**Problem 1.1**: RMS Values

The **rms** (root-mean-square) value of a periodic signal is defined to be

where ** T** is defined to be the signal's

**period**: the smallest positive number such that

**s (t)= s (t +**

*T***)**.

- What is the period of s (t)= Asin (2πf
_{0}t + φ)? - What is the rms value of this signal? How is it related to the peak value?
- What is the period and rms value of the depicted (Figure 1.6)
**square wave**, generically denoted by**sq (t)**? - By inspecting any device you plug into a wall socket, you'll see that it is labeled "110 volts AC". What is the expression for the voltage provided by a wall socket? What is its rms value?

**Problem 1.2**: Modems

The word "modem" is short for "modulator-demodulator." Modems are used not only for connecting computers to telephone lines, but also for connecting digital (discrete-valued) sources to generic channels. In this problem, we explore a simple kind of modem, in which binary information is represented by the presence or absence of a sinusoid (presence representing a "1" and absence a "0"). Consequently, the modem's transmitted signal that represents a single bit has the form

Within each bit interval ** T**, the amplitude is either

**or zero.**

*A*- What is the smallest transmission interval that makes sense with the frequency
*f*_{0}? - Assuming that ten cycles of the sinusoid comprise a single bit's transmission interval, what is the datarate of this transmission scheme?
- Now suppose instead of using "on-of" signaling, we allow one of several
**different**values for the amplitude during any transmission interval. If N amplitude values are used, what is the resulting datarate? - The classic communications block diagram applies to the modem. Discuss how the transmitter must interface with the message source since the source is producing letters of the alphabet, not bits.

**Problem 1.3**: Advanced Modems

To transmit symbols, such as letters of the alphabet, RU computer modems use two frequencies (1600 and 1800 Hz) and several amplitude levels. A transmission is sent for a period of time
** T** (known as the transmission or baud interval) and equals the sum of two amplitude-weighted carriers.

We send successive symbols by choosing an appropriate frequency and amplitude combination, and sending them one after another.

- What is the smallest transmission interval that makes sense to use with the frequencies given above? In other words, what should T be so that an integer number of cycles of the carrier occurs?
- Sketch (using Matlab) the signal that modem produces over several transmission intervals. Make sure you axes are labeled.
- Using your signal transmission interval, how many amplitude levels are needed to transmit ASCII characters at a datarate of 3,200 bits/s? Assume use of the extended (8-bit) ASCII code.

*A*and

_{1}*A*. If we have

_{2}*N*values for amplitude

_{1}*A*, and

_{1}*N*values for

_{2}*A*, we have

_{2}*N*possible symbols that can be sent during each T second interval. Toconvert this number into bits (the fundamental unit of information engineers use to qualify things), compute log

_{1}N_{2}_{2}(

*N*).

_{1}N_{2}- 1045 reads