Faculty of engineering sciences
communication and electronics department
communication theory (1)
Sheet (6)
1. The figure bellow (Fig.1)shows a scheme for coherent demodulation. Show that
this scheme can demodulate the AM signal [A+ m(t)] cos 𝑤𝑐 t regardless of the
value of A
Fig.1
2. Sketch the AM signal [𝐴 + 𝑚(𝑡)]𝑐𝑜𝑠𝑤𝑐 𝑡 for the periodic triangle signal 𝑚(𝑡)
shown in the Fig.2 corresponding to the modulation index: (a)𝜇 = 0.5; (b) 𝜇 = 1;
(c) 𝜇 = 2; (d) 𝜇 = ∞. How do you interpret the case 𝜇 = ∞?
Fig.2
3. For the AM signal in prob 2 with 𝜇 = 0.8:
(a) Find the amplitude and power of the carrier.
(b) Find the sideband power and the power efficiency 𝜂.
4.
(a) Sketch the DSB_SC signal corresponding to 𝑚(𝑡) = 𝑐𝑜𝑠2𝜋𝑡.
(b)This DSB_SC signal 𝑚(𝑡) cos 𝑤𝑐 𝑡 is applied at the input of an envelop
detector. Show that output of the envelop detector is not 𝑚(𝑡), but |𝑚(𝑡)|. Show
that ,in general, if an AM signal [𝐴 + 𝑚(𝑡)]𝑐𝑜𝑠𝑤𝑐 𝑡 is envelop-detected, the
output is |𝐴 + 𝑚(𝑡)|. Hence ,show that the condition for recovering 𝑚(𝑡)from
the envelop detector is 𝐴 + 𝑚(𝑡) > 0 for all t.
Faculty of engineering sciences
communication and electronics department
communication theory (1)
5. Show that any scheme that can be used to generate DSB-SC can also generate
AM . Is the converse true? Explain.
6. Show that any scheme that can be used to demodulate DSB_SC can also
demodulate AM. Is the converse true? Explain.
7. In thee textbook, mentioned in the syllabus, the power efficiency of AM for a
sinusoidal m(t) was derived. Carry out a similar analysis when m(t) is a random
binary signal as shown in the figure(Fig.3) and 𝜇 = 1. Sketch the AM signal with
𝜇 = 1, find the sideband’s power and the total power (power of the AM signal)
as well as their ratio (the power efficiency 𝜂.
Fig.3
8. A modulated signal 𝑚(𝑡) is given by:
(a) 𝑚(𝑡) = 𝑐𝑜𝑠100𝑡.
(b) 𝑚(𝑡) = 𝑐𝑜𝑠100𝑡 + 2𝑐𝑜𝑠300𝑡.
(c) 𝑚(𝑡) = 𝑐𝑜𝑠100𝑡 𝑐𝑜𝑠500𝑡.
Determine 𝜙𝐿𝑆𝐵 (𝑡) and 𝜙𝑈𝑆𝐵 (𝑡) if the carrier frequency 𝑤𝑐 = 1000. Hint: if
𝑚(𝑡) is sinusoidal , its Hilbert transform 𝑚ℎ (𝑡) is the sinusoidal 𝑚(𝑡) phase
𝜋
delayed by 2 𝑟𝑎𝑑.
9. Find 𝜙𝐿𝑆𝐵 (𝑡) and 𝜙𝑈𝑆𝐵 (𝑡) for the modulated signal 𝑚(𝑡) = 𝑠𝑖𝑛𝑐 (2𝜋𝐵𝑡) with
𝐵 = 1000 and carrier frequency 𝑤𝑐 = 10,000𝜋 follow these do it yourself steps:
(a) Sketch spectra of m(t) and the corresponding DSB_SC signal 2𝑚(𝑡)𝑐𝑜𝑠𝑤𝑐 𝑡.
(b) To find the LSB spectrum , suppress the USB. in the DSB_SC spectrum
found in (a).
(c) Find the LSB signal 𝜙𝐿𝑆𝐵 (𝑡), which is the inverse Fourier transform of the
LSB spectrum found in part (b). follow a similar procedure to find 𝜙𝑈𝑆𝐵 (𝑡).
10. An USB signal is generated by using the phase-shift method .if the input is
𝑚ℎ (𝑡) instead of 𝑚(𝑡), what will be the output? Is this signal still an SSB signal
with equal to that of 𝑚(𝑡)? Can this signal be demodulated [to get back 𝑚(𝑡)]? If
so, how?