover X(f ) from Z(f ). 3. (15 points) Text, Chapter 2, Number 2.38 4. (10 points) Text, Chapter 2, Number 2.41 5. (15 points) Text, Chapter 2, Number 2.47 6. (15 points) Text, Chapter 2, Number 2.49 7. (15 points) We have seen in class that modulating a n...