Comp 346/446 Homework 1

Due Friday March 1

1. What is the Shannon channel capacity for a teleprinter channel with a 300-Hz band-width and a signal-to-noise ratio of 3 dB? [3.15]


2. A digital signaling system is required to operate at 9600 bps. [3.16]
a. If a signal element (symbol) encodes 4 bits at a time, what is the minimum required band-width of the channel? Use the multi-level Nyquist formula.
b. Repeat part (a) for the case of 8 bits at a time.


3. Suppose a phone line has a 4 kHz audio band-width and a nominal SNR of 56 dB (400,000). What is the (Shannon) theoretical maximum channel capacity (kbps) of the line? What is the maximum channel capacity if the SNR falls to 100 (20 dB)? [3.18]


4. Consider a channel with a 1-mHz band-width and an SNR of 63. [3.20]
(a). What is the upper limit to the data rate that the channel can carry?
(b). The result of part (a) is the upper limit. However, as a practical matter, better error performance will be achieved at a lower data rate. Assume we choose a data rate of 2/3 of the maximum theoretical limit. How many signal levels are needed to achieve this data rate?


5. A square wave alternating between +1 and -1, with a frequency f, can be expressed as a sum of sine waves as follows: [3.22]

    sqr(ft) = (4/𝜋) [ 1*sin(2𝜋ft) + (1/3)sin(2𝜋*3ft) + (1/5)sin(2𝜋*5ft) + (1/7)sin(2𝜋*7ft) + ... ]

The power of a square wave with amplitude 1 is the same as the power of a DC current with amplitude 1: it is simply 1. The power in a sine wave of the form A sin(gt) is A²/2 (note that it does not depend on frequency)

Find the total power in

a. The first term above, (4/𝜋) sin(2𝜋ft)
b. The first two terms above
c. The first three terms above
d. The first four terms above


6. A telephone line is known to have a power loss of 20 dB. The input signal power is measured as 0.5 W, and the output noise level is measured as 4.5 mW. Using this information, calculate the output signal-to-noise ratio in dB. [4.2]


7. Consider a stream of binary data consisting of a long sequence of 1s followed by a zero followed by a long string of 1s. Draw the waveform for this sequence using each of the following encodings: [5.8]
a. NRZ-L
b. Bipolar-AMI
c. Pseudoternary


8. The bipolar-AMI waveform representing the binary sequence 0100101011 is transmitted over a noisy channel. The received waveform, converted to 0, +1, -1, is as follows; it contains a single error.

    0 +1 0 0 -1 0 -1 0 -1 +1

Locate the position of this error and explain your answer. [5.9]