ECE300 Communication Theory

Problem Set IV: Analog Communications

1. This problem reviews basic parameters for FM. Consider a tone signal with amplitude 2V at frequency 5kHz that is frequency modulated using a VCO with frequency

sensitivity 10kHz=V . Compute: (1)the frequency deviation; (2)the ; (3)the bandwidth estimate given by Carsonís rule. Also specify whether this should be considered

narrowband or wideband FM.

2. In this problem, you will be synthesizing and studying various analog modulations of

a speciÖc signal:

m (t) = 1

1 + (t 10)2

which you will represent over the time span 0 t 20 sec. The spectrum is:

M (f) = ej40f e

2jfj

The 90% energy containment bandwidth B (i.e., 90% of the energy is contained in the

range jfj B) is given by:

B =

ln 10

4

= 0:1832Hz

Some other information you will Önd useful for this problem: the Hilbert transform is:

m^ (t) = t 10

1 + (t 10)2

and the integral is:

v (t) = Z t

=1

m ( ) d =

2

+ tan1

(t 10)

where here tan1

is the principal branch, returning values in the range =2 to =2.

For example, v (1) = 0, v (10) = =2, v (1) = .

We are going to use a carrier frequency of fc = 2Hz, and a sampling rate of fs = 10Hz.

(a) In MATLAB, set up a time vector t with values 0 t 20 at sample times (i.e.,

integer multiples of 1=fs). All your time domain signals should use this t vector.

You will note t has length approximately (but not exactly) 2000.

(b) The way you are going to compute spectra is to use § t of length N = 4096, via the

usual zero padding. After applying § tshift, the proper frequency vector should

span roughly from fs=2 to fs=2. Generate the correct frequency vector f. All

you frequency domain plots should use this f vector (unless speciÖed otherwise

below). Also, for any of the frequency plots in decibels (not all will be in dB),

make sure the vertical bounds are set so interesting features are visible (i.e., donít

let is go down to 1million dB or s