# 1.5: Introduction Problems

Learning Objectives

- Problems for introduction to signals and systems.

### RMS Values

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

\[s = \sqrt{\frac{1}{T}\int_{0}^{T}s^{2}(t)dt}\]

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

\[s(t)= A\sin (2\pi f_{0}t+\varphi )\]

- 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
**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?

### 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

\[x(t)= A\sin (2\pi f_{0}t), 0\leq t\leq T\]

Within each bit interval the amplitude is either

- 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-off" 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 data rate? - 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.

### 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.

\[x(t)= A_{1}\sin (2\pi f_{1}t) + A_{2}\sin (2\pi f_{2}t), 0\leq t\leq T\]

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.

Note

We use a discrete set of values for A_{1}_{ }and A_{2}. If we have N_{1} values for amplitude A_{1}, and N_{2} values for A_{2}, we have N_{1}N_{2 }possible symbols that can be sent during each T second interval. To convert this number into bits (the fundamental unit of information engineers use to qualify things), compute:

\[\log _{2}(N_{1}N_{2})\]

### Contributor

- ContribEEOpenStax