# Communication Systems and Signal Processing Problems

## 1) Communication Systems and Signal Processing Problems

### 1) Armstrong FM Modulator Design

The total amount of frequency multiplication required is: 75 / 1.5 = 50. If the narrowband FM signal is input to a frequency multiplier with factor 50, then the carrier frequency gets converted to 5 MHz. So at the output of the frequency multiplier, we need a mixer, which shifts the carrier frequency by 99 MHz to 104 MHz.

### 2) Binary Channel Decision Rule

(a) Since 0 and 1 are sent with equal probabilities, the optimal decision rule is the maximum likelihood rule. If p > 1/3, the optimal decision rule is to decide that 1 was sent if 0 is received and to decide that 0 was sent if 1 was received. If p < 1/3, then the optimal rule is to decide that 0 was sent if 0 was received and to decide that 1 was sent if 1 was received. If p = 1/3, then the optimal rule is to decide that 0 was sent with probability 1/2 and that 1 was sent with probability 1/2, regardless of whether 0 or 1 was received.

(b) If p > 1/3, then the probability of error is: P(e) = 1 – 3p/2. If p < 1/3, then the probability of error is: P(e) = 3p/2. If p = 1/3, then the probability of error is 1/2.