1.4 Introduction problems

Rms values

The rms (root-mean-square) value of a periodic signal is defined to be $$s=\sqrt{\frac{1}{T}\int_0^T s(t)^2\,d t}$$ where $T$ is defined to be the signal's period : the smallest positive number such that $s(t)=s(t+T)$ .

1. What is the period of $s(t)=A\sin (2\pi f_{0}t+\phi )$ ?
2. What is the rms value of this signal? How is it related to the peak value?
3. What is the period and rms value of the depicted square wave , generically denoted by $\mathrm{sq}(t)$ ?
4. By inspecting any device you plug into a wall socket, you'll see that it is labeled "110 volts AC". What isthe 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 totelephone lines, but also for connecting digital (discrete-valued) sources to generic channels. In thisproblem, we explore a simple kind of modem, in which binary information is represented by the presence orabsence of a sinusoid (presence representing a "1" and absence a "0"). Consequently, the modem's transmittedsignal that represents a single bit has the form $$x(t)=A\sin (2\pi f_{0}t)\text{,}0\le t\le T$$ Within each bit interval $T$ , the amplitude is either $A$ or zero.

1. What is the smallest transmission interval that makes sense with the frequency $f_0$ ?
2. Assuming that ten cycles of the sinusoid comprise a single bit's transmission interval, what is thedatarate of this transmission scheme?
3. 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 datarate?
4. The classic communications block diagram applies to the modem. Discuss how the transmitter mustinterface 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, RUcomputer modems use two frequencies (1600 and 1800 Hz) and several amplitude levels. A transmission is sentfor a period of time $T$ (known as the transmission or baud interval) and equals the sum of two amplitude-weighted carriers. $$x(t)=A_1\sin (2\pi f_{1}t)+A_2\sin (2\pi f_{2}t)\text{,}0\le t\le T$$ We send successive symbols by choosing an appropriate frequency and amplitude combination, and sending themone after another.

1. 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?
2. Sketch (using Matlab) the signal that modem produces over several transmission intervals. Make sure youaxes are labeled.
3. Using your signal transmission interval, how many amplitude levels are needed to transmit ASCIIcharacters at a datarate of 3,200 bits/s? Assume use of the extended (8-bit) ASCII code.