(Solved): [20] Modulation formats in digital communication systems, and the choice of modulation format affec ...

[20] Modulation formats in digital communication systems, and the choice of modulation format affects the probability of a bit error (mistaking a " 0 " for a " 1 " or vice versa). One format, which was discussed in class, is called binary phase shift keying (BPSK), where we transmit a positive voltage to represent a " 1 " and a negative voltage of equal magnitude to represent a "0". [Just as an FYI: We don't directly transmit DC voltages, but rather transmit them using a sinusoidal carrier wave. Then the receiver extracts the DC voltage $+\mathrm{A}$ or $?\mathrm{A}$.] As we discussed in class, the voltage extracted by the receiver has additive noise. Suppose we use BPSK for a message where 1's and 0 's are equally likely. Suppose the signal voltages are $+\mathrm{A}$ and $?\mathrm{A}$ and the noise is zero-mean Gaussian with a standard deviation of $?$ volts. If our decision threshold is zero volts, a positive voltage is interpreted as a " 1 " and a negative voltage a " 0 ". Write an expression for the probability of a noise-induced bit error (either kind) and plot it as a function of the voltage signal-to-noise ratio $\mathrm{A}/?$.