HW 1

pdf

School

Princeton University *

*We aren’t endorsed by this school

Course

306

Subject

Electrical Engineering

Date

Oct 30, 2023

Type

pdf

Pages

Uploaded by ColonelBravery12436

ECE 206/COS 306: Contemporary Logic Design, Fall 2023 HW 1: Due 9/18/2023 at 11:59 pm 1. [20pts] In a digital circuit, the term Noise Margin refers to the maximum noise that a signal can tolerate before its value is corrupted. That is, it quantifies how far a signal can deviate from its ideally digitized value in either direction while still being correctly classified. A digital circuit’s noise margin is a common metric for robustness; the anticipated level of noise for specific applications can be used to select a necessary margin in circuit design. Consider a digital signal represented using voltage values from 0 to 5V (both inclusive). What is the noise margin if the signal is: a. Ternary (Three valued)? b. n valued? Be sure to clearly state any additional assumptions you make, e.g., which voltages the signal takes. 2. [20 pts] The term Quantization Error refers to the maximum error that results from using a finite encoding of an analog signal in digital form. For example, if the value “ 1 ” is used to represent any signal between 0.5 and 1.5, the quantization error would be 0.5, as the maximum loss of precision is 0.5. The quantization error thus quantifies how precise an encoding is, serving as an important guideline in the design of analog-to-digital converters. Consider an electrical analog signal that can take values from 0 to 5V. a. What is the maximum quantization error if 8 bits are used to encode this signal? b. Generalize this to a code with n bits. As in Problem 1, be sure to clearly state any additional assumptions you make. 3. [45 pts] Besides binary and hexadecimal, Octal (base 8) representations are sometimes used in digital systems. a. How many “Octal digits” are needed to encode a variable with N possible values? b. How would you convert a decimal number D to an n “Octal digits” number o n-1 o n-2 … o 1 o 0 ? Explain any equations/formulas that you are using. c. Conversely, what is the decimal value of an n “Octal digits” number o n-1 o n-2 … o 1 o 0 ? (As an aside, it has been observed that OCT 31 = DEC 25.) 4. [15 pts] Besides those discussed in class, another important advantage of digital systems is error detection and correction . That is, a bit that gets corrupted through excessive noise can

still be recovered if appropriate error detection and correction mechanisms are in place. This is what enables us, for example, to continue to recover information from scratched CDs. In one specific error detection and correction scheme, each bit is represented with three bits: 0 is represented as 000, and 1 is represented as 111. The three-bit signal 101, then, is represented as x = 111 000 111. Suppose that in the presence of noise, this value of x becomes 101 001 111. Can you recover the original signal? What kinds of errors can you fully recover from? Explain.

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version