ECE-S 306 Homework#3 Solution

The carrier power does not change in any of the experiments. However, the sideband power is greatly affected by the choice of modulation index, as expected from the theory. It is known that it is desirable to have modulation index close to 1 to have as much power in the sideband as possible. Note that the magnitude spectra plots clearly show this, as the amplitude of the impulses at 90 Hz and 110 Hz (and the corresponding impulses at -90 Hz and -110 Hz) is substantially larger for modulation index of 0.9 compared to the other two cases (modulation index of 0.2 and 0.5). However, the magnitude of the impulse at 100 Hz (carrier frequency) is unchanged and thus carrier power remains constant. Finally, the bandwidth required for transmission of the modulated signal is exactly the same for all simulations and is equal to 20 Hz. This is the result predicted from the theory, i.e. BW = 2B, where B=10Hz is the bandwidth of the message signal. The bandwidth cannot be changed by selecting a different value for the modulation index.

As expected from the theory, as we increase Kp, the bandwidth of the PM signal grows substantially, as indicated by the increasing number of nonzero impulses in the vicinity of 100 Hz and -100 Hz. From the magnitude spectra plots, the bandwidth can be determined as:

Note that depending on where you decide the last nonzero impulse is affects your estimate of the bandwidth from the magnitude spectra plot. As we would expect, Carson's rule provides estimated bandwidths that are close to those estimated directly from the plots.

m_p'=max{x(t)'}=max{-0.75*20pi*sin[20pi*t]}=15pi

delta_f= (k_p m_p')/2pi = 7.5 k_p

Carson's rule bandwidth BW = 2(delta_f+B), where B is the bandwidth of the message signal, B = 10 Hz. Plugging in the numbers, we get the following estimates

Note that the bandwidth calculated from the Carson's rule is very close to that determined directly from the magnitude spectra plots. Of course, it is only expected that the theoretical predictions should match the simulation results.

For Kp=3pi/4 the magnitude spectrum of the phase modulated signal is shown in the figure below. In order to determine bandwidth, we find frequencies at which XPM(f) is nonzero. Clearly, XPM is nonzero for f=70,80,90,100,110,120, and 130 Hz (and thus for the corresponding negative frequencies). Assuming that XPM is approximately zero for f = 60 and 140 Hz, the resulting bandwidth is B=130-70=60 Hz which is close to the Carson's rule bandwidth estimated to be 55.3 Hz. If we assume that XPM is nonzero for f = 60 and 140 Hz, the bandwidth is then B=140-60=80 Hz.

Finally, it should be noted that the choice of k_p affects the bandwidth of the phase-modulated signal substantially. For the selected values of k_p, phase-modulated signals require significantly larger channel bandwidth for transmission than the AM signal.