Page 143 - Fundamentals of Communications Systems

Complex Baseband Representation of Bandpass Signals 4.19

Problem 4.2. Find the form of x I (t) and x Q (t) for the following x c (t)
(a) x c (t) = sin(2π(f c − f m )t)
(b) x c (t) = cos(2π(f c + f m )t)
(c) x c (t) = cos(2π f c t + φ p )

Problem 4.3. If the lowpass components for a bandpass signal are of the form
x I (t) = 12 cos(6πt) + 3 cos(10πt)
and

x Q (t) = 2 sin(6πt) + 3 sin(10πt)
(a) Calculate the Fourier series of x I (t) and x Q (t).
(b) Calculate the Fourier series of x z (t).
(c) Assuming f c = 40 Hz calculate the Fourier series of x c (t).
(d) Calculate and plot x A (t). Computer might be useful.
(e) Calculate and plot x P (t). Computer might be useful.

Problem 4.4. A bandpass filter has the following complex envelope representation
for the impulse response

⎧
⎨2 1 exp − t + j 2 1 exp − t t ≥ 0
2 2 4 4
h z (t) = (4.41)
0 elsewhere
⎩
(a) Calculate H z (f ).
Hint: The transforms you need are in a table somewhere.
(b) With x z (t) from Problem 4.3 as the input, calculate the Fourier series for
the filter output, y z (t).
(c) Plot the output amplitude, y A (t), and phase, y P (t).
(d) Plot the resulting bandpass signal, y c (t) using f c = 40 Hz.

Problem 4.5. The picture of a color television set proposed by the National Tele-
vision System Committee (NTSC) is composed by scanning in a grid pattern
across the screen. The scan is made up of three independent beams (red, green,
and blue). These independent beams can be combined to make any color at a
particular position. In order to make the original color transmission compati-
ble with black and white televisions the three color signals (x r (t), x g (t), x b (t))
are transformed into a luminance signal (black and white level), x L (t), and two
independent chrominance signals, x I (t) and x Q (t). These chrominance signals
are modulated onto a carrier of 3.58 MHz to produce a bandpass signal for
transmission. A commonly used tool for video engineers to understand these
coloring patterns is the vectorscope representation shown in Figure 4.14.