In this lab, we will develop the digital signal code for a very basic, notional codee. A specialized application has been developed which needs to quantize analog signals varying between -3.5 volts and +3.5 volts (see figure). The proper sampling frequency has been decermined, and the sensitivity requires that each sample be within 0.5 volts of the true analog signal. Your job is to provide a digital (binary) code, using the minimum number of bits necessary, to sepresent each voltage, in 0.5-volt inerements, from -3.5 volts to 3.5 volts. When the blank lines on the left of the figure are filled in with your digital code, the graph will be similar to (but not the same as) graphs in figure 3 20 of the text. Turn-in Requirements: To complete the assignment, upload a Word (docx) or Adobe (.pdf) file with answers to the 2 questions below; followed by a table or list indicating your digital code representation for each volage level as your submission to the M03 Lab assignment in Blackboard. 1. (5 pts) How many different voltage levels must be reptesented? 2. (10 pts) How many bits are needed to represent ar least this many levels? What about 17 levels? Show your work
The Correct Answer and Explanation is :
To address the tasks in the Signal Coding Lab, let’s break down each part step by step:
1. How many different voltage levels must be represented?
The signal must represent voltage levels from -3.5 volts to +3.5 volts, with increments of 0.5 volts. To find the number of voltage levels:
- The range of voltages is from -3.5 V to +3.5 V, inclusive.
- Start by dividing the voltage range into increments of 0.5 V:
[
\text{Number of voltage levels} = \frac{\text{Voltage range}}{\text{Step size}} + 1
]
[
\text{Number of voltage levels} = \frac{(3.5 – (-3.5))}{0.5} + 1 = \frac{7}{0.5} + 1 = 14 + 1 = 15 \text{ voltage levels}
]
Thus, there are 15 distinct voltage levels.
2. How many bits are needed to represent at least this many levels? What about 17 levels?
For 15 levels:
To represent a specific number of levels in digital form, the number of bits required can be calculated using the formula:
[
\text{Number of bits} = \lceil \log_2(\text{number of levels}) \rceil
]
For 15 levels:
[
\log_2(15) \approx 3.91
]
Thus, the minimum number of bits required is:
[
\lceil 3.91 \rceil = 4 \text{ bits}
]
For 17 levels:
Similarly, for 17 levels:
[
\log_2(17) \approx 4.09
]
So, the minimum number of bits required is:
[
\lceil 4.09 \rceil = 5 \text{ bits}
]
3. Fill in the table
Here is the digital representation for each voltage level using 4 bits (for 15 levels):
| Voltage (V) | Binary Code (4 bits) |
|---|---|
| -3.5 | 0000 |
| -3.0 | 0001 |
| -2.5 | 0010 |
| -2.0 | 0011 |
| -1.5 | 0100 |
| -1.0 | 0101 |
| -0.5 | 0110 |
| 0.0 | 0111 |
| 0.5 | 1000 |
| 1.0 | 1001 |
| 1.5 | 1010 |
| 2.0 | 1011 |
| 2.5 | 1100 |
| 3.0 | 1101 |
| 3.5 | 1110 |
Explanation (300 words):
In this lab, we are tasked with quantizing an analog signal that ranges from -3.5V to +3.5V, with voltage levels spaced 0.5V apart. The primary goal is to create a digital binary code that can represent these voltage levels using the minimum number of bits.
The first step in the process is to determine how many voltage levels must be represented. Since the signal varies from -3.5V to +3.5V in steps of 0.5V, we calculate that the total number of voltage levels is 15. This is done by taking the total voltage range (7V) and dividing it by the step size (0.5V), then adding one to include both endpoints.
To represent these 15 levels in binary form, we need to determine the minimum number of bits. The formula for this is the base-2 logarithm of the number of levels, rounded up. For 15 levels, we find that we need 4 bits. This means that we can represent each of the 15 voltage levels using a 4-bit binary code.
When considering 17 levels, we calculate that 5 bits would be required to represent this number. This demonstrates how the number of bits needed increases as the number of levels increases.
In the table, each voltage level is represented by a binary code. The binary code increments by 1 for each successive voltage level, with the lowest level (-3.5V) represented by 0000 and the highest level (3.5V) represented by 1110. This ensures that each voltage is quantized and can be accurately represented with the minimum number of bits necessary.
Conclusion:
The lab demonstrates how to quantize an analog signal and convert it into a digital format. By calculating the number of voltage levels and determining the minimum number of bits required, we ensure that the analog signal can be represented with sufficient accuracy and efficiency.